A method for evaluating the resilience of a water-electricity coupled system considering drought risk evolution

Abstract

The application discloses a kind of water-electricity coupling system resilience assessment method considering drought risk evolution, it is related to water-electricity coupling system technical field. Including: S1, water-electricity coupling system network modeling;S2, water-electricity coupling network drought risk propagation modeling based on seepage;S3, water-electricity coupling system network is simulated and resilience is evaluated based on the above two models basis system.This application constructs heterogeneous weighted double-layer network model, develops cross-scale dynamics modeling method, combines seepage theory and physical flow dynamics, establishes bidirectional seepage equation of disaster cascade propagation, proposes resilience assessment framework based on seepage theory and reveals the law of different coupling effects of water-electricity system, which can provide scientific basis for risk assessment, resilience improvement and emergency management strategy formulation of key infrastructure.

Description

Technical Field

[0001] This invention belongs to the technical field of water-electric coupling systems, and more specifically, relates to a method for assessing the resilience of water-electric coupling systems that takes into account the evolution of drought risk. Background Technology

[0002] Hydro-electric coupling network systems refer to complex network systems in which water networks and power grids are interdependent and influence each other, serving as vital lifelines for maintaining people's production and daily life. Water resources are the fundamental data for electricity production, playing a crucial role, especially in hydropower generation; conversely, electricity is the driving force for water resource supply, with water pumps alone consuming 20% ​​of the total electricity consumption. In existing research on hydro-electric coupling network systems, at the topological level, complex network theory has been widely applied and yielded relatively mature research results in various transmission networks (such as power grids, transportation networks, and water supply pipelines). Some researchers have analyzed China's power grid and confirmed its scale-free network characteristics, namely, the existence of several important central nodes that play a key role in ensuring system stability and connectivity. Researchers have also studied water supply pipelines and urban transportation networks, confirming the prevalence of similar scale-free network structures in different transmission systems. Based on complex network theory and the properties of natural gas pipelines, researchers have developed evaluation indicators based on factors such as network type, overall topological characteristics, and path-dependent topological characteristics, and conducted a quantitative comparison and analysis of the topological structures of natural gas pipeline networks in two locations.

[0003] The above studies all focus on single-layer networks. In multi-layer networks, domestic and international research primarily explores the coupling relationships, node centrality, and inter-layer interaction mechanisms at different levels. Some researchers have considered the overlap of multiplexed network layers and extended the clustering coefficient to multiplexed networks. Others have defined PageRank for multi-layer networks based on inter-layer interactions, directly considering the impact of these interactions on node importance. Regarding layer centrality, some have proposed a new eigenvector centrality metric based on the tensor definition of the network; others have proposed calculating layer centrality based on edge betweenness centrality and shortest path; still others have integrated layer and node information to determine layer and node centrality. However, these studies largely remain at the theoretical level, lacking integration with real-world coupled systems.

[0004] In the topology analysis of two-layer coupled networks (such as water-electricity coupled networks), there are interdependencies between multiple different types of systems. In particular, in the coupling of water resources and power systems, the multi-layered nature and cross-domain dependencies of the systems make the topology analysis more complex.

[0005] The unique network structure characteristics of water networks and power grids, along with their different node associations and interactions, can alter the robustness and phase transition threshold of two-layer networks. The first challenge of this research is how to mathematically represent two-layer interdependent networks, construct a water-electricity coupled weighted network model, and analyze the point strength, node clustering, and other network characteristics of single-layer networks based on this model.

[0006] In water-electricity coupled systems of different scales, nodes are assigned different physical meanings. For example, at the urban distribution network scale, water and electricity are coupled through pumping stations, while at the regional main grid scale, they are coupled through hydropower stations. Similarly, the physical constraints, temporal granularity, and spatiotemporal relationships of nodes differ across systems of different scales. The second challenge in this research is how to construct a dynamic model of the water-electricity coupled network by combining the physical meanings of nodes and weighted edges.

[0007] Compared to single-layer networks, hydro-electric coupled networks exhibit multi-center characteristics. The cascading propagation of disasters in two-layer networks necessitates modeling and analyzing the network's robustness in conjunction with the failure cascading process. The third challenge of this research is how to analyze the phase transition threshold of hydro-electric coupled networks from a spatial correlation perspective, comprehensively considering factors such as the single-layer network characteristics, inter-layer cascading characteristics, and disaster attack methods, thereby revealing the robustness characteristics of hydro-electric coupled networks.

[0008] Significant progress has been made in risk modeling and measurement of power systems under extreme weather conditions in recent years. Some researchers have studied the impact of extreme weather on power system reliability from a probabilistic perspective, classifying it solely by the degree of weather extremeity without distinguishing specific weather types, and calculating the equivalent reliability parameters of components under different weather conditions. However, more research focuses on specific extreme weather events, extracting meteorological characteristic parameters and simulating meteorological paths to establish predictive models of the reliability parameters of various power system components under that meteorological environment, thereby assessing the overall risk of the system. Some researchers have predicted the real-time failure probability of transmission lines and towers under ice storms; others have proposed predictive models of the failure probability of distribution equipment under extreme rainstorms. Existing research mainly focuses on single power network scenarios such as distribution and transmission networks, lacking research on risk measurement for complex systems like hydro-electric coupling networks.

[0009] The resilience of complex systems refers to their ability to absorb, adapt to, and recover from sudden and significant shocks, and is an essential requirement for ensuring system safety. Understanding the system's response to disturbances from the perspectives of structural and dynamic stability is crucial, and percolation theory provides an effective framework. As a widely used framework in statistical physics for studying system behavior and its critical response to disturbances, percolation theory reveals the connectivity and functional maintenance capabilities of networks under random failures or targeted attacks by simulating the gradual failure process of nodes or edges in a network. Some researchers have proposed a general framework and studied the percolation characteristics of interdependent networks. Others have discussed the percolation theory of 3D network topologies from the perspective of spatial topological differences among nodes. Taking a road network as an example, a small local disturbance can lead to large-scale system failure at the critical point of the entire road network. Some researchers have examined the percolation process of multi-layered networks with hierarchical structures, showing that the resilience of the hierarchical structure depends on the number of community structures in each layer, the degree of nodes, the proportion of nodes removed, and the coupling strength between layers. Some researchers have focused on the iterative percolation process of multilayer networks, exploring the role of history-dependent mechanisms. They have shown that continuous percolation phase transitions may be formed by the iterative action of several processes, while infinite iterative percolation processes can change the emergence mode of giant branches in the system, thus exhibiting discontinuous phase transitions.

[0010] While percolation theory has been widely applied in the resilience analysis of complex networks, traditional percolation models have certain limitations. They mainly focus on the topological characteristics of the network, neglecting the functional indicators of task dependencies in real-world networks. Some researchers have considered that in real-world complex networks, when the main connected component splits, the system can still function if the remaining subnetworks can handle the main traffic demand. To address this, a new theoretical framework based on the Unaffected Demand (UD) indicator has been proposed, improving upon traditional percolation theory. This framework has been applied to a city public transportation system, emphasizing the importance of functional indicators in assessing network resilience. On the other hand, traditional percolation theory limits network components to only binary states: present (functional) or removed (failed), neglecting the multiple states that may exist in real-world applications. Based on the network layer propagation dynamics of virus transmission and power flow characteristics, some researchers have proposed multi-state node models and state transition models in power network-communication network coupled systems, enabling percolation theory to better simulate the multi-state interdependencies in real-world applications.

[0011] In summary, addressing the critical strategic need to enhance the resilience and security of water-electricity coupled network systems, this study combines statistical modeling, simulation optimization, and artificial intelligence technologies. By modeling the water-electricity coupled system as a complex network and studying the risk propagation mechanism of the network under drought conditions based on seepage theory, the robustness characteristics of the network are analyzed, revealing its vulnerability and anti-interference capabilities. This provides a scientific basis for the resilience assessment and improvement of water-electricity coupled systems and has significant practical implications for safeguarding people's lifelines. Summary of the Invention

[0012] The technical problem to be solved by this invention is to provide a resilience assessment method for water-electricity coupled systems that considers the evolution of drought risk. This method constructs a heterogeneous weighted two-layer network model, develops a cross-scale dynamic modeling method, combines seepage theory with physical flow dynamics, establishes a two-way seepage equation for the cascading propagation of disasters, proposes a resilience assessment framework based on seepage theory, and reveals the laws of different coupling effects in water-electricity systems. This method can provide a scientific basis for risk assessment, resilience enhancement, and emergency management strategy formulation for critical infrastructure.

[0013] To solve the above-mentioned technical problems, the technical solution adopted by the present invention is: a method for assessing the resilience of a water-electricity coupled system considering the evolution of drought risk, comprising the following steps:

[0014] S1. Network modeling of the water-electricity coupled system;

[0015] S2. Drought risk propagation modeling based on seepage-induced water-electricity coupled networks;

[0016] S3. Based on the above two models, system simulation and resilience assessment of the water-electricity coupled system network are carried out.

[0017] Preferably, in step S1, the process of establishing the water-electricity coupled system network model includes the following steps:

[0018] S11. Characterizing the key elements of the water-electricity coupling system: Representing the water network model as follows .in It is the set of nodes in a water network. For the edge set in the water network, Let be the edge weights of the nodes in the water network;

[0019] S12. Reservoir and hydropower station network modeling, including the following steps:

[0020] S121. Node dataset construction and watershed topology association;

[0021] S122. Construct an edge generation mechanism based on watershed topology;

[0022] Based on the two major geospatial datasets HydroBASINS and HydroRIVERS, a quantitative characterization of the hydraulic connections between hydropower stations is achieved through the spatial correlation of multi-scale watershed units and river networks.

[0023] S123. Establish a side weight calculation model that simultaneously considers the watershed isomorphism and dynamic time-delay coupling characteristics;

[0024] Watershed isomorphism coefficient Representation from node arrive The hydrological connectivity cost is based on the watershed hierarchy in the HydroBASINS dataset, and on the spatial association results between hydropower stations and sub-watersheds obtained by Algorithm 1, according to the hydropower station and The watershed matching situation is used to define the watershed isomorphism coefficient. The classification table; known For the edge A list of HYRIV_IDs for the main rivers; a river transport time delay model is constructed based on hydrodynamic principles to calculate the key parameters of each river, including river meander, average slope, hydraulic radius, flow velocity, and transport time delay;

[0025]

[0026] S13. Power Network Modeling: The IEEE-118 node system is selected as the power network. A DC power flow model is used to model the steady-state and dynamic behavior of the power system;

[0027] S14, Water-Electricity Coupled Network Modeling

[0028] The water-electricity coupled network consists of a water network and power grid Composition. Power Grid Nodes representing generator sets, substations, and / or power loads The system consists of links between nodes via transmission lines and transformer branches. Connections; Reservoir and hydropower station networks Hydropower station nodes with upstream and downstream relationships Composed of nodes connected by rivers or waterways The water flow along the water network is controlled by the reservoir's scheduling, and the power grid... WaterNet They are connected by a hydropower conversion link, forming an interdependent two-layer coupled network;

[0029] Hydropower Conversion Link The energy conversion process of hydropower generation is defined, which is the core channel for transmitting electrical energy from the water network to the power grid, starting from the water network nodes and ending at the power generation nodes of the power grid; the hydropower generation power is expressed as a function of the power generation flow, the hydropower station head, and the hydropower station output coefficient:

[0030]

[0031] in, Let g be the density of water, g be the acceleration due to gravity, Q be the power generation flow rate, and H be the effective head height. To determine the overall hydropower efficiency coefficient, the head height H has dynamic characteristics and is calculated from the difference between the upstream and downstream water levels of the hydropower station.

[0032] Based on the above, and using complex network theory, the water-electricity coupled system is constructed as a two-layer dependent heterogeneous network. :

[0033]

[0034] Preferably, step S121 includes the following steps:

[0035] S1211, Define the node attributes of the hydropower station:

[0036] This service utilizes the Python Requests module to call the Google Elevation API for batch data collection, obtaining elevation information of hydropower stations and enabling 3D spatial positioning of hydropower station nodes. Based on multi-source remote sensing elevation data fusion technology, it allows developers to obtain elevation data for specific locations on the Earth's surface based on given latitude and longitude coordinates. When the requested coordinates lack a directly accurate elevation measurement, the Google Elevation API estimates the elevation using a bilinear interpolation algorithm, calculating the average from the four nearest known points to ensure continuous elevation data for any location. The core parameters of the API call include the `locations` parameter and the `key` parameter; the `locations` parameter contains the latitude and longitude coordinates, and the `key` parameter is the API key used for authentication and quota control. The returned JSON data contains the elevation information of the requested location.

[0037] S1212. Constructing the mapping relationship between hydropower stations and river basins:

[0038] Based on the HydroBASINS dataset, hydropower station nodes are associated with watershed topology to achieve spatial association between hydropower stations and watersheds.

[0039] Preferably, achieving spatial correlation between hydropower stations and sub-basins mainly includes the following three steps:

[0040] S12121, Coordinate Transformation: Convert the WGS84 latitude and longitude coordinates of the hydropower station to ESRI:54009 world cylindrical projection to eliminate spherical distance distortion;

[0041] S12122, Spatial Query: Use the contains() method of the Shapely library to perform point-to-face containment detection;

[0042] S12123, Hierarchical Matching: Prioritize searching within Pfafstetter level 5-7 sub-basins; if a match fails, expand to adjacent levels.

[0043] Preferably, step S122 includes the following steps:

[0044] S1221. Watershed topology path search: Based on the determined hydropower station-watershed mapping relationship, and the hierarchical watershed division system of HydroBASINS, establish the topological connection relationship between watershed units.

[0045] S1222, Main River Path Extraction; The HydroRIVERS dataset is used to map watershed paths to physical rivers; for confirmed connected paths... Extract the path of the main river :

[0046]

[0047] In the formula, ℝ represents the set of Hydrorivers. To filter the river and ensure that only the main channel is selected;

[0048] S1223, Rules for constructing topological edges

[0049] The final two hydroelectric power station nodes Connecting edges The generation must satisfy two constraints:

[0050] I. Hydraulic connectivity: There exists a collection of non-empty main rivers.

[0051] II. Topological Exclusivity: There are no intermediate hydropower station nodes in path p, i.e.

[0052]

[0053] When the above conditions are met, from node arrive directed edges And record its attribute tuple: ,in For the edge A list of HYRIV_IDs for the main rivers.

[0054] 6. A method for assessing the resilience of a hydro-electric coupled system considering drought risk evolution, as described in claim 5, characterized in that, for any two hydropower station nodes... The spatial correlation is determined through the following process:

[0055] S12211, Locating the watershed unit: Through spatial point inclusion analysis, determine the HYBAS_ID of the watershed to which each hydropower station belongs, denoted as... , ;

[0056] S12212, Tracing downstream paths: From Starting from a given watershed, move to its downstream watershed based on the NEXT_DOWN field within that watershed. Construct potential connection paths ;

[0057] S12213. Determine connectivity: If the path contains the target watershed... Then determine for The upstream node; if it is still not found after traversing to the end of the watershed (NEXT_DOWN=0). If no direct hydraulic connection is found, this process is implemented using Algorithm 2:

[0058]

[0059] Preferably, in step S2, the drought risk propagation modeling process based on the water-electricity coupling network of seepage is as follows:

[0060] S21. First, establish seepage models within the water network and the power grid respectively: the water network model focuses on the state evolution based on water balance and control strategies; the power grid model focuses on link failures and dynamic power rebalancing based on overload.

[0061] The seepage model of the water network should take into account the time-delay propagation mechanism of hydropower station power generation capacity degradation. Based on the principle of watershed hydrology balance, the reservoir of the hydropower station can be modeled as a storage unit with a certain capacity and daily inflow to describe the water volume change of the reservoir.

[0062] Water balance equation for upstream hydropower station:

[0063] ;

[0064] Water balance equation for downstream hydropower station:

[0065]

[0066] It represents the set of nodes of the upstream hydropower station;

[0067] It represents the set of nodes of the downstream hydropower station;

[0068] and These represent the water volume of hydropower station i at time t and t+1, respectively;

[0069] It is the natural inflow of water into the reservoir of a hydropower station at time t;

[0070] It is the amount of water used by a hydroelectric power station to generate electricity at time t;

[0071] It is the amount of water discharged by the hydropower station at time t;

[0072] It is the set of directly upstream nodes affecting hydropower station j. ;

[0073] It is the water flow lag time from the upstream hydropower station to the downstream hydropower station j, where , ;

[0074] Multi-state model of hydropower station nodes:

[0075] Define nodes The storage capacity ratio is A five-level drought classification system was established based on the reservoir capacity ratio, with the drought level of node i as the reference. Classification criteria:

[0076]

[0077] in, This represents the maximum reservoir capacity of node i in the hydropower station. This represents the drought level of node i at time t; based on the drought level defined above according to the reservoir capacity ratio, a multi-state model of the hydropower station node is established; for the hydropower station node... From the perspective of drought level, its state space is as follows: Let represent the drought level state set of hydropower station node i; at time t, the state of hydropower station node i is... From the state space Take the value from;

[0078] Water volume regulation strategy for water network nodes: power generation water volume Based on the current drought level The proportional adjustment reflects the practical strategy of reducing power generation to conserve water resources under drought conditions; the adjustment rules are as follows:

[0079] This represents the baseline water generation capacity of hydropower station i at time t; under normal, drought-free conditions... hour, This can be considered as the maximum power generation capacity during normal operation. , This is a proportionality coefficient determined based on the drought level;

[0080] The overall state update of the system follows the discrete-time recursive principle, with the water network state at input time t. The system state at time Δt can be obtained. Its dynamic equation is:

[0081]

[0082] In the formula The state transition function for the water network has the following input parameters:

[0083] Network Topology : Describes the set of reservoir nodes Connecting edge to water flow and corresponding weights ,

[0084] Water volume operation status ;

[0085] Node status ;

[0086] The output state variables are defined as follows:

[0087]

[0088]

[0089] Seepage process in power grid:

[0090] In power grid model Among them Let E be the set of nodes and E be the set of links. We assume that each node in the power grid is either a generator node or a link node. It is either a demand node (load), represented as and define , , ;

[0091] S22. By establishing a mapping relationship between the state of water network nodes and the state of power grid nodes, the risk transmission process between layers is simulated, and finally the seepage dynamics equation of the coupled network is formed, realizing the cascading failure propagation model driven by physical mechanism.

[0092] By coupling the hydrodynamic processes in the water network with the power flow-based fault propagation in the power grid, the seepage equation of the water-electric coupled two-layer dependent network is obtained:

[0093] The seepage equation of the network:

[0094]

[0095] The simulation continues until the following termination condition is met, indicating that the system state has reached steady state:

[0096] .

[0097] Preferably, the power grid operation status update process under the influence of the water network includes the following steps:

[0098] Step 1: Adjust the ramp rate of the restricted nodes;

[0099] Step 2: Recalculate the power output boundary of the restricted node;

[0100] Step 3: Remove the failed node;

[0101] Step 4: Update the power grid operating status.

[0102] Preferably, step S3 includes a resilience index system for water-electricity coupled networks and a water network resilience assessment based on seepage theory;

[0103] The resilience index system of hydro-electric coupled networks includes topological resilience index and functional resilience index. The functional resilience index includes power grid functional resilience index and hydro network functional resilience index.

[0104] Topology resilience index: The relative size of the largest connected component GC and the second largest connected component SC is used as the topology resilience index.

[0105] Relative size of the largest connected component GC Defined as:

[0106]

[0107] in, It is the number of nodes contained in the largest connected component of the network. This represents the total number of nodes in the original network; the evaluation is conducted before and after the perturbation. The change in can quantify the degree of loss in the network topology;

[0108] The relative size of the second largest connected component Defined as:

[0109]

[0110] in, It is the number of nodes contained in the second largest connected component of the network. Changes in the size of the network can reveal the degree of network fragmentation and structural fragility;

[0111] Power grid resilience index: assesses how much power demand the system can still meet after a cascading failure. Defined as the proportion of total electricity demand that can still be met after a cascading failure, it reflects the service maintenance capacity of the power system.

[0112]

[0113] in It is the set of all nodes with electricity demand in the initial network. It is a node electricity demand, It is the set of nodes with remaining power demand after a cascading failure process triggered by an attack. It is the remaining demand node Power requirements;

[0114] Water network functional resilience index: assesses the level of hydropower generation capacity maintained by the water network during cascading failures. Defined as the proportion of water that can still be used for power generation under disturbance to the total water volume used for power generation, reflecting the hydropower potential.

[0115] The resource retention capability of an electrical system, namely:

[0116]

[0117] in This represents the set of all hydropower station nodes in the initial water network. For nodes The amount of water used for power generation This refers to the set of remaining hydropower station nodes that continue to operate normally despite the disturbance. For the remaining nodes The actual available water volume for power generation;

[0118] Water network resilience assessment based on seepage theory includes analytical methods based on improved seepage theory, water network resilience analysis based on structural seepage, and water network resilience analysis based on functional seepage.

[0119] An analysis method based on improved seepage theory: For reservoir-hydropower station nodes, the reservoir capacity ratio is a key indicator for measuring their current water storage status and potential power generation capacity. The selection of the reservoir capacity ratio for each hydropower station node is crucial. As its "quality" indicator at time step t; each seepage analysis is performed at a specific time snapshot, in which the state of each node remains unchanged during the analysis, that is, at each snapshot t, the reservoir capacity ratio of the node is... The parameters remain fixed; specifically, based on a specific time snapshot in the current system dynamics simulation process, for any time step t of the network, Algorithm 7 is executed, and the seepage control parameters are adjusted accordingly. The specific process of regulating network topology evolution is as follows:

[0120] Node quality characterization: The reservoir capacity ratio of node i in the hydropower station. Mapped to node quality metrics, when A node is considered to be functioning normally if it is in a normal state, otherwise it is considered to be a node with impaired function.

[0121] Dynamic adjustment of connecting edges: Remove edges associated with failed nodes;

[0122] Progressive failure simulation: by Starting from 0 and gradually increasing to 1, the system studies the response process of the network under stress scenarios such as persistent drought. Each... The value corresponds to a percolation subnetwork. By tracking the phase transition patterns of indicators such as the size of its maximum connected component, the resilience of the system can be quantitatively assessed.

[0123]

[0124] Water network resilience analysis based on structural seepage: Structural resilience focuses on the change in the integrity of the network topology during node removal. In seepage analysis, the node topology resilience index is used. By tracking the changes in the relative size of the largest and second largest connected components with the seepage parameter ρ, the resilience of the network topology is systematically evaluated.

[0125] When control parameters Reaching a certain critical value At this point, the macroscopic properties of the system will change drastically, reaching the critical seepage threshold. Defined as the second largest connected component size When it reaches its peak Value; in At that time, there exists a giant connected component in the network; when achieve At that time, the giant component began to disintegrate, leading to Increase; when At that time, the network further fragmented. Consequently, it decreases; when analyzing the phase transition in the network seepage process, the method of analyzing phase transition in statistical physics is used as a reference, that is, the order parameter is examined. For seepage parameters Derivative: First derivative Second derivative

[0126] Water network resilience analysis based on functional seepage: Based on the functional resilience index of the water network, the functional seepage index of the water network is defined as the index under a given seepage threshold. Below, all "surviving" nodes are At the current time step Total hydropower generation ;

[0127]

[0128] in It is the set of all nodes in the water network. It is a node At time step The actual hydropower generation; comparing the resilience of different time steps or different networks, using relative functional indicators, i.e., the current total hydropower generation versus the initial seepage flow ( The ratio of total hydropower generation (at any given time):

[0129]

[0130] The range of values ​​is This indicates that when removing storage capacity ratios below a certain level... After the node, the remaining power generation capacity of the network accounts for the proportion of the original total power generation capacity; Follow The changing curves illustrate the process by which the function of the hydropower network decreases as the intensity of disturbance increases;

[0131] When the demand in the network meets the homogeneous condition, use The area under the curve is used as the current time step. The overall resilience index of the network is denoted as :

[0132]

[0133] It can comprehensively reflect the network's functional maintenance level at different times. The larger the value, the better. The more robust the network snapshot at a given time step is to the overall power generation capacity of nodes under drought disturbances, the stronger its resilience. This is achieved by calculating different time steps. of This allows us to track the evolution of the resilience of hydropower networks over time.

[0134] The beneficial effects of adopting the above technical solution are as follows:

[0135] 1. This invention constructs a heterogeneous weighted two-layer network model: By integrating multi-layer network theory and watershed topology association mechanism, a water-electricity coupled two-layer dependent network model is established. For the first time, it realizes the integrated characterization of node heterogeneity, edge weight dynamics and inter-layer coupling relationship, providing a theoretical tool for topology analysis of complex coupled systems.

[0136] 2. This invention develops a cross-scale dynamic modeling method: Combining power flow equations and water flow balance calculations, a differential equation modeling framework considering time delay characteristics and load dynamics is proposed, which solves the problem of dynamic simulation of coupled systems with multiple time granularities and multiple physical constraints in urban distribution networks and regional main networks.

[0137] 3. This invention innovates a robust analysis paradigm driven by seepage: by combining seepage theory with physical flow dynamics, a two-way seepage equation for the cascading propagation of disasters is established, revealing the phase transition threshold and toughness critical conditions of the water-electricity coupled network, and providing a new method for risk quantification assessment under extreme events.

[0138] 4. This invention forms a full-chain risk analysis system: Based on the coupled case of IEEE-118 node power grid and watershed network, a simulation platform for the entire process from drought risk triggering, cross-network propagation to system collapse is constructed, which verifies the effectiveness of the model in guiding drought relief scheduling and resilience optimization.

[0139] 5. This invention not only deepens the understanding of the complex response mechanism of water-electricity coupled systems under extreme drought events, but also proposes a resilience assessment framework based on seepage theory and reveals the laws of different coupling effects of water-electricity systems, which can provide a scientific basis for risk assessment, resilience enhancement and emergency management strategy formulation for critical infrastructure. Attached Figure Description

[0140] Figure 1 This is a flowchart illustrating the present invention;

[0141] Figure 2 This is an example of generating topological connections between nodes in a hydropower station;

[0142] Figure 3 This is a schematic diagram of a two-layer interdependent heterogeneous network framework for a water-electricity coupled system.

[0143] Figure 4 This refers to the daily flow rates of major reservoirs and hydropower stations in the Dadu River basin from June to October 2022.

[0144] Figure 5 The seepage index of the water network structure during the early stages of the drought (July 30th) varies with the seepage threshold. Changes;

[0145] Figure 6 These are the first and second derivatives of the seepage index of the water network structure in the early stage of the drought (July 30);

[0146] Figure 7 It refers to the functional resilience of the power grid under different coupling strategies;

[0147] Figure 8 These are the results of seepage analysis of water network structures under different drought stages;

[0148] Figure 9 This refers to the topological evolution during the seepage process on July 30, 2022.

[0149] Figure 10 This refers to the topological evolution during the seepage process on August 30, 2022.

[0150] Figure 11 It represents the results of seepage and functional seepage in the water network structure under different drought stages;

[0151] Figure 12 This refers to the water network seepage threshold from June to October 2022. and functional resilience indicators The daily changes;

[0152] Figure 13 It refers to the resilience of the power grid structure under different coupling strategies;

[0153] Figure 14 It is a power grid resilience indicator and The time distribution of the occurrence of the minimum value;

[0154] Figure 15 This refers to the water network seepage threshold from June to October 2022. and functional resilience indicators Monthly statistics. Detailed Implementation

[0155] 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 of ordinary skill in the art without creative effort are within the scope of protection of this application.

[0156] This invention will be developed from the following three aspects:

[0157] 1. Construction of a Two-Layer Network Model for Water-Electric Coupling. This study utilizes multilayer network theory to establish a two-layer weighted network model for regional water-electric coupling systems. This model treats water supply and demand points in the water network, and power sources, energy storage, and load points in the power grid as heterogeneous nodes, and sets the rated flow of the water network and the maximum transmission power of the power grid as edge weights, thus fully reflecting the heterogeneity of nodes, connections, and edge weights.

[0158] 2. Dynamic Modeling of Water-Electric Coupling. In multilayer network theory, the dynamic processes between nodes are often characterized as first-order differential equations through dimensionality reduction. In actual water-electric networks, the dynamics of water-electric coupling are influenced by the infrastructure functions of different nodes, the time delay characteristics such as water flow lag, and the changes in load curves. Therefore, the dynamic model of water-electric coupling needs to be combined with the above physical characteristics, and the flow rates of nodes and connections need to be determined by combining power flow calculations and water flow balance calculations.

[0159] 3. Network Robustness Analysis Based on Seepage Theory. To investigate the impact of interdependencies between networks on cascading failures and network robustness, the seepage model models the probability of node cascading failures, thereby revealing the phase transition characteristics of multilayer networks. In water-electric coupling networks, it is necessary to combine water and electrodynamic modeling to improve the seepage model of nodes and connections, integrating its probability generation with the physical flow rates of water and electrodynamics. Further analysis is conducted on the node clustering characteristics and network robustness of water-electric coupling networks.

[0160] like Figure 1 As shown, the resilience assessment method for water-electricity coupled systems considering drought risk evolution includes the following steps:

[0161] S1. Network modeling of the water-electricity coupled system;

[0162] S2. Drought risk propagation modeling based on seepage-induced water-electricity coupled networks;

[0163] S3. Based on the above two models, perform system simulation and resilience assessment on the water-electricity coupled system network.

[0164] In step S1, the process of establishing the network model of the water-electricity coupled system includes the following steps:

[0165] S11. Characterizing the key elements of a water-electricity coupling system

[0166] The water network model is represented as .in It is the set of nodes in a water network. For the edge set in the water network, Let be the edge weight of the node in the water network.

[0167] S12, Network Modeling of Reservoirs and Hydropower Stations

[0168] S121. Node Dataset Construction and Watershed Topology Association

[0169] S1211, Define the node attributes of the hydropower station

[0170] Hydropower stations can be classified into two categories based on their regulation capabilities: reservoir hydropower stations and run-of-river hydropower stations. These two types differ significantly in their engineering structure and operating mechanisms. Reservoir hydropower stations utilize large reservoirs with regulation capabilities to redistribute natural runoff over time. By adjusting reservoir capacity, they adapt to changes in electricity demand; when inflow exceeds demand, the reservoir stores water; when inflow is insufficient, the reservoir replenishes water. Therefore, reservoir hydropower stations can actively control hydrological processes, improving the efficiency of natural runoff utilization. Run-of-river hydropower stations, on the other hand, are a general term for hydropower stations without regulation capabilities over natural runoff. They rely on natural runoff processes, lack significant regulation capabilities, and their power output is highly correlated with instantaneous inflow, resulting in low guaranteed output and a large proportion of seasonal electricity generation. When the incoming water flow exceeds the turbine's capacity, the hydropower station operates at full capacity. The excess water is not used for power generation but is discharged directly downstream through the spillway, a process known as water wastage. When the incoming water flow is low, all the water is used for power generation, but some of the installed capacity remains unused due to water shortage.

[0171] In this embodiment, we focus on 24 large-scale hydropower stations in the Yangtze River Basin. These hydropower stations are all reservoir-type hydropower stations, whose seasonal inflow mainly originates from surface runoff processes in the upstream catchment areas. They have large installed capacity, strong power generation capabilities, and are of great significance to the regional power supply. Spatially, these hydropower stations are mainly distributed in western China, located between 97°42′E and 111°16′E and 22°3′N and 36°7′N, covering administrative regions including Chongqing Municipality, Sichuan Province, Guizhou Province, Yunnan Province, and Hubei Province, with a total area of ​​1.313 million square kilometers, accounting for 13.67% of my country's land area. The main rivers where these hydropower stations are located include the Yangtze River, Jinsha River, and Yalong River, among others. These rivers have large drainage areas, complex topography, deep river systems, and abundant surface runoff.

[0172] In addition, we used the Python Requests module to call the Google Elevation API for batch data collection, obtaining the altitude information of hydropower stations to achieve 3D spatial positioning of hydropower station nodes. This service is based on multi-source remote sensing elevation data fusion technology, allowing developers to obtain elevation data of specific locations on the Earth's surface based on given latitude and longitude coordinates. This API is widely used in mapping applications, hiking navigation, and real estate analysis. When the coordinates requested by the user do not have a direct and accurate elevation measurement, the Google Elevation API estimates the altitude using a bilinear interpolation algorithm, calculating the average from the four nearest known points to ensure continuous elevation data for any location. The core parameters of the API call include the `locations` parameter and the `key` parameter. The `locations` parameter is the latitude and longitude coordinates, and the `key` parameter is the API key used for authentication and quota control. The JSON data returned by the request contains the elevation information of the requested location, in meters. Specific geographic information is shown in Table 2.1.

[0173]

[0174] Table 2.1 Summary of Geographic Information for 24 Large Hydropower Stations

[0175] (Latitude and longitude are in degrees, and altitude is in meters.)

[0176] S1212, Constructing the mapping relationship between hydropower stations and river basins

[0177] In constructing the water network node dataset, in addition to the basic geographic information of hydropower stations, we further associate hydropower station nodes with watershed topology based on the HydroBASINS dataset to achieve spatial association between hydropower stations and watersheds. The HydroBASINS dataset is a series of polygonal layers used to characterize watershed boundaries and sub-watershed delineation globally, providing consistent-sized and hierarchically nested sub-watersheds at different scales (from tens of square kilometers to millions of square kilometers), achieving seamless global coverage. The following algorithm is used to achieve spatial association between hydropower stations and sub-watersheds, mainly including the following three steps:

[0178] S12121, Coordinate Transformation: Convert the WGS84 latitude and longitude coordinates of the hydropower station to ESRI:54009 world cylindrical projection to eliminate spherical distance distortion;

[0179] S12122, Spatial Query: Use the contains() method of the Shapely library to perform point-to-face containment detection;

[0180] S12123, Hierarchical Matching: Prioritize searching within the Pfafstetter levels 5-7 sub-basins; if a match fails, expand to adjacent levels. The specific process is shown in Algorithm 1.

[0181]

[0182] S122. Constructing an edge generation mechanism based on watershed topology.

[0183] Based on the two major geospatial datasets HydroBASINS and HydroRIVERS, this method achieves a quantitative characterization of the hydraulic connections between hydropower stations through the spatial correlation of multi-scale watershed units and river networks. Specifically, this method mainly includes the following three steps:

[0184] S1221, Watershed Topology Path Search: Based on the hydropower station-watershed mapping relationship determined in step S1212, and using the HydroBASINS hierarchical watershed division system, establish the topological connection relationship between watershed units. For any two hydropower station nodes... The spatial correlation is determined through the following process:

[0185] S12211, Locating the watershed unit: Through spatial point inclusion analysis, determine the HYBAS_ID of the watershed to which each hydropower station belongs, denoted as... , See Algorithm 1 for details.

[0186] S12212, Tracing downstream paths: From Starting from a given watershed, move to its downstream watershed based on the NEXT_DOWN field within that watershed. Construct potential connection paths ;

[0187] S12213. Determine connectivity: If the path contains the target watershed... Then determine for The upstream node; if it is still not found after traversing to the end of the watershed (NEXT_DOWN=0). If no direct hydraulic connection is found, it is determined that there is no direct hydraulic connection. This process is implemented through Algorithm 2, which effectively avoids the complex topographic data processing in traditional hydrological analysis methods and directly utilizes standardized watershed topology to improve computational efficiency.

[0188]

[0189] S1222, Extraction of main river paths; obtaining watershed paths between hydropower stations. Later, in order to describe the hydropower station nodes in more detail... and To determine the hydraulic transmission relationship between the hydropower stations, it is necessary to further extract the main river path in order to realize the physical hydraulic connection between the hydropower stations.

[0190] The HydroRIVERS dataset was used to map watershed paths to physical rivers. The HydroRIVERS dataset contains a vectorized network of rivers worldwide, all meeting certain hydrological conditions (catchment area of ​​at least 10 square kilometers or average flow rate of at least 0.1 cubic meters per second). It covers 8.5 million rivers with an average length of 4.2 kilometers and a total length of 35.9 million kilometers. The dataset includes not only geometric information but also hydrological and hydraulic information such as river segment length, distances to upstream sources and ocean outlets, river order, and estimates of long-term average flow. Each river segment is also collectively registered with its corresponding sub-watershed in the HydroBASINS database (via a shared ID). See Table 2.2 for details.

[0191]

[0192] Table 2.2 Attributes of the HydroBASINS dataset

[0193] For the confirmed connectivity path Extract the path of the main river .

[0194]

[0195] In the formula, ℝ represents the set of Hydrorivers. This system filters the river, ensuring that only the main channel is selected. This filtration mechanism effectively eliminates interference from tributaries, highlighting the main water conveyance channels between watersheds.

[0196] S1223, Rules for constructing topological edges

[0197] The final two hydroelectric power station nodes Connecting edges The generation must satisfy two constraints:

[0198]

[0199] I. Hydraulic connectivity: There exists a collection of non-empty main rivers.

[0200] II. Topological Exclusivity: There are no intermediate hydropower station nodes in path p, i.e.

[0201]

[0202] When the above conditions are met, a slave node is established. arrive directed edges And record its attribute tuple: ,in For the edge A list of HYRIV_IDs for the main rivers. (e.g.) Figure 2 The image shows an example of generating topological connections between nodes in a hydropower station.

[0203] S123. Establish a side weight calculation model that simultaneously considers the characteristics of watershed isomorphism and dynamic time-delay coupling.

[0204] Watershed isomorphism coefficient Representation from node arrive The hydrological connectivity cost is based on the watershed hierarchy in the HydroBASINS dataset and the spatial association results between hydropower stations and sub-watersheds obtained by Algorithm 1, according to the hydropower station and The watershed matching situation is used to define the watershed isomorphism coefficient. The classification table is 2.3.

[0205]

[0206] Table 2.3 Watershed isomorphism coefficients based on HydroBASINS Classification table

[0207] Known For the edge A list of HYRIV_IDs for the main rivers. A river transport time delay model is constructed based on hydrodynamic principles to calculate key parameters for each river, including river bend, average slope, hydraulic radius, flow velocity, and transport time delay.

[0208] I. River section curvature

[0209] River meandering is an indicator of how tortuous a river is, defined as the ratio of the actual length of the river's path to the straight-line distance from its starting point to its endpoint. A straight river has a meandering value of 1, and a higher meandering value indicates a more meandering river. Meandering affects flow velocity, sediment transport, and bank erosion, and is particularly important in geomorphological and hydrological studies. For example, rivers with high meandering typically have slower flow rates, providing rich habitats for life and supporting the formation of diverse ecosystems.

[0210] For each section of the river In the HydroRIVERS dataset, the geometric information is shapely.LineString, which is a one-dimensional geometric shape type composed of one or more line segments, consisting of several coordinate points, with a line segment connecting every two adjacent points. Let... Let be the k-th coordinate point in the geometry of river segment r, and geod(⋅) represent the geodetic distance between two points. Then, the formula for calculating the sinuosity of this river segment is:

[0211]

[0212] II. Hydraulic Gradient Slope

[0213] The average slope is the ratio of the elevation change of a river reach to its actual length. It affects flow velocity and sediment transport capacity, and is a direct reflection of the river's energy gradient. In hydraulics, slope usually refers to the riverbed slope, but in some cases it may involve energy slope or surface slope. For steady-state uniform flow, the riverbed slope is usually used as an approximation. It is calculated by dividing the elevation difference Δℎ between the starting and ending points by the actual length of the river reach.

[0214] III. Water Conservancy Radius

[0215] The hydraulic radius R is an important parameter of a river channel cross-section, referring to the ratio of the flow area of ​​a water conveyance section to the side length of the water conveyance pipe in contact with the water (i.e., wetted perimeter). It is related to the cross-sectional shape and is often used to calculate the water conveyance capacity of channels and tunnels. According to the literature, for the common trapezoidal cross-section of natural rivers, the average water depth can only be used to replace the hydraulic radius for flow calculation when the width-to-depth ratio W / D ≥ 66.3. Therefore, we still need to estimate the hydraulic radius here. Leopold and Madison proposed an empirical formula for estimating the hydraulic radius, indicating that it can be estimated based on the river flow and upstream area:

[0216]

[0217]

[0218]

[0219]

[0220]

[0221] Let Q be the upstream catchment area (square kilometers), Q be the historical average flow rate (cubic meters per second), W be the river width (meters), estimated based on the classic river width index, and D be the average river depth (meters). Considering that the natural river channel is more similar to a trapezoid, a trapezoidal slope coefficient Z = 1.5 is added.

[47] ， A and P represent the corrected trapezoidal cross-sectional area (square meters) and wetted perimeter (meters), respectively; R is the final estimated hydraulic radius (meters).

[0222]

[0223] Table 2.4 Verification of River Parameter Prediction Accuracy

[0224] IV. Estimation of water flow velocity based on the Manning-Strickler Formula;

[0225] The Manning equation, as the core equation of open channel fluid mechanics, provides a practical method for converting geometric and frictional channel parameters into flow velocity predictions in cubic meters per second based on channel characteristics (such as cross-sectional area, hydraulic radius, channel slope, and surface roughness). It is widely used in engineering applications ranging from urban drainage systems to natural river analysis.

[0226]

[0227] •u is the average flow velocity across the cross section (m / s)

[0228] •k is the dimension conversion factor, which is 1 in the International System of Units (SI units).

[0229] • n is the Manning coefficient (or Manning roughness coefficient), with units of _____.

[0230] R is the hydraulic radius (m), defined as the ratio of the cross-sectional area A to the wetted perimeter P.

[0231] • S is the gradient of the water flow, which is dimensionless and is the ratio of the drop height of the water flow to the horizontal distance of the water flow.

[0232] V. In summary, for each river section The water flow velocity was calculated. Based on water flow velocity and river length The transport time delay of water flow in river segment r was calculated. :

[0233]

[0234] From node arrive Directed edge Transmission delay It is defined as the sum of the transmission time delays of all river segments along the edge, reflecting the cumulative effect of water flow transmission time.

[0235]

[0236] S13, Power Network Modeling

[0237] The IEEE-118 node system was selected as the power network. The system consists of 118 nodes and 186 branches, and the specific parameters are given in Table 2.6.

[0238]

[0239] Table 2.6 Number of nodes, links, generator nodes, and demand nodes in the IEEE-118 bus system

[0240] We selected the IEEE-118 node system as the power network. , where the node set It includes generation nodes and load nodes. The node attributes are defined as follows:

[0241] • Generation node: Baseline active power reactive power Voltage amplitude and constraints on regulatory capacity;

[0242] • Load node: Active power demand reactive power demand Voltage amplitude ;

[0243] • Connection node: Voltage phase angle As a system reference benchmark.

[0244] Edge set This includes transmission lines and transformer branches, each side It has the following attributes:

[0245] • Series resistor With reactance , forming a complex impedance ;

[0246] • Parallel admittance (Line charging effect);

[0247] • Thermally stable capacity , which represents the maximum permissible transmission power.

[0248] In the weight matrix In the construction, edge weights are usually... Defined as the absolute value of line susceptance:

[0249]

[0250] This weight reflects the natural tendency of a line in power allocation; lines with lower reactance have higher power transmission priority.

[0251] A DC power flow model is used to model the steady-state and dynamic behavior of the power system. Node voltage phase angle vectors are defined. With active power injection vector Its physical relationship is determined by the weighted Laplace matrix. Establish: The elements of the susceptance matrix B , The construction satisfies:

[0252]

[0253] Line power flow Driven by phase angle difference, its endpoint phase angle difference is proportional:

[0254]

[0255] Obtained by calculating the Moore-Penrose pseudoinverse of the Laplace matrix. :

[0256]

[0257] Implicit nonlinear dependence on network topology B when a certain line Exceeding capacity At that time, the following dynamic process is triggered:

[0258] I. Overloaded circuit Removed, update topology ;

[0259] II. Reconstructing the susceptance matrix And recalculate ;

[0260] Third, the redistribution of power flow according to equation (2.19) may cause cascading overloads on adjacent lines.

[0261] This process exhibits significant nonlocality: a single line fault can induce power flow oscillations across the entire network through topological correlations, and the computational complexity increases superlinearly with network size. While this cascaded model based on the DC approximation simplifies the electromagnetic transient process, it effectively captures the macroscopic statistical characteristics of fault propagation, making it significant for risk propagation and dynamic analysis in hydro-electric coupled networks.

[0262] S14, Water-Electricity Coupled Network Modeling

[0263] The water-electricity coupled network consists of a water network and power grid Composition. Power Grid Nodes representing generator sets, substations, and / or power loads The system consists of links between nodes via transmission lines and transformer branches. Connection. Simultaneously, the reservoir and hydropower station network. Hydropower station nodes with upstream and downstream relationships Composed of nodes connected by rivers or waterways The water flow along the water network is controlled by reservoir scheduling. In addition, the power grid... WaterNet These networks are interconnected via hydropower conversion links, forming an interdependent, two-layer coupled network. For example... Figure 3 As shown.

[0264] Hydropower Conversion Link The energy conversion process of hydropower generation is defined, serving as the core channel for transmitting electricity from the hydropower network to the power grid. It begins at a node in the hydropower network and terminates at a node in the power grid. Hydropower generation power can be expressed as a function of the power generation flow, the hydropower station head, and the hydropower station's output coefficient.

[0265]

[0266] in Let be the density of water (1000 kg / m³), g be the acceleration due to gravity (9.81 m / s²), Q be the power generation flow rate (m³ / s), and H be the effective head height (m). The comprehensive hydropower efficiency coefficient is (0.75-0.92). The head height H has dynamic characteristics and is calculated from the difference between the upstream and downstream water levels of the hydropower station.

[0267] Based on the above, and using complex network theory, we construct the water-electricity coupled system as a two-layer dependent heterogeneous network. :

[0268]

[0269] Network heterogeneity manifests itself in three dimensions:

[0270] 1. Node heterogeneity

[0271] Water network nodes represent hydropower stations at various levels, while power grid nodes include power generation nodes and load nodes. The nodes are connected by transmission lines to form a network structure.

[0272] 2. Edge heterogeneity

[0273] The nodes of the water network are distributed in series or parallel along the rivers, constructed based on watershed topological relationships. The edge weights of the water network comprehensively consider the watershed isomorphism of the nodes, the time delay effect of water flow transmission, and spatial gradient differences. The nodes of the power grid are connected through transmission lines and transformer branches, and the edge weights reflect the natural tendency of the lines in power distribution. Inter-layer connections. This depicts the energy exchange relationship between the two infrastructure systems: the water network and the power grid.

[0274] 3. Dynamic heterogeneity

[0275] The dynamic response mechanisms of the water network and the power grid subsystems differ significantly when faced with external disturbances. The upstream and downstream relationships between nodes in the water network determine the direction of water flow and the allocation of available water; the scheduling decisions of upstream hydropower stations directly affect the inflow of water and the power generation capacity of downstream hydropower stations. In the power system, node or line faults can affect load matching and supply-demand balance, causing link overload, leading to power transmission interruptions, and subsequently triggering a cascading failure in the power grid.

[0276] In addition, through inter-layer connections The water network and the power grid form an inter-layer energy transmission channel, causing external disturbances to not only trigger chain reactions within the subsystems but also create feedback loops between layers, leading to the cross-layer propagation of risks. It is worth noting that the physical constraints, temporal granularity, and spatiotemporal relationships of nodes differ between the water network and the power grid subsystems. Therefore, a dynamic model of the water-electricity coupled network needs to be constructed by combining the physical meaning of nodes and weighted edges to more accurately characterize the complex dynamic behavior of the system.

[0277] By constructing a water-electricity coupled two-layer network model that simultaneously considers nodes, edges, and dynamic heterogeneity, the limitations of traditional single-layer and homogeneous networks are overcome, thus more realistically depicting the complex water-electricity coupled system.

[0278] In step S2, the drought risk propagation modeling process based on the water-electricity coupling network of seepage is as follows:

[0279] S21. First, establish seepage models within the water network and the power grid respectively: the water network model focuses on the state evolution based on water balance and control strategies; the power grid model focuses on link failures and dynamic power rebalancing based on overload.

[0280] The seepage process in the water network:

[0281] I. The seepage model of the water network should consider the time-delay propagation mechanism of hydropower station power generation capacity degradation. Based on the principle of watershed hydrological balance, the reservoir of the hydropower station can be modeled as a storage unit with a certain capacity (in terms of water volume) and daily inflow to describe the water volume change of the reservoir.

[0282] Water balance equation for upstream hydropower station: ;

[0283] Water balance equation for downstream hydropower station:

[0284]

[0285] It represents the set of nodes of the upstream hydropower station.

[0286] It represents the set of nodes of the downstream hydropower station.

[0287] and These represent the water volume of hydropower station i at time t and t+1, respectively.

[0288] It is the natural inflow of water into the reservoir of a hydroelectric power station at time t.

[0289] It is the amount of water used by a hydroelectric power station to generate electricity at time t.

[0290] It is the amount of water discharged by the hydropower station at time t.

[0291] It is the set of directly upstream nodes affecting hydropower station j.

[0292] It is the water flow lag time from the upstream hydropower station to the downstream hydropower station j, where ,

[0293] By introducing Incorporating this into the water balance equation allows for a more accurate simulation of the water transfer process between reservoirs, i.e., the upstream node i in... Actions at any moment ( and water discharge ), needs to go through It will take time before it affects the water storage of downstream node j.

[0294] By using historical hydrological data and statistical analysis methods, a probability distribution model of the natural inflow of water into the reservoir is established to simulate the changes in water volume in the reservoir under drought conditions.

[0295] II. Multi-state model of hydropower station nodes

[0296] Define nodes The storage capacity ratio is A five-level drought classification system was established based on the reservoir capacity ratio, with the drought level of node i as the reference. Classification criteria:

[0297]

[0298] in, Represents the maximum reservoir capacity of node i in the hydropower station. Let represent the drought level of node i at time t. After discretization, the lower the reservoir capacity ratio of the hydropower station, the less water in the reservoir, and the higher the corresponding drought level; conversely, the higher the reservoir capacity ratio of the hydropower station, the more water in the reservoir, and the lower the corresponding drought level.

[0299] Based on the drought level defined by the reservoir capacity ratio, a multi-state model of the hydropower station nodes is established. For the hydropower station nodes... From the perspective of drought level, its state space is as follows: Let represent the set of drought level states for hydropower station node i. The state of hydropower station node i at time t is... From the state space Take the value from the middle.

[0300] III. Water Quantity Regulation Strategies for Hydropower Network Nodes: The action space defines the decision variables and adjustment rules for each hydropower station node at each time step t. The action variables for each hydropower station include the amount of water used for power generation. and water discharge .for and The solution is obtained by adjusting the value using pre-defined control rules, which are based on the current state of the node. That is, drought level The water balance equation is used to determine the action space. The following is a detailed description of the action space.

[0301] A. Hydropower generation

[0302] This represents the amount of water used for power generation at hydropower station i at time t, and the amount of water released through the turbine for power generation (unit: cubic meters). Power generation water volume Based on the current drought level The proportional adjustment reflects the practical strategy of reducing power generation to conserve water resources under drought conditions. The adjustment rules are as follows:

[0303]

[0304] The baseline water generation capacity of hydropower station i at time t is typically determined based on the station's installed capacity, historical operating data, or electricity demand. This is under normal, drought-free conditions... hour, This can be considered as the maximum power generation capacity during normal operation. . The proportional coefficients are determined based on drought levels, and the specific values ​​are shown in the table below.

[0305]

[0306] Table 3.1 Power generation ratio under different drought levels

[0307] This is related to the amount of water used for power generation. The adjustment rules are in line with the actual strategy of hydropower stations to reduce the amount of water used for power generation in order to ensure reservoir storage and downstream water supply needs. Under normal conditions... Under normal conditions, the amount of water used for power generation can reach the benchmark value, making full use of water resources for power generation; while in drought conditions... If necessary, the amount of water used for power generation will be gradually reduced to prioritize ensuring water storage in reservoirs and water supply needs downstream.

[0308] B. Wastewater volume

[0309] This represents the amount of water discharged by hydropower station i at time t, and represents the amount of excess water actively released, which usually occurs under the following circumstances: (1) the current water volume of the reservoir. Approaching maximum storage capacity (2) The inflow exceeds the maximum power generation capacity and available reservoir capacity of the hydropower station. Based on the above characteristics, the adjustment rules for the amount of water to be discharged take into account the current status of the hydropower station and the impact of drought level on the water discharge strategy. The specific calculation steps are as follows:

[0310] 1. Calculate potential new storage capacity

[0311] Potential new storage capacity This indicates that the water volume of the reservoir at the next moment can be calculated without discarding water.

[0312] • Upstream hydropower station:

[0313]

[0314] • Downstream hydropower station:

[0315]

[0316] 2. Calculate the amount of water to be discarded.

[0317] If the potential new storage capacity exceeds the maximum storage capacity In such cases, water must be released to maintain the safety of the reservoir.

[0318] • Upstream hydropower station:

[0319]

[0320] • Downstream hydropower station:

[0321]

[0322] The above rules ensure that reservoirs and hydropower stations minimize water discharge during drought conditions, and only discharge water when the reservoir overflows.

[0323] IV. Water Network Operation Status Update

[0324] By integrating water balance and regulation strategies, the hydrological state evolution of a hydropower station network under drought conditions was simulated. The overall system state update follows a discrete-time recursive principle, with the water network state at input time t. The system state at time Δt can be obtained. Its dynamic equation is:

[0325]

[0326] In the formula The state transition function for the water network has the following input parameters:

[0327] Network Topology : Describe the set of reservoir nodes Connecting edge to water flow and corresponding weights

[0328] Water volume operation status

[0329] Node status

[0330] The output state variables are defined as follows:

[0331]

[0332]

[0333] Seepage process in power grid:

[0334] Seepage flow in a power grid refers to the disturbance propagation process based on power flow. Under extreme drought conditions, transmission lines in a power grid may experience cascading failures due to fluctuations in power generation and load or link overload, thereby affecting the stable operation of the system. This embodiment constructs a cascading failure model of the power grid based on an overload threshold-based link tripping mechanism, combined with a power redistribution mechanism, an islanding balancing strategy, and a dynamic state update mechanism, to simulate power grid seepage flow under extreme drought conditions.

[0335] In power grid model Among them Let E be the set of nodes (i.e., buses), and let E be the set of links (i.e., transmission lines). We assume that each node in the power grid is either a generator node, denoted as... It is either a demand node (load), represented as and define , , .

[0336] A. Link tripping mechanism based on overload threshold

[0337] In power network relay protection devices, a tripping mechanism is triggered based on the real-time relationship between link power and transmission capacity. For any link... Its overload level is defined as:

[0338]

[0339] in For the current power flow, This represents the maximum transmission capacity. When... At this time, the link enters an overload state.

[0340] Based on the link overload level defined above, a link... Overload severity :

[0341]

[0342] in, It is the start moment of each time step. and These are links Real-time current and maximum transmission capacity.

[0343] like In time Exceeding the threshold The relay protection device will automatically trip the link, meaning the link... The tripping time is The system trip time can be obtained by calculating the trip times of all links. :

[0344]

[0345] if If the value is greater than the time step Δt, then no link is tripped within that time step, and the power system remains in steady state; conversely, if the value is less than the time step Δt, then no link is tripped within that time step, and the power system remains in steady state. If the value is less than the time step Δt, it means that at least one link has tripped within that time step. The system then employs a two-stage verification strategy to handle this, implementing overload protection in stages according to priority:

[0346] 1. First priority (overload trigger):

[0347] If an overloaded link exists, the most severely overloaded link should be disconnected first to quickly eliminate the greatest risk.

[0348]

[0349] 2. Second priority (time-triggered):

[0350] If multiple links reach their maximum simultaneously Then select the link that first enters the overload state.

[0351] Trip

[0352]

[0353] B. Dynamic power rebalancing mechanism

[0354] After disconnecting an overloaded link, the generation and load in the power grid may become unbalanced, necessitating a rescheduling process to redistribute power and establish a new generation-load balance. (For the surviving subgraph) Define power imbalance:

[0355]

[0356] in and These are sets of generating nodes and load nodes, respectively. and These represent the power of the generating node and the load node, respectively.

[0357] Given the slope rate parameter This indicates the magnitude of power fluctuations at each power generation node. upper limit of power and lower limit Set to:

[0358] and

[0359] in , ,and These are the power generation nodes. The maximum, minimum, and actual power output. Achieving power generation-load balance through iterative adjustment, and thus dynamic power rebalancing, is mathematically essentially about solving...

[0360] The following is a constrained optimization problem:

[0361] ,

[0362] Based on the power imbalance, the system executes a power reallocation strategy to restore power balance by increasing or decreasing power generation or demand. The detailed mechanism of power reallocation is shown in Algorithm 4.

[0363]

[0364]

[0365] The process of updating the power grid operation status under the influence of water networks includes the following steps:

[0366] When the power load is greater than the power generation, that is (Insufficient power generation), the output of all generator sets is based on a given ramp rate. Increase. If all units reach their maximum output level. If the load demand still cannot be met, the excess power load is eliminated by successively cutting off the load of the least important node. The upscaling capacity of the power generation node is subject to two constraints: (1) output adjustment constraint. (2) Physical capacity constraints The actual adjustable range is the minimum of the two:

[0367]

[0368] When the power load is less than the power generation, that is (Excess power generation), all units output at a given ramp rate Reduce. If all unit outputs reach the minimum level. If power output balance cannot be achieved afterward, excess power generation is eliminated by shutting down the units at the least important nodes. The amount of power generation that can be reduced at a power generation node is subject to two constraints: (1) power output adjustment constraints. (2) Physical capacity constraints The actual adjustable range is the minimum of the two:

[0369]

[0370] C. Island Balancing Strategy

[0371] Due to the tripping of dangerous overloaded links, a fully connected network may be split into multiple connected components, i.e., islands. Unlike the giant component failure model in traditional seepage theory, each island in a power network can maintain operation with independent topology and operating state. Therefore, after the network is split into multiple connected components, for each island... implement:

[0372] 1. Topology integrity verification: If a subgraph does not contain any generator nodes or contains only one node, it is assumed that all nodes in it have failed, and it is determined to be a failed island and removed from the network.

[0373] 2. Power balance detection: calculation ,like This triggers the power redistribution mechanism and calculates the STT (Solution Time Limit) for the corresponding island according to the formula. If the STT of all islands is greater than the time step Δt, no link is tripped and the power system remains in steady state; otherwise, overloaded links will be tripped for protection.

[0374] 3. Multi-island coordination: Each island establishes a local adjacency matrix, independently performs power flow calculations, and updates the system state.

[0375]

[0376] in The reduced susceptance matrix, This is the net injected power vector.

[0377] D. Power grid operation status update

[0378] By combining network topology and power mechanism, a dynamic state update mechanism for power networks was realized, and the cascading fault model constructed is summarized as follows:

[0379] Step 1: System Initialization

[0380] Establishing the power grid topology Initialize the parameters of each node:

[0381] • Power generation nodes Set upper and lower power limits ,

[0382] • Load node Loading reference power

[0383] • Calculate the initial power flow distribution It satisfies Kirchhoff's laws.

[0384] Step 2: Disturbance Injection

[0385] Based on the specific event, determine the set of nodes that are disturbed. And inject disturbances.

[0386] Step 3: Subgraph Detection

[0387] After a disturbance occurs, the power grid may break down into several isolated subgraphs. If a subgraph does not contain any generating nodes or contains only one node, it is assumed that all nodes in it have failed and it is removed from the power grid.

[0388] Step 4: Power Balance Adjustment

[0389] After step 3, if some isolated islands If there is an imbalance between power generation and load, a power reallocation strategy is implemented to restore power balance, and the power flow of each link in the island is recalculated and updated.

[0390] Step 5: Link overload detection

[0391] Link overload detection is performed based on the mechanism described above. If an overloaded link is found, a tripping mechanism is executed, and then the process proceeds to step 3 to continue the risk propagation process. Otherwise, a determination is made as to whether the system state has converged.

[0392]

[0393] If the condition is met, the iteration stops, and the cascading failure process terminates.

[0394] By combining the above steps, the simulation of cascading fault propagation in a power network is achieved. The overall state evolution of the system follows the discrete event-driven principle, with the power grid state at input time t. It can obtain the operating status of the power grid system after a time step Δt. The dynamic equation is:

[0395]

[0396]

[0397] in, It is the power grid's operating state transition function, with the current power flow operating state of the power grid as the input. Node status and network topology The output is the updated power flow operating status. Node status and network topology .

[0398] Power Flow Operation Status Includes power generation output Power load Admittance matrix Link operation status and voltage phase angle The state vector is defined as:

[0399]

[0400] The seepage process from the water network to the power grid:

[0401] E. Multistate model of hydropower nodes in power grid

[0402] To characterize the seepage process of drought risk propagating from the water network to the power grid, we defined a multi-state model for the power generation nodes in the power grid connected to hydropower stations in the water network. Based on the availability of water resources in the water network and its impact on power generation capacity, the models are divided into three operating states:

[0403] Normal state

[0404] The power generation nodes operate without restrictions and can adjust their power output according to dispatch instructions. This state corresponds to the hydropower station being unaffected by drought (i.e., drought level). At this point, the power generation capacity meets the benchmark level. .

[0405] Restricted state

[0406] Due to drought, water resources in the water network are insufficient, limiting the amount of water available for power generation at hydropower station nodes. This reduces the power generation capacity of nodes in the power grid that rely on hydropower stations, specifically manifested in their gradient rate. (i.e., the power output adjustment rate) is limited.

[0407] This occurred when the hydropower station faced ordinary drought conditions (i.e.) When the gradient is calculated based on the power generation ratio, the gradient rate is adjusted accordingly. Scaling (e.g., hour , hour (etc.), drought level and The relationships between them are defined in Table 3.1.

[0408] Failure status

[0409] The generating node completely loses its power generation capacity and is unable to contribute electricity to the power system. This state is triggered when a hydroelectric power station completely loses its power generation capacity, for example, during extreme drought (e.g., , In the event of a malfunction in the hydropower station equipment.

[0410] The multi-state model of hydropower nodes in the power grid defined above describes the direct impact of water availability on the operational performance of power system nodes, laying the foundation for analyzing the propagation of risk from the water network to the power grid during drought.

[0411] F. State transitions of hydropower nodes in the power grid

[0412] Driven by changes in drought levels at hydropower stations within a water network, the state of hydropower nodes in the power grid will change. The state transition rules are as follows:

[0413] 1. From normal to restricted

[0414] When the drought level rises to At that time, the power generation capacity of the hydropower station decreases. The ramp rate of the grid node is proportional. The adjustment reflects a weakening of its ability to regulate power output.

[0415] 2. Restricted to normal

[0416] When the drought level returns to At that time, the hydropower station's power generation capacity was fully restored. The grid node's ramp rate returned to its nominal value, and normal operation resumed.

[0417] 3. From limitation to failure

[0418] If the drought level worsens significantly (e.g.) If the threshold is reached, the power generation capacity will be completely lost, and the node will switch to a failure state.

[0419] 4. From failure to limitation

[0420] As the drought subsides and hydropower stations gradually resume generating power, grid nodes may transition from a failed state to a constrained state, and then potentially return to a normal state.

[0421] These conversion rules establish a dynamic link between water network conditions and power grid functions, which helps to track fault propagation.

[0422] G. Update of power grid operation status under the influence of water network

[0423] The power grid status update is based on the dynamic changes of hydropower stations in the water network, affecting the operational status of generation nodes and the network topology. The update process includes the following steps:

[0424] Step 1: Adjust the ramp rate of the restricted nodes

[0425] For nodes that are in or have just transitioned to a restricted state, the ramp rate According to drought level Corresponding power generation ratio Scaling:

[0426]

[0427] This adjustment reflects the limited ability of nodes to respond to changes in electricity demand.

[0428] Step 2: Recalculate the power output boundary of the confined node

[0429] Update the constrained power generation nodes using the adjusted ramp rate. Upper and lower limits of power output:

[0430]

[0431]

[0432] in, ,and These represent the maximum, minimum, and actual power output, respectively. These boundaries ensure that the node operates within its limited capacity.

[0433] Step 3: Remove failed nodes

[0434] Nodes in a failed state and their connected links are removed from the power grid, and the network topology is updated.

[0435]

[0436] in, It is a set of failed power generation nodes. It is a set of links connected to these nodes.

[0437] Step 4: Update the power grid operating status

[0438] Update the power grid's operating status using the methods described above, including power generation output, load, and power flow distribution.

[0439] Based on the above discussion, we can understand the seepage process from the water network to the power grid as a process in the water energy conversion link. Under the influence of [the system], the drought risk in the water network is propagated to the power grid, thereby changing the state of relevant power generation nodes in the power grid (see Table 3.2 for details), affecting the overall operating state of the power grid. The dynamic equation of the system can be described as:

[0440]

[0441] in, Refers to the link Update the power grid's operating status; the input is time. Water network node status The output is time. State vector of power grid system .

[0442] Table 3.2 Status Classification and Description of Hydropower Nodes in the Power Grid

[0443]

[0444] S22. By establishing a mapping relationship between the state of water network nodes and the state of power grid nodes, the transmission process of risk between layers is simulated, and finally the seepage dynamics equation of the coupled network is formed, realizing the cascading failure propagation model driven by physical mechanism.

[0445] By coupling the hydrodynamic processes in the water network with the power flow-based fault propagation in the power grid, the seepage equation of the water-electric coupled two-layer dependent network is obtained:

[0446]

[0447] The simulation continues until the following termination condition is met, indicating that the system state has reached steady state:

[0448]

[0449] Step S3 includes case-based simulation and verification, and a resilience index system for water-electricity coupled networks.

[0450] H. Simulation Settings

[0451] 1. Network topology

[0452] The reservoir-hydropower station network topology constructed earlier is adopted. The nodes of each hydropower station were clearly defined. and their upstream and downstream connections The edge attributes include the calculated water flow propagation time delay. Key parameters, etc.

[0453] 2. Drought Scenario Input

[0454] The daily natural inflow of each hydropower station node from June 1 to October 31, 2022, obtained by reconstructing the data using the improved quantile mapping method (e.g., ...) Figure 4 shown) sequence As a key input.

[0455] 3. Node operating parameters

[0456] Each hydropower station node The operating parameters are set based on publicly available datasets, including the maximum storage capacity. Initial storage capacity and the historical average hydropower generation during the same period Meanwhile, according to the storage capacity ratio defined in Formula 3.4 With drought level The mapping relationship, and the adjustment coefficients for the power generation ratio based on different drought levels. (Table 3.1) adjusts the power generation capacity of hydropower station nodes under different drought conditions.

[0457] 4. Dynamic Model

[0458] The dynamic state evolution of the water network system follows the dynamic equations defined in Section 3.2.4:

[0459]

[0460]

[0461] Simulation Result Analysis

[0462] The simulation uses a time step on the order of hours. The simulation (hourly) covers a complete period of 153 days, from June 1st to October 31st, 2022. Through simulation, detailed hourly operational status data of major hydropower stations in the upper reaches of the Yangtze River can be obtained during the continuous drought in summer and autumn of 2022. This data includes reservoir capacity at each node. Storage capacity ratio Actual hydropower generation Wastewater volume Actual natural inflow and the assessed drought level Key time series data are collected to provide a data foundation for subsequent analysis of the response mechanism of hydropower systems to drought events, assessment of model accuracy, and empirical verification.

[0463] J. Resilience Index System of Water-Electric Coupling Networks

[0464] 1. Topological resilience index

[0465] In terms of topology, resilience assessment typically focuses on changes in the integrity and connectivity of the system's structure before and after a disturbance and cascading failure. A commonly used assessment metric is comparing the size of connected components before and after the disturbance, particularly the changes in the size of the largest connected component (GC) and the second largest connected component (SC), thus reflecting the network's ability to maintain its structure and function. To eliminate the influence of different network sizes, we use the relative sizes of the largest connected component (GC) and the second largest connected component (SC) as a resilience indicator for the topology.

[0466] Relative size of the largest connected component (GC)

[0467] Relative size of the largest connected component Defined as:

[0468]

[0469] in, It is the number of nodes contained in the largest connected component (for a directed graph, it usually refers to the largest weakly connected component). It represents the total number of nodes in the original network. This directly reflects the network's ability to maintain large-scale connectivity even after the removal of nodes or edges. A higher... A value (especially after a perturbation) indicates that the main parts of the network remain connected, the network has good structural integrity, and stronger topological resilience against fragmentation. Evaluate before and after the perturbation. The change in the value can quantify the degree of loss in the network's topology.

[0470] The relative size of the second largest connected component (SC)

[0471] The relative size of the second largest connected component Defined as:

[0472]

[0473] in, It is the number of nodes contained in the second largest connected component of the network. Changes in the magnitude of these changes can reveal the degree of network fragmentation and structural vulnerability. When a network is attacked or malfunctions, if... While reducing A significant increase indicates that the network is rapidly splitting into multiple distinct components, suggesting a sharp decline in overall network connectivity and an impending collapse. Therefore, monitoring... Changes in these characteristics help identify the critical points of network resilience. One under perturbation... Smaller networks typically exhibit better topology resilience.

[0474] 2. Functional resilience index

[0475] Functional resilience measures the degree to which a system retains its expected functionality after being attacked and experiencing cascading failures. For critical infrastructure networks like hydro-electric coupling systems, functionality is primarily reflected in the ability to meet service demands. Hydro-electric coupling systems consist of a power grid (electricity transmission network) and a water network (reservoir and hydropower station network) coupled through a generation-transmission relationship. Functional resilience needs to be quantified separately for both the power grid and the water network subsystems. The core function of the power grid is to meet the electricity demand of end users, and its functional resilience is reflected in the service maintenance capability of electricity demand nodes. The core function of the water network is to maintain the water supply required for hydropower generation, and its functional resilience is reflected in the water supply maintenance capability of hydropower station nodes. The Unaffected Demand (UD) proposed by Hamedmoghadam et al. provides a unified analytical method for quantifying the functional resilience of hydro-electric coupling systems.

[0476] (1) Power grid functional resilience index

[0477] For power grid subsystems, the typical assessment is how much power demand the system can still meet after a cascading failure. Defined as the proportion of total electricity demand that can still be met after a cascading failure, it reflects the service maintenance capacity of the power system.

[0478]

[0479] in It is the set of all nodes with electricity demand in the initial network. It is a node The electricity demand. It is the set of nodes with remaining power demand after a cascading failure process triggered by an attack. It is the remaining demand node The power requirements.

[0480] (2) Water network functional resilience index

[0481] For the water network subsystem, the functional resilience index assesses the level at which the water network maintains its hydropower generation capacity during cascading failures. Defined as the proportion of water that can still be used for power generation under disturbance, out of the total water volume used for power generation, it reflects the resource retention capacity of a hydropower system, i.e.:

[0482]

[0483] in This represents the set of all hydropower station nodes in the initial water network. For nodes The amount of water used for power generation This refers to the set of remaining hydropower station nodes that continue to operate normally despite the disturbance. For the remaining nodes The actual available water volume for power generation.

[0484] 3. Water network resilience assessment based on seepage theory

[0485] An analysis method based on improved seepage theory: For reservoir-hydropower station nodes, the reservoir capacity ratio is a key indicator for measuring their current water storage status and potential power generation capacity. Therefore, the reservoir capacity ratio of the hydropower station node is selected accordingly. As a measure of its "quality" at time step t.

[0486] Furthermore, since the state of the water network changes dynamically over time under the influence of drought events, a seepage analysis needs to be performed at each time step t during the simulation. In other words, each seepage analysis is conducted at a specific snapshot time. Therefore, the state of each node in this snapshot remains unchanged throughout the analysis process; that is, at each snapshot t, the reservoir capacity ratio of the nodes remains constant. Fixed and unchanging. Specifically, based on a specific time snapshot during the current system dynamics simulation. For a network at any time step t, execute Algorithm 7, controlling the seepage parameters... The specific process of regulating network topology evolution is as follows:

[0487] Node quality characterization:

[0488] Hydropower station nodes Storage capacity ratio Mapped to node quality metrics, when A node is considered to be functioning normally if it is in a normal state, otherwise it is considered to be a node with impaired function.

[0489] Dynamic adjustment of connecting edges:

[0490] Removing only the edges associated with the failed nodes can reflect the impact of node functional degradation on network connectivity while preserving the integrity of the network's physical structure.

[0491] Progressive failure simulation:

[0492] By By gradually increasing from 0 to 1, the response process of the network under stress scenarios such as persistent drought can be systematically studied. The value corresponds to a percolation subnetwork. By tracking the phase transition patterns of indicators such as the size of its maximum connected component, the resilience of the system can be quantitatively assessed.

[0493]

[0494] Unlike traditional seepage models, this algorithm has two significant features: first, the node status is determined by the reservoir capacity ratio. Achieve continuous quantization, rather than simple binary partitioning, by using seepage parameters. The increment from 0 to 1 simulates the gradual failure of nodes due to a decrease in reservoir capacity (e.g., a drop in reservoir water level caused by drought). Secondly, failed nodes are not directly deleted, but functional isolation is achieved by disconnecting their connecting edges. This improvement is more in line with the actual operating mechanism of infrastructure networks, because even if a hydropower station temporarily fails due to insufficient reservoir capacity, its physical nodes still exist in the network and may re-enter system operation in the future as water storage recovers.

[0495] Water network toughness analysis based on structural seepage:

[0496] Structural resilience focuses on the change in the integrity of the network topology during node removal. In percolation analysis, the previously defined topological resilience index is used, which is determined by tracking the relative sizes of the largest and second largest connected components as a function of percolation parameters. Changes The system assesses the resilience of the network topology.

[0497] Another core concept in seepage theory is phase transition. When the control parameters... Reaching a certain critical value At that time, the macroscopic properties of the system (such as The critical seepage threshold can change drastically. Defined as the second largest connected component size When it reaches its peak Value. In At that time, there exists a giant connected component in the network; when achieve At that time, the giant component began to disintegrate, leading to Increase; when At that time, the network further fragmented. Consequently, it decreases. When analyzing phase transitions in network seepage processes, methods from statistical physics are often used to analyze phase transitions, namely, examining the order parameter. For seepage parameters Derivative: First derivative Second derivative .

[0498] Figure 5 The seepage index of the water network structure during the early stages of the drought (July 30th) varies with the seepage threshold. The changes are shown in the figure. The figure illustrates the relative size of the largest connected components. The relative size of the second largest connected component Critical threshold Depend on Peak value determined.

[0499] Regarding the change of the derivative Figure 6 show, right first derivative at the seepage threshold The value first increases rapidly and then decreases rapidly in the vicinity, creating a cusp; second derivative. at the seepage threshold The surrounding area exhibited a faster, more "explosive" process of change, suggesting that the water network model may have generated a quasi-discontinuous phase transition. It is important to note that since the seepage analysis is based on discrete... The derivatives obtained from the calculations are all finite values, therefore it is impossible to test whether the first derivative can reach infinity, forming a typical problem in Landau's theory. A phase transition does occur, but the very large second derivative suggests this possibility exists. To determine the type of phase transition more precisely, further analysis at the critical point is needed. The surrounding area uses a denser The values ​​are sampled and analyzed.

[0500] Water network resilience analysis based on functional seepage: Besides structural integrity, practical applications often prioritize the network's ability to maintain function under drought impacts. Based on the water network functional resilience index, the functional seepage index of the water network is defined as the ability to maintain function under a given seepage rate.

[0501] Flow threshold Below, all "surviving" nodes (i.e. At the current time step Total hydropower generation .

[0502]

[0503] in It is the set of all nodes in the water network. It is a node At time step The actual hydropower generation. To facilitate comparison of the resilience of different time steps or different networks, a relative function index is typically used, i.e., the current total hydropower generation versus the initial seepage flow ( The ratio of total hydropower generation (at any given time):

[0504]

[0505] The range of values ​​is This indicates that when removing storage capacity ratios below a certain level... After a node is reached, the remaining power generation capacity of the network accounts for the proportion of the original total power generation capacity. Follow The changing curves visually demonstrate how the function of the hydropower network diminishes as the intensity of disturbance increases.

[0506] To obtain a single scalar to measure the overall functional resilience of the network throughout the seepage process, Hamed-moghadam et al. proposed that when the demands in the network satisfy the homogeneity condition, the following scalar can be used: The area under the curve is used as the current time step. The overall resilience index of the network is denoted as... :

[0507]

[0508] It can comprehensively reflect the network's functional maintenance level at different times. The larger the value, the better. The more resilient a network snapshot is at any given time to drought disturbances, the stronger its overall power generation capability becomes. This is achieved by calculating data at different time steps. of It can track the evolution of the functional resilience of hydropower networks over time (e.g., during drought processes).

[0509] Power grid resilience analysis under different coupling strategies:

[0510] Hydropower Coupling Strategy Construction: In a hydropower coupling network, effectively coupling hydropower station nodes with power generation nodes in the power grid is crucial for drought analysis. The hydropower station network contains 24 hydropower station nodes. To ensure the rationality of the coupling, we match each hydropower station node with 2 or 3 power generation nodes in the power grid, based on the principle of even distribution. To systematically evaluate the potential impact of the coupling method on power grid resilience, five different coupling strategies were designed, covering three modes: random, assortative coupling, and disassortative coupling.

[0511] Stochastic Coupling: Stochastic coupling involves completely randomizing the pairing of hydropower station nodes with grid generation nodes. This strategy serves as a baseline scenario to measure the impact bias caused by other specific coupling strategies.

[0512] Based on matching of generating capacity: First, hydropower station nodes are ranked from highest to lowest drought severity, and then grid generating nodes are ranked from highest to lowest maximum generating capacity. Then, the hydropower station node most severely affected by drought is matched with the generating node with the highest maximum output. This strategy aims to simulate a "high vulnerability-high load" correlation, where critical (high-output) generating units rely on water sources that may face severe water shortage risks. This configuration could lead to direct damage to large-capacity generating units during droughts, resulting in significant power supply gaps, increasing the compensation pressure on other units, and potentially inducing cascading failures. This helps assess the network's resilience under extreme adverse conditions.

[0513] Degree-centrality-based matching: First, hydropower station nodes are sorted from highest to lowest drought severity. However, power grid generating nodes are then matched based on their degree centrality within the grid topology. Nodes with high degree centrality are typically key nodes in the network. Under this strategy, the hydropower stations most severely affected by drought will be associated with critical key generators in the grid. Therefore, the failure of critical water sources can directly impact the core structure of the power grid, potentially triggering wider and deeper cascading failures. This also helps in assessing network resilience under extreme adverse conditions.

[0514] Heterogeneous coupling based on power generation: Hydropower station nodes are sorted from highest to lowest drought severity, but grid power generation nodes are sorted from lowest to highest (i.e., in reverse order) before matching. This forms an asymmetric matching of "high vulnerability - low load", that is, hydropower stations with high drought risk are associated with power generation nodes with lower output, while hydropower stations with low risk are associated with power generation nodes with higher output.

[0515] Heterogeneous coupling based on degree centrality: Hydropower station nodes are sorted from highest to lowest drought severity, and power generation nodes in the power grid are sorted from lowest to highest degree centrality (in reverse order) before matching. This also forms a "strong-weak" asymmetric matching, that is, hydropower stations with high drought risk are associated with power generation nodes with lower importance (low degree centrality) in the power grid.

[0516] By comparing and analyzing the grid response under the five coupling strategies described above, we can reveal the influence of the coupling structure on the resilience of the hydro-electric coupled network under drought disturbances and identify coupling configurations that are more sensitive to drought disturbances. To ensure the statistical robustness of the results, 2000 Monte Carlo simulations were performed for each coupling strategy, and the resilience of the grid under drought impacts under different coupling strategies was evaluated from two dimensions: structural resilience and functional resilience.

[0517] Power grid structural toughness analysis

[0518] Experimental results show that, under the simulated 2022 drought scenario, regardless of the coupling strategy employed, the power grid did not experience large-scale continuous line tripping (cascading faults) leading to network collapse (split into multiple islands). The faults primarily manifested as the passive disconnection of small-scale load nodes due to triggered load shedding strategies. This preliminarily demonstrates that, under the drought impact intensity and power grid parameters set in this study, the overall structure of the power grid exhibits good resilience.

[0519] Since no large-scale network collapse occurred, we primarily used the relative magnitude of the largest connected component in the power grid structural resilience analysis. To quantitatively assess structural changes.

[0520] As shown in the figure, there are significant differences in the degree of topological damage among different strategies. Under the power-coordinated coupling strategy, The minimum value is the lowest, approximately 0.975; followed by the degree centrality isomorphic coupling strategy. The minimum value is approximately 0.992. In comparison, under both random coupling and the two heterogeneous coupling strategies, the power grid structure remains basically stable. The value is close to 1, indicating that network fragmentation has not occurred. This suggests that co-location, especially capacity-based co-location, poses a greater potential threat to the integrity of the power grid structure.

[0521] From the perspective of the dynamic changes in structural toughness, The timing of the onset of decline and the arrival of the lowest point also vary depending on the strategy. The structural effects of the power-coordinated coupling strategy appear earliest, beginning a gradual decline around the end of July, at the beginning of the drought.

[0522] And it reaches its minimum during the peak of the drought (late August to early September). It is worth noting that... The probability distribution of the minimum occurrence time exhibits a multi-peak pattern, with peaks occurring in late July, late August, and early September, with the highest probability in early September. The impact of degree centrality iso-matching strategies on the structure appears relatively late. It only began to decline at the end of August, and reached its lowest point in mid-to-late September; The distribution of the time of occurrence of the minimum value is approximately normal between July 20 and October 20, with the peak concentrated in early September.

[0523] Power grid functional resilience analysis

[0524] Experimental results show that drought significantly reduces the function of the power grid, leading to a significant power supply gap, and the functional resilience performance varies significantly under different coupling strategies.

[0525] like Figure 7As shown, power matching coupling The most significant decline occurred in mid-July, with a rapid rate of decrease between July 1st and September 9th, reaching its minimum around September 9th, at approximately [value missing]. Subsequently, it gradually recovered from late September to early October, eventually returning to around 0.95. The second factor is the degree-centrality isomorphic coupling strategy. The significant decline began at the end of July, and also reached its minimum around September 9th, at approximately... After that, it began to recover from October, basically returning to a level close to 1. The functional resilience of stochastic coupling was also somewhat affected. The minimum value is approximately Under the two heterogeneous coupling strategies, There was almost no significant decline, demonstrating strong functional resilience.

[0526] right Analysis of the distribution of the minimum occurrence times reveals that both capacity matching and degree centrality matching strategies exhibit a significant bimodal distribution. The first peak occurs in early September, coinciding with the peak of the drought; the second peak occurs at the end of September, reflecting the cumulative effect of the drought. The random matching strategy... The timing of the minimum value shows an approximately normal distribution, with the peak occurring in mid-to-late September.

[0527] Resilience assessment of water-electric coupled networks under drought risk

[0528] Based on the seepage analysis method described above, taking the 2022 Yangtze River Basin summer-autumn drought as an example, this study delves into the resilience evolution of hydropower networks under real drought risks, focusing on two key time sections: (July 30, 24:00, early stage of drought) and (August 30, 24:00, peak of drought) The resilience of network structure and function is analyzed.

[0529] Dynamic evolution of network bottlenecks

[0530] With seepage parameters As the network size increases, the network at each time point exhibits a decrease in the size of the largest connected component (GC) and an increase in the size of the second largest connected component (SC). This is due to network fragmentation caused by the breaking of key bottlenecks in the network. Therefore, by analyzing the evolution of the network topology during the seepage process, structural bottlenecks in the network at different stages can be identified.

[0531] like Figure 8 a and Figure 9 As shown, in the early stages of drought, when Increase to the seepage threshold At that time, the first point of failure was the Dongfeng Reservoir on the upper reaches of the Wujiang River; when When the value increased to 0.57, the Wujiangdu Reservoir downstream of the Dongfeng Reservoir and the Ahai Reservoir upstream of the Jinsha River successively became ineffective; with With further increases, more nodes have a storage capacity ratio lower than [a certain value]. The network structure began to break down, and most of these nodes were concentrated in the Wujiang River basin. This was consistent with the fact that the Wujiang River basin had experienced a "reverse drought during the flood season" and severely low water levels as early as June, indicating that the network's structural vulnerability was mainly concentrated in the upper reaches of the Wujiang and Jinsha Rivers at this time.

[0532] After the drought peaks in August (such as...) Figure 8 b and Figure 10 (As shown), the network's percolation threshold The failure rate dropped from 0.48 to 0.10, with the first node to fail shifting to the Shenxigou Hydropower Station in the Dadu River basin; when When the flow rate increased to 0.35, upstream nodes such as the Luding node also failed; in contrast, in the seepage analysis in July, the nodes in the Dadu River basin... The failure occurred only after a period of time. This indicates that as the drought continued and spread, its impact expanded, and even the previously relatively stable Dadu River basin faced immense pressure, becoming a new structural bottleneck. This phenomenon of network structural bottlenecks dynamically shifting with the spatiotemporal evolution of drought is key to characterizing the risk of cascading failures in large infrastructure networks.

[0533] Asynchronousness and decoupling of structural toughness and functional toughness

[0534] Figure 11 This study demonstrates the different behaviors of structural and functional seepage in a water network during the 2022 drought event, revealing the asynchronicity and decoupling between structural and functional resilience. Simulation data shows the network seepage threshold on July 30 (the early stage of the drought). Corresponding structural indicators The mutation; however, when The seepage threshold has been reached. hour, The overall network functional resilience index remains as high as 0.9995, meaning that although the network structure has begun to break down, its power generation function is almost unaffected, with over 99% of the power generation capacity remaining unaffected. This indicates that the network's functional resilience significantly lags behind the structural phase transition. Entering the peak of the drought in August, although... A drop to 0.1 reflects increased structural fragility, but The value is as high as 0.998, and the overall network function remains basically normal.

[0535] This illustrates that the traditional approach based on structural seepage thresholds... The existing network resilience assessment methods have limitations and may severely underestimate the actual carrying capacity of the system when faced with node function degradation. Therefore, it is necessary to introduce a functional percolation index. To more comprehensively assess the network's resilience. The hydropower network exhibits strong functional resilience; even if the reservoir-to-capacity ratio of some nodes decreases or even falls below the structural seepage threshold, as long as most nodes with high reservoir-to-capacity ratios and large installed capacities remain operational, the overall power generation capacity of the network can be maintained at a high level. Overall functional resilience indicators Daily variations during drought ( Figure 12 This also confirms this point, although It began to decline at the end of June, reached its lowest point at the end of September, and then gradually rebounded in October, reflecting the erosion of overall functional resilience by drought. However, its absolute value remains high, and the decline is much smaller than that of other drought-affected systems. The magnitude of the change is consistent with the conclusions of the demand service network resilience analysis in the literature, namely that the functional loss of the network often lags behind its structural disintegration.

[0536] Unlike water networks, power grids exhibit the opposite characteristics under drought conditions: their structural resilience is relatively strong, while their functional resilience is more fragile. For example... Figure 13 As shown, even under the most unfavorable power matching coupling strategy, the relative size of the maximum connected components of the power grid is... The minimum value remains around 0.975, indicating that the power grid structure has not experienced a large-scale collapse. However, as... Figure 7 As shown, the corresponding functional resilience index The power density decreased significantly, dropping to approximately 0.65 and 0.80 under the power matching and degree centrality matching strategies, respectively. Meanwhile, the functional degradation and structural damage of the power grid exhibited different characteristics at different stages of the drought. In the early stages of the drought, the functional degradation of the power grid was mainly caused directly by the reduction in hydropower output, while structural damage occurred relatively later. For example, under the power matching strategy, It started to decline in mid-July, and The decline began at the end of July.

[0537] However, as the drought intensifies, in addition to reduced output from hydropower stations, load nodes in the power grid may be passively disconnected due to load shedding strategies, further exacerbating the grid's functional degradation. For example... Figure 14 As shown, the functional resilience of the power grid The minimum value occurred later than the structural toughness (around September 9th). (Power matching: late August to early September). This may indicate that although the initial functional degradation (directly caused by reduced hydropower output) occurs relatively early, the most severe functional losses only appear after line overloads and cascading load shedding faults occur in the power grid, reflecting the complex dynamic process of power grid operation.

[0538] Network resilience evolution under drought risk

[0539] By conducting simulations over the entire period from June to October 2022 and performing seepage analysis at each time step, we obtained the seepage threshold. and functional resilience indicators Time series ( Figure 12 Furthermore, the monthly statistical characteristics of the indicators were calculated (Table 4.3 and...). Figure 15 This depicts the dynamic evolution of the resilience of hydropower station networks as drought progresses.

[0540] Table 4.3 Monthly Statistics on Seepage Threshold and Functional Resilience of Hydropower Networks

[0541]

[0542] Resilience deterioration phase (June-September)

[0543] seepage threshold The size can be considered a measure of network structural resilience: higher... This means the network can maintain its overall structure when most nodes are functioning well (high storage capacity); conversely, a lower capacity means... This indicates that the network is very fragile, and even removing only a few nodes with extremely poor functional status (very low capacity ratio) can cause the network structure to collapse.

[0544] As can be seen from Table 4.3, The monthly average value continued to decline from 0.773 in June to 0.611 in September (a decrease of 21.0%). Only decreased during the same period This data indicates that the network structure resilience is more sensitive to drought stress, and the functional redundancy mechanism of hydropower station networks has a significant buffering effect on structural damage caused by extreme drought.

[0545] For the power grid, its functional resilience This stage also shows a clear downward trend, especially under the same coupling strategy. Figure 7 It is worth noting that the decline in the functional resilience of the power grid exhibits a significant lag. (The text then abruptly shifts to discussing the functional resilience indicators of hydropower networks.) It began to decline from the end of June to the beginning of July. Figure 12 ), while the functional resilience index of the power grid Under the same power distribution strategy, the drought began to decline in mid-July, while under other coupling strategies (same degree distribution, random distribution, and different distribution), a significant decline did not appear until the end of July. This lag indicates that the transmission of drought risk from the hydropower network to the power grid is not instantaneous. The hydropower network system buffers the impact of the early drought to some extent. Only when the drought continues to intensify and hydropower output declines significantly does its impact fully transfer to the power grid, leading to a significant decrease in the grid's functional resilience.

[0546] Resilience recovery phase (September-October)

[0547] Starting from the end of September, the drought intensity eased, and the seepage threshold... and functional resilience indicators And it began to recover as well. Despite It recovered significantly to 0.978 (+1.03%), but The October monthly average only recovered to 0.614 (up 0.49% from September). This asymmetric recovery is due to the fact that the recovery of the network structure is constrained by the drought conditions of network bottlenecks (such as the Dongfeng Reservoir and Wujiangdu Reservoir). The Wujiang River basin, where these reservoirs and hydropower stations are located, remained drought-stricken in October, with precipitation only... Even lower than September (rainfall) (less than the multi-year average) The drought was not effectively alleviated until significant rainfall occurred at the end of October.

[0548] In contrast to the diurnal resilience fluctuations exhibited in transportation networks, hydropower station networks display unique monthly-scale gradual variation characteristics. Figure 15 This difference stems from the fundamental dynamic distinction between the two types of systems: hydropower networks are dominated by climate influences, and their resilience changes synchronously with drought processes; while transportation networks are subject to human scheduling and regulation, and experience rapid fluctuations under short-term pressures such as morning and evening rush hours.

[0549] Power grid functional resilience This stage also showed a clear recovery trend. Figure 7 Unlike the hysteresis of the attenuation phase, the recovery of power grid functional resilience is similar to that of water network functional resilience. The recovery of both hydropower and the power grid showed good consistency in time, with both gradually recovering from late September to early October. This indicates that when upstream hydropower output begins to recover, its positive effects can be quickly transmitted to the power grid, promoting the recovery of the grid's power supply capacity. The synchronicity shown by the hydropower network and the power grid during the recovery phase reflects the synergistic effect of the hydro-electric coupling system in mitigating the impact of external disturbances and restoring normal operation.

[0550] The study found that: (1) Improved seepage analysis can effectively identify the structural bottlenecks and dynamic transfer characteristics of water networks at different stages of drought; (2) Structural resilience and functional resilience exhibit significant asynchrony and decoupling. For water networks, functional loss usually lags behind structural damage, indicating that relying solely on structural indicators may underestimate system resilience and functional seepage indicators need to be introduced for comprehensive evaluation. In the power grid, functional loss and structural damage have different characteristics at different stages of drought; (3) Water network resilience exhibits a gradual monthly scale during drought, and its attenuation and recovery process can be tracked by the dynamic changes of structural seepage threshold and functional resilience indicators; (4) Water-electricity coupling strategy has a significant impact on power grid resilience. Coupling with the same type (especially based on power generation) will amplify the negative impact of drought, leading to more severe structural damage and functional loss, while coupling with different types shows stronger robustness.

[0551] In summary, this invention not only deepens our understanding of the complex response mechanisms of water-electricity coupled systems under extreme drought events, but also proposes a resilience assessment framework based on seepage theory and reveals the patterns of different coupling effects in water-electricity systems. This can provide a scientific basis for risk assessment, resilience enhancement, and emergency management strategy development for critical infrastructure.

[0552] The above are merely preferred embodiments of the present invention, but the scope of protection of the present invention is not limited thereto. Any equivalent substitutions or modifications made by those skilled in the art within the scope of the technology disclosed in the present invention, based on the technical solution and inventive concept of the present invention, should be covered within the scope of protection of the present invention.

Claims

1. A method for assessing the resilience of a water-electricity coupled system considering the evolution of drought risk, characterized in that, Includes the following steps: S1. Network modeling of the water-electricity coupled system; S2. Drought risk propagation modeling based on seepage-induced water-electricity coupled networks; S3. Based on the above two models, system simulation and resilience assessment of the water-electricity coupled system network are performed. In step S2, the drought risk propagation modeling process based on the water-electricity coupling network of seepage is as follows: S21. First, establish seepage models within the water network and the power grid respectively: the water network model focuses on the state evolution based on water balance and control strategies; the power grid model focuses on link failures and dynamic power rebalancing based on overload. The seepage model of the water network should take into account the time-delay propagation mechanism of hydropower station power generation capacity degradation. Based on the principle of watershed hydrology balance, the reservoir of the hydropower station can be modeled as a storage unit with a certain capacity and daily inflow to describe the water volume change of the reservoir. Water balance equation for upstream hydropower station: ； Water balance equation for downstream hydropower station: ； It represents the set of nodes of the upstream hydropower station; It represents the set of nodes of the downstream hydropower station; and These represent the water volume of hydropower station i at time t and t+1, respectively; It is the natural inflow of water into the reservoir of a hydropower station at time t; It is the amount of water used by a hydroelectric power station to generate electricity at time t; It is the amount of water discharged by the hydropower station at time t; It is the set of directly upstream nodes affecting hydropower station j. ; It is the water flow lag time from the upstream hydropower station to the downstream hydropower station j, where , ; Multi-state model of hydropower station nodes: Define nodes The storage capacity ratio is A five-level drought classification system was established based on the reservoir capacity ratio, with the drought level of node i as the reference. Classification criteria: ； in, Represents the maximum reservoir capacity of node i in the hydropower station. This represents the drought level of node i at time t; based on the drought level defined above according to the reservoir capacity ratio, a multi-state model of the hydropower station node is established; for the hydropower station node... The drought level state space is Let represent the drought level state set of hydropower station node i; at time t, the state of hydropower station node i is... From the state space Take the value from; Water volume regulation strategy for water network nodes: power generation water volume Based on the current drought level The proportional adjustment reflects the practical strategy of reducing power generation to conserve water resources under drought conditions; the adjustment rules are as follows: ; This represents the baseline water generation capacity of hydropower station i at time t; under normal, drought-free conditions... hour, This can be considered as the maximum power generation capacity during normal operation. , This is a proportionality coefficient determined based on the drought level; The overall state update of the system follows the discrete-time recursive principle, with the water network state at input time t. Obtain the system state at time Δt. Its dynamic equation is: ； In the formula The state transition function for the water network has the following input parameters: Network Topology : Describes the set of reservoir nodes Connecting edge to water flow and corresponding weights , Water volume operation status ; Node status ; The output state variables are defined as follows: ； ； Seepage process in power grid: In power grid model Among them Let E be the set of nodes and E be the set of links. Assume that each node in the power grid is either a generator node, denoted as... It is either a demand node, represented as ; and define , , ; S22. By establishing a mapping relationship between the state of water network nodes and the state of power grid nodes, the risk transmission process between layers is simulated, and finally the seepage dynamics equation of the coupled network is formed, realizing the cascading failure propagation model driven by physical mechanism. By coupling the hydrodynamic processes in the water network with the power flow-based fault propagation in the power grid, the seepage equation of the water-electric coupled two-layer dependent network is obtained: The seepage equation of the network: ； The simulation continues until the following termination condition is met, indicating that the system state has reached steady state: 。 2. The method for assessing the resilience of a water-electricity coupled system considering drought risk evolution according to claim 1, characterized in that, In step S1, the process of establishing the network model of the water-electricity coupled system includes the following steps: S11. Characterizing the key elements of the water-electricity coupling system: Representing the water network model as follows ,in It is the set of nodes in a water network. For the edge set in the water network, Let be the edge weights of the nodes in the water network; S12. Reservoir and hydropower station network modeling, including the following steps: S121. Node dataset construction and watershed topology association; S122. Construct an edge generation mechanism based on watershed topology; Based on the two major geospatial datasets HydroBASINS and HydroRIVERS, a quantitative characterization of the hydraulic connections between hydropower stations is achieved through the spatial correlation of multi-scale watershed units and river networks. S123. Establish a side weight calculation model that simultaneously considers the watershed isomorphism and dynamic time-delay coupling characteristics; Watershed isomorphism coefficient Representation from node arrive The hydrological connectivity cost is based on the watershed hierarchy in the HydroBASINS dataset and the spatial association results between hydropower stations and sub-watersheds obtained by the algorithm, according to the hydropower station and The watershed matching situation is used to define the watershed isomorphism coefficient. The classification table; known For the edge A list of HYRIV_IDs for the main rivers; a river transport time delay model is constructed based on hydrodynamic principles to calculate the key parameters of each river, including river meander, average slope, hydraulic radius, flow velocity, and transport time delay; S13. Power Network Modeling: The IEEE-118 node system is selected as the power network. A DC power flow model is used to model the steady-state and dynamic behavior of the power system; S14, Water-Electricity Coupled Network Modeling The water-electricity coupled network consists of a water network and power grid Composition, power grid Nodes representing generator sets, substations, and / or power loads The system consists of links between nodes via transmission lines and transformer branches. Connections; Reservoir and hydropower station networks Hydropower station nodes with upstream and downstream relationships Composed of nodes connected by rivers or waterways The water flow along the water network is controlled by the reservoir's scheduling, and the power grid... WaterNet They are connected by a hydropower conversion link, forming an interdependent two-layer coupled network; Hydropower Conversion Link The energy conversion process of hydropower is defined as the core channel for transmitting electrical energy from the water network to the power grid, starting at the water network nodes and ending at the power generation nodes of the power grid; the power generation capacity of hydropower is expressed as a function of the power generation flow, the head of the hydropower station, and the output coefficient of the hydropower station. ， in, Let g be the density of water, g be the acceleration due to gravity, Q be the power generation flow rate, and H be the effective head height. To determine the overall hydropower efficiency coefficient, the head height H has dynamic characteristics and is calculated from the difference between the upstream and downstream water levels of the hydropower station. Based on complex network theory, the water-electricity coupled system is constructed as a two-layer dependent heterogeneous network. : 。 3. The method for assessing the resilience of a water-electricity coupled system considering drought risk evolution according to claim 2, characterized in that, Step S121 includes the following steps: S1211, Define the node attributes of the hydropower station: This system uses the Python Requests module to call the Google Elevation API for batch data collection, obtaining the altitude information of hydropower stations and enabling 3D spatial positioning of hydropower station nodes. When the user-requested coordinates do not have a direct and accurate elevation measurement, the Google Elevation API estimates the altitude using a bilinear interpolation algorithm, calculating the average from the four nearest known points to ensure continuous elevation data for any location. The core parameters of the API call include the locations and key parameters; the locations parameter is the latitude and longitude coordinates, and the key parameter is the API key used for authentication and quota control. The JSON data returned by the request contains the elevation information of the requested location. S1212. Constructing the mapping relationship between hydropower stations and river basins: Based on the HydroBASINS dataset, hydropower station nodes are associated with watershed topology to achieve spatial association between hydropower stations and watersheds.

4. The method for assessing the resilience of a water-electricity coupled system considering drought risk evolution according to claim 3, characterized in that, Achieving spatial correlation between hydropower stations and sub-basins mainly involves the following three steps: S12121, Coordinate Transformation: Convert the WGS84 latitude and longitude coordinates of the hydropower station to ESRI:54009 world cylindrical projection to eliminate spherical distance distortion; S12122, Spatial Query: Use the contains() method of the Shapely library to perform point-to-face containment detection; S12123, Hierarchical Matching: Prioritize searching in Pfafstetter level 5-7 sub-basins, and expand to adjacent levels if a match fails.

5. The method for assessing the resilience of a water-electricity coupled system considering drought risk evolution according to claim 3, characterized in that, Step S122 includes the following steps: S1221. Watershed topology path search: Based on the determined hydropower station-watershed mapping relationship, and the hierarchical watershed division system of HydroBASINS, establish the topological connection relationship between watershed units. S1222, Main River Path Extraction; The HydroRIVERS dataset is used to map watershed paths to physical rivers; for confirmed connected paths... Extract the path of the main river : ， In the formula, ℝ represents the set of Hydrorivers. To filter the river and ensure that only the main channel is selected; S1223, Rules for constructing topological edges The final two hydroelectric power station nodes Connecting edges The generation must satisfy two constraints: 1) Hydraulic connectivity: There exists a collection of non-empty main rivers. ; 2) Topological Exclusivity: There are no intermediate hydropower station nodes in path p, that is: ， When the above conditions are met, from node arrive directed edges And record its attribute tuple: ,in For the edge A list of HYRIV_IDs for the main rivers.

6. The method for assessing the resilience of a water-electricity coupled system considering drought risk evolution according to claim 5, characterized in that, For any two hydropower station nodes The spatial correlation is determined through the following process: S12211, Locating the watershed unit: Through spatial point inclusion analysis, determine the HYBAS_ID of the watershed to which each hydropower station belongs, denoted as... , ; S12212, Tracing downstream paths: From Starting from a watershed, move to its downstream watershed based on the NEXT_DOWN field in that watershed. Construct potential connection paths ; S12213. Determine connectivity: If the path contains the target watershed... Then determine for The upstream node; if no upstream node is found when traversing to the end basin. If so, it is determined that there is no direct hydraulic connection.

7. The method for assessing the resilience of a water-electricity coupled system considering drought risk evolution according to claim 1, characterized in that, The process of updating the power grid operation status under the influence of water networks includes the following steps: Step 1: Adjust the ramp rate of the restricted nodes; Step 2: Recalculate the power output boundary of the restricted node; Step 3: Remove the failed node; Step 4: Update the power grid operating status.

8. The method for assessing the resilience of a water-electricity coupled system considering drought risk evolution according to claim 1, characterized in that, Step S3 includes the resilience index system of water-electricity coupled network and the resilience assessment of water network based on seepage theory; The resilience index system of hydro-electric coupling network includes topological resilience index and functional resilience index. The functional resilience index includes power grid functional resilience index and water network functional resilience index. Topology resilience index: The relative size of the largest connected component GC and the second largest connected component SC is used as the topology resilience index. Relative size of the largest connected component GC Defined as: ; in, It is the number of nodes contained in the largest connected component of the network. This represents the total number of nodes in the original network; the evaluation is conducted before and after the perturbation. The change in can quantify the degree of loss in the network topology; The relative size of the second largest connected component Defined as: ； in, It is the number of nodes contained in the second largest connected component of the network. The magnitude of the change reveals the degree of network fragmentation and structural fragility; Power grid resilience index: assesses how much power demand the system can still meet after a cascading failure. Defined as the proportion of total electricity demand that can still be met after a cascading failure, it reflects the service maintenance capacity of the power system. ； in It is the set of all nodes with electricity demand in the initial network. It is a node electricity demand, It is the set of nodes with remaining power demand after a cascading failure process triggered by an attack. It is the remaining demand node Power requirements; Water network functional resilience index: assesses the level of hydropower generation capacity maintained by the water network during cascading failures. Defined as the proportion of water that can still be used for power generation under disturbance to the total water volume used for power generation, reflecting the hydropower potential. The resource retention capability of an electrical system, namely: ； in This represents the set of all hydropower station nodes in the initial water network. For nodes The amount of water used for power generation This refers to the set of remaining hydropower station nodes that continue to operate normally despite the disturbance. For the remaining nodes The actual available water volume for power generation; Water network resilience assessment based on seepage theory includes analytical methods based on improved seepage theory, water network resilience analysis based on structural seepage, and water network resilience analysis based on functional seepage. An analysis method based on improved seepage theory: For reservoir-hydropower station nodes, the reservoir capacity ratio is a key indicator for measuring their current water storage status and potential power generation capacity. The selection of the reservoir capacity ratio for hydropower station nodes is crucial. As its "quality" indicator at time step t; each seepage analysis is performed at a specific time snapshot, in which the state of each node remains unchanged during the analysis, that is, at each snapshot t, the reservoir capacity ratio of the node. The parameters remain fixed; specifically, based on a specific time snapshot in the current system dynamics simulation process, for any time step t of the network, Algorithm 7 is executed, and the seepage control parameters are adjusted accordingly. The specific process of regulating network topology evolution is as follows: Node quality characterization: The reservoir capacity ratio of node i in the hydropower station. Mapped to node quality metrics, when A node is considered to be functioning normally if it is in a normal state, otherwise it is considered to be a node with impaired function. Dynamic adjustment of connecting edges: Remove edges associated with failed nodes; Progressive failure simulation: by Starting from 0 and gradually increasing to 1, the system studies the response process of the network under stress scenarios such as persistent drought. Each... The value corresponds to a percolation subnetwork. By tracking the phase transition patterns of indicators such as the size of its maximum connected component, the resilience of the system can be quantitatively assessed. Water network resilience analysis based on structural seepage: Structural resilience focuses on the changes in the integrity of the network topology during node removal. In seepage analysis, the node topology resilience index is used. By tracking the changes in the relative size of the largest and second largest connected components with the seepage parameter ρ, the resilience of the network topology is systematically evaluated. When control parameters Reaching a certain critical value At this point, the macroscopic properties of the system will change drastically, reaching the critical seepage threshold. Defined as the second largest connected component size When it reaches its peak Value; in At that time, there exists a giant connected component in the network; when achieve At that time, the giant component began to disintegrate, leading to Increase; when At that time, the network further fragmented. Consequently, it decreases; when analyzing the phase transition in the network seepage process, the method of analyzing phase transition in statistical physics is used as a reference, that is, the order parameter is examined. For seepage parameters Derivative: First derivative Second derivative ; Water network resilience analysis based on functional seepage: Based on the functional resilience index of the water network, the functional seepage index of the water network is defined as the index under a given seepage threshold. Below, all "surviving" nodes are At the current time step Total hydropower generation ; ； in It is the set of all nodes in the water network. It is a node At time step The actual hydropower generation; comparing the resilience of different time steps or different networks, using relative function indicators, i.e., the current total hydropower generation versus the initial seepage flow. The ratio of total hourly hydropower generation: ； The range of values ​​for is This indicates that when removing storage capacity ratios below a certain level... After the node, the remaining power generation capacity of the network accounts for the proportion of the original total power generation capacity; Follow The changing curves illustrate the process by which the function of the hydropower network decreases as the intensity of disturbance increases; When the demand in the network meets the homogeneous condition, use The area under the curve is used as the current time step. The overall resilience index of the network is denoted as... : ； It can comprehensively reflect the network's functional maintenance level at different times. The larger the value, the better. The more robust the network snapshot at a given time step is to the overall power generation capacity under drought disturbances, the stronger the resilience of the nodes; this is achieved by calculating different time steps. of To track the evolution of the resilience of hydropower networks over time.

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