A time-delay closed-loop coupled power cyber-physical system cascading failure evaluation method

Abstract

The application discloses a time-delay closed-loop coupled power cyber-physical system cascade failure evaluation method, which comprises the following steps: firstly, a double-layer coupled model of the information physical power system is constructed, wherein the power network is used as the physical layer and the communication network is used as the information layer; then, a time-delay endogenous closed-loop cascade failure model is established based on the double-layer coupled model, and the system is iterated and simulated round by round by using the time-delay endogenous closed-loop cascade failure model, and the communication time delay is embedded into the time-delay endogenous closed-loop cascade failure model in each iteration; finally, the robustness is evaluated according to the supply conservation ratio LSR and the dispatching fidelity error DTE, and the sensitivity of the tolerance margin, the time-delay severity and the respective components of the uplink measurement channel time delay and the downlink execution channel time delay is analyzed. The method can endogenize the communication time delay in the cascade closed-loop process, distinguish the measurement and execution double-channel time delay effects and provide the unit-level evaluation granularity.

Description

Technical Field

[0001] This invention relates to the field of robustness assessment technology for cyber-physical power systems, specifically a method for assessing cascading failures in cyber-physical power systems with a time-delay-endogenous closed loop. Background Technology

[0002] Modern power systems achieve wide-area monitoring and control through tight coupling with communication infrastructure: field sensors transmit measurement data to the control center, and dispatch decisions are then sent down to generators and circuit breakers. The physical layer and information layer together constitute the cyber-physical power system.

[0003] The reliability of a Cyber-Physical Power System (CPPS) depends not only on the accuracy of the information but also on its timeliness. When disturbances remove communication assets, the survival path becomes longer, congestion worsens, measurements reaching the control center become outdated, and commands sent to generators are delayed. The system faces the risk of cascading failures induced or exacerbated by communication delays.

[0004] Currently, scholars both domestically and internationally have conducted some research on cascade failures in CPPS (Concurrent Power Supply System). For example, existing literature typically constructs a two-layer coupled model of power and communication networks, simulating cross-layer fault propagation by removing faulty nodes or links, and using tools such as DC power flow to calculate branch overload conditions, evaluating system resilience using indicators such as load loss or power supply ratio. Regarding the handling of communication delay, existing research generally adopts an observational approach: that is, calculating the end-to-end delay on a fixed topology, giving it as an average delay or worst-case delay, and then adding it as a scalar penalty to a cascade trajectory predetermined by a purely structural or purely electrical process.

[0005] However, the aforementioned existing technologies have the following shortcomings:

[0006] First, in existing models, regardless of whether the communication system experiences delay degradation, the fault sequence experienced by the system is the same as that of an ideal communication system; the only difference is the application of a scalar penalty value at the end. This approach removes the key link in the delay-causing harm. The harm of delay is essentially generated by a closed-loop feedback process: communication degradation reduces generator controllability, the resulting power imbalance triggers new branch overloads, new structural faults further remove communication assets, and the damaged topology lengthens the delay in the next round. Simplifying this closed loop into an open-loop penalty not only sacrifices assessment accuracy but also incorrectly characterizes the qualitative nature of the delay impact.

[0007] Second, most models compress the entire communication path into a single delay value and a single penalty curve. However, from a control and scheduling perspective, the truly important quantity is not a single delay, but rather the delays of two different channels: the uplink delay of the measurement reaching the decision-maker, and the downlink delay of the decision command reaching the executor. These two channels carry different message sizes—measurement messages are larger, command messages are smaller—and undergo different queuing mechanisms and processing hardware before entering the generator control loop through different mechanisms. An indiscriminate single-delay model cannot explain why a specific delay is harmful, nor can it guide which physical interface should be upgraded to achieve maximum resilience.

[0008] Third, existing assessment indicators mainly focus on load retention rate, but this indicator cannot reflect the extent to which the actual output of surviving units deviates from the dispatch plan. It may mask hidden degradation at the unit level. The system may be supplying load from units far from the set point when the load indicators appear safe, leading to overly optimistic assessment results.

[0009] In summary, existing methods for cascading failure modeling and robustness assessment of cyber-physical power systems urgently need a technical solution that can endogenize communication delay within the cascading closed-loop process, distinguish between the delay effects of measurement and execution channels, and provide unit-level assessment granularity. Summary of the Invention

[0010] The purpose of this invention is to overcome the problems of systematic optimism in delay-free models and the inability to attribute single delay penalties, and to propose a delay-endogenous closed-loop method for assessing cascaded failures in cyber-physical power systems.

[0011] To achieve the above objectives, the technical solution specifically adopted by the present invention is as follows:

[0012] A time-delay-endogenous closed-loop cyber-physical power system cascading failure assessment method includes the following steps:

[0013] Step 1: Construct a two-layer coupling model of the cyber-physical power system: using the power network as the physical layer and the communication network as the information layer, determine the control center node in the communication network, and couple the physical layer nodes and information layer nodes one by one in a positive betweenness correlation manner; set the capacity of physical layer branches (the physical layer branches are not physical layer nodes, but transmission lines connecting two bus nodes in the power network, which are directly determined by the topology data of the power network) as well as information layer nodes and information layer links according to the same tolerance margin.

[0014] Cascading failures in cyber-physical power systems involve both physical power flow and information interaction. Electrical measurements at the physical layer must be relayed to the control center via nodes in the information layer, and dispatch commands from the control center must also be sent to various actuators via the information layer. Establishing a two-layer coupling model is the basic physical architecture for delay assessment. The physical and information layers are coupled in a positive betweenness correlation manner. This aims to match the most critical nodes in the electrical structure of the physical layer to the most central nodes in the communication structure of the information layer, so that the coupling relationship reflects the correspondence in structural importance between the two types of networks. All components are configured with the same tolerance margin, providing a unified comparable benchmark for evaluating system robustness at different margin levels. When α=0, the system has no redundancy; the larger α is, the stronger the system's tolerance to single faults.

[0015] Step 2: Based on the two-layer coupling model, establish a time delay endogenous closed-loop cascade failure model, and perform iterative simulation of the system using the time delay endogenous closed-loop cascade failure model. In each iteration, the communication delay is embedded into the time delay endogenous closed-loop cascade failure model.

[0016] In each iteration, the uplink measurement channel delay and downlink execution channel delay are recalculated for each surviving generator on the current damaged communication network topology. The calculated communication delay is mapped to the scaling factor of the planned output of the generator through the control efficiency factor on the one hand, and used to determine the load reduction depth of low-frequency load reduction on the other hand, so that the communication delay becomes an endogenous state quantity that evolves with the cascade process and reacts on the cascade process.

[0017] The impact of communication latency on power systems is not a static, one-way scalar penalty. Damaged communication links lead to longer paths, increased congestion, and reduced timeliness of measurements and commands. Generators, unable to respond to dispatch instructions in a timely manner, experience a decrease in actual controllable output. This output reduction causes power redistribution at the physical layer and may trigger new branch overloads. Overload-induced circuit breaker actions further remove communication assets, and the damaged topology further exacerbates communication latency in the next round. This invention places latency within a cascading iterative inner loop, recalculating it on the current topology in each round and using the actual output after latency degradation as the input for the next power flow, thus comprehensively characterizing the causal closed loop of latency, output degradation, power flow redistribution, topology change, and latency deterioration. This mechanism allows each latency scenario to evolve along its own distinct trajectory, rather than a scaled copy of a common, latency-free trajectory, fundamentally different from existing methods that treat latency as a post-hoc annotation.

[0018] Step 3: Evaluate robustness based on the supply conservation ratio (LSR) and scheduling fidelity error (DTE), and perform sensitivity analysis on the tolerance margin, delay severity, and the components of uplink measurement channel delay and downlink execution channel delay.

[0019] A single delay value cannot distinguish the different harmful mechanisms of information uplink lag and control downlink lag, and a single load preservation index cannot reflect the fidelity of surviving units executing the scheduling plan. This invention measures the proportion of load that the system can still effectively supply after cascading termination by using the supply conservation ratio, and measures the degree of deviation of the actual output of surviving units from the capacity-weighted scheduling plan by using the scheduling fidelity error. The two are complementary: the former answers "how much load survives," and the latter answers "at what level of fidelity do surviving units operate." Sensitivity analysis, by applying single-parameter mitigation actions to each delay component under the most severe delay scenario, identifies the physical interface that contributes most to the robustness of the recovery system, providing an actionable optimization direction for improving engineering resilience.

[0020] Preferably, in step 1, the communication network is a scale-free network; the control center node is selected in the following manner: calculate the structural importance score of each node, which is equal to the sum of the node's normalized betweenness number and the node's kernel number divided by the node's clustering coefficient, plus the product of the node's degree and the node's proximity centrality; select the node with the highest structural importance score as the control center node.

[0021] Scale-free networks, with their power-law degree distribution, can reflect the topological characteristics of a few central nodes connecting a large number of edge nodes in real-world communication networks, making them a common choice for modeling the communication layer of cyber-physical power systems. The location of the control center node directly affects the length and reliability of the measurement upload and command download paths for all generators, thus influencing the communication latency of each generator. The structural importance scoring method used in this invention comprehensively considers the betweenness number (global bridging role), core number (hierarchical depth in the network core), clustering coefficient (local clustering degree), degree (breadth of direct connections), and proximity centrality (average distance to all nodes). This multi-dimensional fusion ensures that the selected control center has globally optimal structural importance in the communication topology, guaranteeing that the starting point for latency assessment is representative.

[0022] Preferably, in step 1, the capacity of the physical layer branch is: 1 plus the tolerance margin multiplied by the absolute value of the initial power flow of the branch; the capacity of the information layer node and the information layer link are both: 1 plus the tolerance margin multiplied by the initial bandwidth.

[0023] All components are uniformly configured with capacity margins so that the tolerance capabilities of the physical layer and information layer are compared and scanned at the same leverage level. The physical layer branch capacity is based on the initial power flow, and the information layer capacity is based on the initial bandwidth; both are configured with the same tolerance margin. Synchronous scaling allows for the separation of the individual effects of the margin level and the severity of delay in subsequent sensitivity analysis, revealing the non-uniform distribution characteristics of delay penalties across different margin intervals.

[0024] Preferably, in step 2, the communication delay is calculated along the path: the delay of a single link is equal to the quotient of the message size divided by the link bandwidth plus the quotient of the link length divided by the signal propagation speed; the total delay of a path includes the endpoint transmission and reception processing time and the forwarding time of each intermediate node in addition to the delay of a single link, and the forwarding time of the control center node is less than the forwarding time of the ordinary relay node.

[0025] Communication latency consists of two parts: serialization latency (the ratio of message size to bandwidth) and propagation latency (the ratio of distance to signal speed). The former reflects bandwidth constraints, while the latter reflects physical distance constraints. At the path level, the forwarding time of each intermediate node depends on its role—the control center node has stronger processing capabilities, and its forwarding time is shorter than that of ordinary relay nodes. This distinction allows latency calculations to reflect the actual differences in hardware configuration between the control center and ordinary nodes, avoiding latency estimation bias caused by treating all nodes equally.

[0026] Preferably, in step 2, the uplink measurement channel delay is: the transmission processing delay from the power plant bus to the non-control center node, plus the total transmission delay on the uplink path, plus the queuing delay; the downlink execution channel delay is: the reception processing delay from the non-control center node to the power plant bus, plus the total transmission delay on the downlink path, plus the queuing delay; the uplink channel carries measurement messages, and the downlink channel carries command messages, with different responses from the two channels to the same topology change.

[0027] This invention decomposes end-to-end latency into uplink measurement channels and downlink execution channels based on service functions. The physical basis for this is the different message types carried in the two directions—uplink carries a large amount of measurement data (large messages), while downlink carries scheduling commands (small messages), thus experiencing different serialization latencies. Furthermore, the queuing mechanisms at the control center side differ, and the processing paths on the interface hardware also differ. This decomposition allows any vulnerability to be attributed to a specific channel rather than being absorbed into a single black-box penalty value, thereby providing direction for engineering optimization: whether to upgrade the communication interface on the sensor side or the command issuance interface on the execution mechanism side.

[0028] Preferably, in step 2, the control performance factor is a three-factor product, wherein the three factors are message loss factor, control saturation factor, and stability collapse factor.

[0029] Communication latency impairs generator controllability through three orthogonal physical paths: message loss (reliability dimension), outdated control commands leading to actuator saturation (control bandwidth dimension), and excessive latency causing instability in the closed-loop control system (stability dimension). Each of these three factors corresponds to an independent failure physical axis, and they are combined into a normalized performance factor through a product. This represents the proportion of generator output actually available after experiencing communication delay degradation. The characteristic of this product structure is that when any factor approaches zero, even if other factors are intact, the control effectiveness of the entire generator is cut off. This aligns with the physical fact that any failure mode in a real system can lead to a loss of controllability. The normalization condition ensures that under ideal communication conditions with no delay, all generators participate in power flow calculations at their full planned output, degenerating into a conventional no-delay cascade model, making the model of this invention comparable to existing models on the same benchmark.

[0030] Preferably, the message loss factor is equal to the path hop count raised to the power of the difference between the single-hop packet loss rate and the path hop count, where the path hop count is the sum of the uplink path hop count and the downlink path hop count.

[0031] The control saturation factor is equal to the natural constant. The negative exponent is the uplink channel attenuation coefficient multiplied by the positive part of the difference between the uplink measurement channel delay and the uplink channel dead zone threshold, plus the downlink channel attenuation coefficient multiplied by the positive part of the difference between the downlink execution channel delay and the downlink channel dead zone threshold, where the positive part represents the larger of the value in parentheses and zero.

[0032] The stability collapse factor is equal to one plus the natural constant. The sum of the negative steepness coefficients, divided by a factor plus the natural constant. The following power is applied: steepness coefficient multiplied by the sum of the uplink measurement channel delay and the downlink execution channel delay minus the difference of the critical delay threshold of the i-th generator, and then divided by the critical delay threshold; the critical delay threshold of the i-th generator is equal to the preset maximum critical delay threshold multiplied by the rated capacity of the i-th generator divided by the maximum value of the rated capacity of all surviving generators.

[0033] The three factors are normalized so that the control effectiveness factor equals 1 when all delays are zero.

[0034] The packet loss factor reflects the probability of packet loss at each hop in multi-hop communication. The longer the path (the more hops), the lower the probability of a packet successfully reaching its destination. This factor is determined solely by the number of hops in the path, which is determined by the topology, and is independent of the specific delay amplitude. Therefore, even if there is no congestion in the communication network, the cumulative effect of packet loss in long multi-hop paths has already led to a loss of controllability.

[0035] The control saturation factor reflects the actuator's operating bandwidth limitation: small delays within the sampling or execution cycle have almost no impact on the control effect; however, once the delay exceeds the dead-zone threshold, the actuator's output control quantity begins to deviate from the expected value, and the degree of deviation increases exponentially with increasing delay. Uplink and downlink each have independent dead-zone thresholds and attenuation coefficients, corresponding to the physical limitations of the measurement sampling cycle and command execution cycle, respectively.

[0036] The stability collapse factor reflects the time delay tolerance limit of a closed-loop control system: for each generator, its critical time delay threshold is proportional to its rated capacity. Larger units have greater rotational inertia and slower dynamic response, making them more tolerant of time delay; smaller units have smaller inertia and faster response, so the same absolute time delay causes more severe stability degradation, and their output returns to zero earlier. This heterogeneity allows this invention to characterize the differentiated damage of time delay to different units, which is structurally impossible with a single time delay penalty model.

[0037] Preferably, in step 2, the iteration method for each round is as follows:

[0038] The information layer overload check checks whether the current load of information layer nodes and links exceeds their capacity. If it does, the corresponding node or link fails and the information layer topology fragments. The per-unit delay assessment finds the uplink and downlink paths of the nearest surviving control center for each surviving generator on the current damaged topology and calculates the uplink measurement channel delay and the downlink execution channel delay. The power flow pre-output scaling scales the planned output of each surviving generator based on the communication delay and the control efficiency factor to obtain the actual output after delay degradation. The DC power flow redistribution solves the DC power flow with the actual output after delay degradation, redistributes branch power flow, and determines branch overload. The delay-coupled low-frequency load reduction performs low-frequency load reduction with the depth of load reduction increasing with the current average delay. After each iteration, the branch overload failures and load reduction results generated in this round are propagated across layers through inter-layer coupling and update the two-layer topology. The communication delay is recalculated on the updated topology in the next iteration, thus forming a closed loop through delay feedback between rounds.

[0039] The order of the five sub-steps described above follows a strict causal logic. First, information layer overload is addressed, as information layer failures are the source of delay degradation. Next, delay is evaluated on the damaged topology, which now reflects all structural failures that have occurred in this round. Then, generator output is scaled before power flow calculation—ensuring the power flow calculation "sees" the actual output after delay degradation, rather than the planned output; this is the core aspect of how delay affects physical power flow. Following this, the DC power flow is solved to obtain branch loads and identify overload risks. Finally, delay-coupled low-frequency load shedding is performed, with the shedding depth adaptively increasing as delay deepens, simulating the physical phenomenon that delay-induced frequency response lag necessitates the shedding of more load to maintain power balance. This sequence ensures the correct direction of the causal chain in each round: delay acts on output before power flow, and the power flow result, in turn, determines whether the topology changes in the next round.

[0040] Preferably, in step 3, the supply conservation ratio is equal to the power reduction factor multiplied by the sum of the initial power of the surviving loads divided by the sum of the initial power of all loads, and the power reduction factor is taken as the smaller value among the quotients of the sum of the actual output of all generators divided by the sum of the planned output of all generators; the dispatch fidelity error is equal to the square root of the following: for each surviving generator, multiply its rated capacity by the square of the difference between its actual output and its planned output, then divide by the square of its planned output, sum the above values ​​of all surviving generators, and divide by the sum of the rated capacities of all generators.

[0041] The Load Supply Ratio (LSR) measures the load supply capacity of a system after cascading termination. Its numerator is the sum of the initial power of the surviving loads, and the denominator is the sum of the initial power of all loads. The power reduction factor φ is introduced to handle situations where generator output is insufficient: when the sum of the actual output of all generators is less than the sum of their planned output, even if all load nodes survive, the actual power supply capacity of the system is lower than the original plan. In this case, the power reduction factor φ should be used to reduce the load supply capacity. The LSR value is reduced. This definition allows the LSR to reflect both load availability and generation adequacy.

[0042] Dispatch Tracking Error (DTE) measures the degree of deviation of the actual output of surviving generators from the capacity-weighted dispatch plan. Its physical meaning is that while a system may maintain load supply by adjusting the output of each generator, if these adjustments cause generators to deviate significantly from their planned operating points, then the system's operating state has undergone implicit degradation. LSR and DTE are complementary—the system may maintain a reassuring LSR value, but a significant increase in DTE reveals unit-level functional degradation that load indicators cannot capture. Using both together expands the assessment dimension from "whether power can be supplied" to "how power can be supplied."

[0043] Preferably, in step 3, the sensitivity analysis uses the most severe latency scenario as a baseline, and applies single-parameter mitigation actions to link bandwidth, endpoint processing latency, relay forwarding latency, field measurement interface latency, and field execution interface latency respectively. The recovery contribution of each latency component is measured by the recovery degree, which is equal to the difference between the supply conservation ratio after applying the single-parameter mitigation action and the supply conservation ratio under the most severe latency scenario, divided by the difference between the supply conservation ratio under the no-latency scenario and the supply conservation ratio under the most severe latency scenario. Then, the latency penalty is calculated on the plane composed of the iteration rounds and the tolerance margin to determine the early intervention window.

[0044] In the most severe latency scenario, one latency component is improved sequentially while keeping other components unchanged. The marginal contribution of this component to the system's resilience recovery is measured by the resilience index. The resilience is normalized between the ideal state with no latency and the most severe latency state, taking a value between 0 and 1; a larger value indicates a higher resilience gain from improving that component. This method compares the contributions of each latency component on the same scale, enabling engineering decisions to identify the optimization direction that "achieves the maximum resilience recovery per unit of input." The latency penalty is applied in each round (round number, ...). The distribution on the plane reveals the temporal structure of time-delayed harm—the punishment ignites sharply within a specific round interval, which can determine the time window for early intervention and provide a temporal basis for proactive control strategies in the cascading process.

[0045] Preferably, the cascading failure is triggered by removing the non-control center node with the highest structural importance in the information layer.

[0046] Specifically, the initial perturbation for cascading failure should be applied to the non-control center node with the greatest structural influence in the information layer to simulate the worst-case scenario of an attacker or extreme event damaging critical nodes of the communication network. This triggering condition ensures that all evaluation scenarios have the most severe initial perturbation in a structural sense, guaranteeing that the robustness assessment is a conservative estimate under adversarial premises, rather than a simulation extrapolation under optimistic assumptions.

[0047] This invention has the following characteristics and beneficial effects:

[0048] This invention models communication delay as an endogenous, causal state quantity in a cascading process, rather than a scalar penalty imposed after the fact. Delay is recalculated on the current damaged topology in each iteration and affects generator output before power flow redistribution, causing each delay scenario to evolve along its own distinct trajectory. This design overcomes the fundamental deficiency of existing post-penalty models in failing to reflect the causal loop of delay, output degradation, power flow redistribution, topology changes, and delay deterioration, and eliminates the systematic optimistic bias given by existing delay-free models in the medium-to-high tolerance margin range.

[0049] This invention decomposes end-to-end latency into an uplink measurement channel and a downlink execution channel. These two channels carry measurement messages and command messages respectively, with different message sizes, queuing mechanisms, and processing hardware. Based on this, the invention performs a multiplicative mapping using three performance factors—a message loss factor, a control saturation factor, and a stability collapse factor—corresponding to three orthogonal failure axes: reliability, control bandwidth, and stability. This allows the control performance degradation of each generator to be attributed to a specific channel and a specific physical failure mode, rather than being absorbed into a single black-box penalty curve, providing a clear attribution direction for engineering optimization.

[0050] This invention proposes a dual-index evaluation system for Supply Conservation Ratio (LSR) and Dispatch Fidelity Error (DTE). The former measures the load preservation level, while the latter measures the fidelity of surviving units in executing the dispatch plan. Using both together can reveal hidden functional losses where the LSR appears safe but the DTE has severely degraded. Based on this, this invention identifies the physical interfaces that contribute most to resilience recovery through resilience ranking, and further refines the evaluation system within each round. The time window for early intervention on the plane provides a quantitative basis with engineering guidance value for improving the resilience of cyber-physical power systems.

[0051] In summary, this invention effectively reveals the endogenous harmful mechanism of time delay in cascade failures of cyber-physical power systems, overcomes the problems of systematic optimism and insufficient evaluation granularity in existing models, and has significant theoretical innovation and engineering practical value. Attached Figure Description

[0052] Figure 1 This is an overall flowchart of the method of the present invention;

[0053] Figure 2 The flowchart of the five-step execution of the outer loop of the time-delay endogenous closed-loop cascading failure model;

[0054] Figure 3 This is a schematic diagram of a two-layer coupling model for a cyber-physical power system.

[0055] Figure 4 Uplink measurement channel With the downlink execution channel A schematic diagram of dual-channel delay decomposition;

[0056] Figure 5 This is a magnified diagram illustrating the change in message loss factor with path hop count.

[0057] Figure 6 A schematic diagram of the magnified curve for controlling the change of the saturation factor over time;

[0058] Figure 7 A magnified diagram illustrating the time-delay variation of the stability collapse factor;

[0059] Figure 8 Supply conservation ratio LSR varies with tolerance margin under different delay severity levels. The result of the change is shown in the diagram;

[0060] Figure 9 Scheduling fidelity error (DTE) varies with tolerance margin under different delay severity levels. The result of the change is shown in the diagram;

[0061] Figure 10 For delay penalties in (rounds, A thermal diagram of the spatiotemporal structure on a plane;

[0062] Figure 11 A ranking diagram of the recovery degree of the single-parameter mitigation actions for each time delay component. Detailed Implementation

[0063] The present invention will now be described in detail with reference to specific embodiments. These embodiments will help those skilled in the art to further understand the present invention, but do not limit the invention in any way. It should be noted that, unless otherwise specified, the embodiments and features described in the present invention can be combined with each other.

[0064] A time-delay closed-loop coupled cascaded fault assessment method for power cyber-physical systems (CPS) is proposed. First, a two-layer coupled model of the CPS is constructed. Second, a time-delay-intrinsic closed-loop cascaded failure model is used for iterative simulation round by round. Here, generator output degradation is performed before power flow by using dual-channel time-delay decomposition and a three-factor control efficiency function. Finally, robustness is evaluated using dual-elasticity indices, and sensitivity analysis is conducted. Figure 1 As shown, the technical solution adopted by the present invention to solve its technical problem includes the following steps:

[0065] Step 1: Construct a two-layer coupling model of the cyber-physical power system: using the power network as the physical layer and the communication network as the information layer, determine the control center node in the communication network, and couple the physical layer nodes and information layer nodes one by one in a positive betweenness correlation manner; set the capacity of physical layer branches (the physical layer branches are not physical layer nodes, but transmission lines connecting two bus nodes in the power network, which are directly determined by the topology data of the power network) as well as information layer nodes and information layer links according to the same tolerance margin.

[0066] In this embodiment, as Figure 3 As shown, the physical layer adopts the IEEE 39-node New England system (39 buses, 10 generators, and 46 branches), and uses a DC power flow approximation. , Let P be the nodal susceptance matrix, and P be the nodal injected power vector. The node voltage phase angle vector; branch power flow In the formula, Pe represents the active power flow transmitted on branch e. The branch susceptance connecting bus node i and bus node j. , These represent the voltage phase angles of nodes i and j, respectively. The information layer uses a Barabási-Albert scale-free network with a number of nodes. (Each busbar corresponds to one relay mirror node, plus 8 control centers), by The seed is added per node Strip growth.

[0067] Furthermore, the control center scores structures based on their importance. Select; where, Score the structural importance of information layer node i. Let i be the normalized betweenness of node i. Let i be the number of kernels at node i (i.e., the number of shells obtained from k-shell decomposition). Let be the clustering coefficient of node i. Let i be the degree of node i. Let the proximity centrality of node i be considered. Each bus is mapped through betweenness positive correlation. Coupled with an information layer relay node: the electrically critical node is matched to the communication-centric node. All component capacities are configured with the same tolerance margin. Setting: Branch Road Information layer nodes / links ; For tolerance margin, it is a unified operating margin lever and also the main axis of sensitivity analysis; in the formula, For the capacity of physical layer branch e, For the initial current flow of branch e, The initial bandwidth of the information layer node or information layer link.

[0068] Step 2: Perform iterative simulation of the system using the time-delay-endogenous closed-loop cascaded failure model. The time-delay-endogenous closed-loop cascaded failure model is constructed in this invention. It is a cascaded failure simulation model that uses communication delay as an endogenous state variable in the cascaded process. Its structure is an outer loop iteration executed round by round. Each iteration sequentially executes five sub-steps: information layer overload check, unit-by-unit delay assessment, power output scaling before power flow, DC power flow redistribution, and time-delay-coupled low-frequency load reduction. The rounds form a closed loop through time delay feedback: In each cascaded round, the structural fault propagation across layers is first performed. Then, the communication delay is calculated for each surviving generator on the current damaged communication network topology. Based on the delay, the generator output is scaled by the control efficiency factor to obtain the actual output after time delay degradation. Then, the DC power flow is solved using the actual output after time delay degradation to determine the branch overload. Finally, time-delay-coupled low-frequency load reduction is performed as the time delay deepens. The above delay is recalculated on the current topology in each round and fed back to the next round to form a closed loop.

[0069] The damaged communication network topology refers to the communication network topology formed by the surviving information layer nodes and information layer links after removing the nodes and links that have failed up to the present from the initial two-layer coupled topology.

[0070] Specifically, communication delay is calculated along the path: the delay of a single link equals the quotient of the message size divided by the link bandwidth plus the quotient of the link length divided by the signal propagation speed. The total latency of a path includes, in addition to the latency of a single link, the endpoint transmission and reception processing time and the forwarding time of each intermediate node, and the forwarding time of the control center node is less than that of a normal relay node.

[0071] Furthermore, such as Figure 4 As shown, the uplink measurement channel delay is: the transmission and processing delay from the power plant bus to the non-control center node, plus the total transmission delay on the uplink path, plus the queuing delay, expressed as follows: It carries a large number of measurement messages; the downlink execution channel delay is: the reception and processing delay from the non-control center node to the power plant bus, plus the total transmission delay on the downlink path, plus the queuing delay, expressed as follows: The uplink channel carries smaller command messages, while the downlink channel carries command messages. The two channels respond differently to the same topology change, providing a basis for attributing recoverable damage to specific interfaces.

[0072] Furthermore, in this embodiment, the three-factor control effectiveness factor adopts the form of a three-factor product: When all factors are normalized to make all delays zero The three factors are message loss factor, control saturation factor, and stability collapse factor, respectively.

[0073] like Figure 5 As shown, the packet loss factor is equal to the path hop count raised to the power of the difference between the single-hop packet loss rate and the path hop count, where the path hop count is the sum of the uplink path hop count and the downlink path hop count. , It is the sum of the number of hops in the uplink and downlink paths, determined by the topology rather than the latency magnitude. Therefore, even if there is no latency, long multi-hop paths cannot achieve full retention.

[0074] like Figure 6 As shown, the control saturation factor is equal to the natural constant. The negative exponent is the uplink channel attenuation coefficient multiplied by the positive part of the difference between the uplink measurement channel delay and the uplink channel dead zone threshold, plus the downlink channel attenuation coefficient multiplied by the positive part of the difference between the downlink execution channel delay and the downlink channel dead zone threshold, where the positive part represents the larger of the value in parentheses and zero: Within the dead zone (sampling / execution cycle) of each channel, the impact is negligible; beyond that, the available output decreases exponentially with the sum of the delays of the two channels.

[0075] like Figure 7 As shown, the stability collapse factor is equal to the sum of a negative steepness coefficient plus the natural constant e, divided by a power of the following: steepness coefficient multiplied by the sum of the uplink measurement channel delay and the downlink execution channel delay minus the difference of the critical delay threshold of the i-th generator, then divided by the critical delay threshold; the critical delay threshold of the i-th generator is equal to the preset maximum critical delay threshold multiplied by the rated capacity of the i-th generator divided by the maximum value of the rated capacities of all surviving generators. Unit-related thresholds Randomized scaling of rated capacity means that the same absolute delay has almost no impact on large units but reduces the output of small units to zero. This heterogeneity is the microscopic source of asymmetry and unit-specific damage. The three factors act on the three orthogonal axes of reliability, control bandwidth, and stability, respectively. Therefore, any fault can be directly read from the corresponding factor, which cannot be provided by a single black-box penalty curve.

[0076] Further configuration involves an endogenous closed-loop cascading iteration with latency. Cascading is triggered by removing the non-control center information layer node with the highest betweenness, and all scenarios share the same seed. The inner loop propagates structural faults across layers until no new faults emerge: information subnets disconnected from all control centers are removed, power islands without generators collapse, and cross-layer faults occur with probability p. For example... Figure 2 As shown, each iteration is performed in the following order:

[0077] Step 2-1: Perform an information layer overload check: Check whether the current load of the information layer nodes and information layer links exceeds their capacity. In this embodiment, the capacity value is set as: If the limit is exceeded, the corresponding node or link will fail and the information layer topology will be fragmented.

[0078] Step 2-2: Perform unit-by-unit delay assessment: Find the uplink path to the nearest surviving control center for each surviving generator on the current damaged topology. and downlink path And calculate the uplink measurement channel delay and the downlink execution channel delay;

[0079] Step 2-3: Perform power output scaling before power flow: The actual output of the i-th generator is equal to the control efficiency factor of the i-th generator multiplied by the planned output of the i-th generator. ;

[0080] Steps 2-4: Perform DC power flow redistribution: The node injected power vector equals the susceptance matrix multiplied by the node voltage phase angle vector. Solving this equation yields the current branch power flow. The unbalanced power is absorbed by the balancing machine;

[0081] Steps 2-5: Perform time-delay coupled low-frequency load shedding: The load shedding depth is equal to the smaller of the maximum load shedding depth multiplied by a function of the ratio of the maximum load shedding depth to the low-frequency load shedding reference time delay threshold. The ratio is the current average delay divided by the low-frequency load shedding reference delay threshold, and when the current average delay is zero, it degenerates into a regular single-level load shedding.

[0082] Step 3: Evaluate robustness based on the supply conservation ratio (LSR) and scheduling fidelity error (DTE), and perform sensitivity analysis on the tolerance margin, delay severity, and the components of uplink measurement channel delay and downlink execution channel delay.

[0083] Specifically, the supply conservation ratio is equal to the power reduction factor multiplied by the sum of the initial power of the surviving loads divided by the sum of the initial power of all loads. The power reduction factor is taken as the smaller of the quotients of the sum of the actual output of all generators divided by the sum of the planned output of all generators. The dispatch fidelity error is equal to the square root of the following: for each surviving generator, multiply its rated capacity by the square of the difference between its actual output and its planned output, then divide by the square of its planned output, sum the above values ​​of all surviving generators, and divide by the sum of the rated capacities of all generators.

[0084] Among them, the supply conservation ratio , This measures the proportion of load still being supplied and reduces it by the generation deficit. Dispatch fidelity error. The Load Response Time (LSR) measures the actual output of surviving units relative to the planned capacity weighted normalized root mean square error. The LSR asks how much load is surviving, while the DTE asks the fidelity of the surviving units' scheduling. The system may maintain a seemingly reassuring LSR, while the DTE reveals that its load is supplied by units far from the setpoint, i.e., a hidden degradation that is not visible in the load metric.

[0085] Understandably, the sensitivity analysis uses the most severe latency scenario as a baseline and applies single-parameter mitigation actions to link bandwidth, endpoint processing latency, relay forwarding latency, field measurement interface latency, and field execution interface latency, respectively. The recovery contribution of each latency component is measured by the recovery degree, which is equal to the difference between the supply conservation ratio after applying the single-parameter mitigation action and the supply conservation ratio under the most severe latency scenario, divided by the difference between the supply conservation ratio under the no-latency scenario and the supply conservation ratio under the most severe latency scenario. Furthermore, the latency penalty is calculated round by round on the plane composed of the iteration rounds and the tolerance margin to determine the early intervention window.

[0086] Based on the above technical solution, this embodiment provides the following simulation example:

[0087] Dataset and Experiment Setup

[0088] The physical layer uses the IEEE 39-node system, and the information layer uses... A scale-free BA network containing 8 control centers. Each run is fixed. In a given scenario, a BA network is regenerated and coupled with 39 IEEE nodes. After applying a seed perturbation, a closed-loop cascade is run until termination. The tolerance margin α is... The upper scan (11 levels in total) uses five scenarios to determine latency severity: no latency (0), mild, baseline (1.0), moderate, and severe. All latency parameters are scaled using the same factor for each scenario. For the no-latency scenario, all latency parameters are set to zero, and a normal no-latency cascade is restored as a reference. Each ( The scene combination is averaged over multiple independent implementations. The same random seed is reused across scenes within the same implementation, so that scene differences only reflect time delay rather than sampling noise. The cross-layer propagation probability p is 0.3.

[0089] Experimental results

[0090] The results show that: (1) as Figure 6 As shown, the supply conservation ratio (LSR) decreases monotonically with the severity of latency, with the highest LSR in no-latency scenarios and the lowest LSR in heavily-latency scenarios. Figure 8 The bar chart in the image shows the latency penalty ΔLSR for each latency scenario relative to the no-latency scenario. It can be seen that the penalty difference is not uniformly distributed: in the tight margin ( At a relatively small margin, all scenes crash together, within a certain margin. (1) Physical margin near the point of delay masked the effect of delay, while the delay penalty reached its maximum in the medium-high margin zone—which is precisely the range that delay-free analysis would identify as safe. This indicates that closed-loop modeling located the danger zone in the margin range where operators feel most secure, proving the effectiveness of this invention in eliminating the systematic optimistic bias of the delay-free model; (2) as Figure 9 As shown, the scheduling fidelity error (DTE) is clearly stratified according to the severity of delay across all tolerance margins α, and the curves do not intersect. Even at the seemingly sufficient margin of LSR, the scheduling fidelity degrades smoothly and monotonically with delay, exposing the implicit functional degradation that cannot be captured by a single load index, thus proving the necessity and effectiveness of the dual-index evaluation system of this invention; (3) In multiple independent implementations, the no-delay scenario gives a narrow and concentrated LSR distribution, while the heavy-delay scenario gives a wide distribution with a clear near-total collapse tail, indicating that delay transforms the approximately deterministic robustness into a heavy-tailed, implementation-dependent risk; (4) As Figure 10 As shown, the round-by-round delay penalty ΔLSR is calculated in (cascaded rounds, tolerance margin) The plane presents a clear spatiotemporal structure: the first round has a neutral basic time delay (dark area in the figure), the second round is rapidly ignited, and the third to fifth rounds are magnified and saturated (bright area in the figure). Based on this, a narrow early intervention window can be identified, proving that the present invention can provide a timing basis for active control in the cascade process; (5) as Figure 11 As shown, after applying single-parameter mitigation actions starting from a high-latency scenario, the recovery rates of the two field-side interface actions—speeding up the field execution interface and speeding up the field measurement interface—reached 56.2% and 50.5% respectively, dominating the field. In contrast, the recovery rates of link redundancy backup and endpoint processing upgrades were only 2.2% and 0.6%, respectively. This indicates that upgrading the sensing and execution interfaces provides greater resilience recovery per unit of investment than widening bandwidth, demonstrating the effectiveness of the dual-channel latency decomposition method in engineering attribution and optimization ranking. Figures 8 to 11 The simulation results jointly verify that the method described in this invention can accurately characterize the endogenous closed-loop harm mechanism caused by time delay, and provide attributable and operable robustness assessment conclusions. The form of the three-factor control performance function upon which this method is based is as follows: Figures 5 to 7 As shown: Figure 5 This indicates that the message loss factor decreases monotonically as the number of path hops increases. Figure 6 This indicates that the saturation factor remains at 1 within the dead zone threshold, and decays exponentially with time delay after exceeding it. Figure 7 This indicates that the stability collapse factor exhibits different time delay tolerance limits for large and small generating units.

[0091] The foregoing has shown and described the basic principles, main features, and advantages of the present invention. Those skilled in the art should understand that the present invention is not limited to the above embodiments. The embodiments and descriptions in the specification are merely preferred examples and are not intended to limit the invention. Various changes and modifications can be made to the invention without departing from its spirit and scope, and all such changes and modifications fall within the scope of the present invention as claimed. The scope of protection of the present invention is defined by the appended claims and their equivalents.

Claims

1. A method for assessing cascaded faults in a time-delay closed-loop coupled power cyber-physical system, characterized in that, Includes the following steps: Step 1: Construct a two-layer coupling model of the cyber-physical power system: take the power network as the physical layer and the communication network as the information layer, determine the control center node in the communication network, and couple the physical layer nodes and information layer nodes one by one in a positive betweenness correlation manner, and determine the physical layer branches based on the topology data of the power network. The capacity of physical layer branches, information layer nodes, and information layer links are all set with the same tolerance margin. Step 2: Based on the two-layer coupling model, establish a time delay endogenous closed-loop cascade failure model, and perform iterative simulation of the system using the time delay endogenous closed-loop cascade failure model. In each iteration, the communication delay is embedded into the time delay endogenous closed-loop cascade failure model. The system is simulated iteratively round by round using a time-delay-endogenous closed-loop cascaded failure model. The cascaded failure simulation model takes communication delay as the endogenous state variable of the cascaded process. Its structure is an outer loop iteration executed round by round. Each round of iteration executes five sub-steps in sequence: information layer overload check, unit-by-unit time delay assessment, power output scaling before power flow, DC power flow redistribution, and time delay-coupled low-frequency load reduction. The rounds form a closed loop through time delay feedback: In each round of cascading, the cross-layer propagation of structural faults is first performed. Then, the communication delay is calculated for each surviving generator on the current damaged communication network topology. Based on the time delay, the output of the generator is scaled by the control efficiency factor to obtain the actual output after time delay degradation. Then, the DC power flow is solved with the actual output after time delay degradation to determine the branch overload. Finally, time delay-coupled low-frequency load reduction is performed as the time delay deepens. The aforementioned latency is recalculated on the current topology in each round and fed back to the next round, forming a closed loop; Step 3: Evaluate robustness based on the Supply Conservation Ratio (LSR) and Dispatch Fidelity Error (DTE), and perform sensitivity analysis on the tolerance margin, delay severity, and the components of uplink measurement channel delay and downlink execution channel delay. The supply conservation ratio is equal to the power reduction factor multiplied by the sum of the initial power of the surviving loads divided by the sum of the initial power of all loads, where the power reduction factor is the smaller of the quotient of the sum of the actual output of all generators divided by the sum of the planned output of all generators.

2. The method according to claim 1, characterized in that, In step 1, the communication network adopts a scale-free network; the control center node is selected in the following way: calculate the structural importance score of each node, which is equal to the sum of the normalized betweenness number and the kernel number of the node divided by the clustering coefficient of the node, plus the product of the degree and the proximity centrality of the node.

3. The method according to claim 2, characterized in that, In step 1, the node with the highest structural importance score is selected as the control center node.

4. The method according to claim 1, characterized in that, In step 1, the capacity of the physical layer branch is: 1 plus the tolerance margin multiplied by the absolute value of the initial power flow of the branch; the capacity of the information layer node and the information layer link is: 1 plus the tolerance margin multiplied by the initial bandwidth.

5. The method according to claim 1, characterized in that, In step 2, the method for embedding communication delay is as follows: In each iteration, the uplink measurement channel delay and downlink execution channel delay are recalculated for each surviving generator on the current damaged communication network topology. The calculated communication delay is mapped to the scaling factor of the planned output of the generator through the control efficiency factor on the one hand, and used to determine the load reduction depth of low-frequency load reduction on the other hand, so that the communication delay becomes an endogenous state quantity that evolves with the cascade process and reacts on the cascade process.

6. The method according to claim 5, characterized in that, In step 2, the method for calculating the communication delay is as follows: the delay of a single link is equal to the quotient of the message size divided by the link bandwidth plus the quotient of the link length divided by the signal propagation speed. The total latency of a path includes the endpoint transmission and reception processing time and the forwarding time of each intermediate node in addition to the latency of the single link. The forwarding time of the control center node is less than that of the ordinary relay node.

7. The method according to claim 5, characterized in that, In step 2, the uplink measurement channel delay is: the transmission processing delay from the power plant bus to the non-control center node, plus the total transmission delay on the uplink path, plus the queuing delay; the downlink execution channel delay is: the reception processing delay from the non-control center node to the power plant bus, plus the total transmission delay on the downlink path, plus the queuing delay. The uplink channel carries measurement messages, while the downlink channel carries command messages. The two channels respond differently to the same topology change.

8. The method according to claim 5, characterized in that, In step 2, the control performance factor is a three-factor multiplication form, where the three factors are message loss factor, control saturation factor, and stability collapse factor.

9. The method according to claim 8, characterized in that, The message loss factor is equal to the path hop count raised to the power of the difference between the single-hop packet loss rate and the path hop count, where the path hop count is the sum of the uplink path hop count and the downlink path hop count. The control saturation factor is equal to the negative power of the natural constant e. The exponent is the positive part of the difference between the uplink measurement channel delay and the uplink channel dead zone threshold multiplied by the uplink channel attenuation coefficient, plus the positive part of the difference between the downlink execution channel delay and the downlink channel dead zone threshold multiplied by the downlink channel attenuation coefficient. The positive part represents the larger value between the value in parentheses and zero. The stability collapse factor is equal to the sum of a negative steepness coefficient plus the natural constant e, divided by the following power plus the natural constant e: steepness coefficient multiplied by the sum of the uplink measurement channel delay and the downlink execution channel delay minus the difference of the critical delay threshold of the i-th generator, and then divided by the critical delay threshold; the critical delay threshold of the i-th generator is equal to the preset maximum critical delay threshold multiplied by the rated capacity of the i-th generator divided by the maximum value of the rated capacity of all surviving generators. The three factors are normalized so that the control effectiveness factor equals 1 when all delays are zero.

10. The method according to claim 1, characterized in that, In step 2, the iteration method for each round is as follows: The information layer overload check checks whether the current load of information layer nodes and links exceeds their capacity. If it does, the corresponding node or link fails and the information layer topology is fragmented. The per-generator delay assessment finds the uplink and downlink paths of the nearest surviving control center for each surviving generator on the current damaged topology and calculates the uplink measurement channel delay and the downlink execution channel delay. The power flow pre-output scaling scales the planned output of each surviving generator based on the communication delay and the control efficiency factor to obtain the actual output after delay degradation. The DC power flow redistribution solves the DC power flow with the actual output after delay degradation, redistributes branch power flow, and determines branch overload. The time-delay coupled low-frequency load shedding is a low-frequency load shedding process in which the load shedding depth increases with the current average time delay. After each iteration, the branch overload failure and load reduction results generated in this round are propagated across layers through interlayer coupling and the topology of the two layers is updated. The communication delay is recalculated on the updated topology in the next iteration, thus forming a closed loop through delay feedback between rounds.

11. The method according to claim 1, characterized in that, In step 3, the scheduling fidelity error is equal to the square root of the following: for each surviving generator, multiply its rated capacity by the square of the difference between its actual output and its planned output, then divide by the square of its planned output, sum the above values ​​of all surviving generators, and divide by the sum of the rated capacities of all generators.

12. The method according to claim 1, characterized in that, In step 3, the sensitivity analysis uses the most severe latency scenario as the baseline and applies single-parameter mitigation actions to link bandwidth, endpoint processing latency, relay forwarding latency, field measurement interface latency, and field execution interface latency, respectively. The recovery contribution of each latency component is measured by the recovery degree, which is equal to the difference between the supply conservation ratio after applying the single-parameter mitigation action and the supply conservation ratio under the most severe latency scenario, divided by the difference between the supply conservation ratio under the no-latency scenario and the supply conservation ratio under the most severe latency scenario. Then, the latency penalty for each round is calculated on the plane composed of the iteration rounds and the tolerance margin to determine the early intervention window.

13. The method according to claim 1, characterized in that, The cascading failure is triggered by the removal of the non-control center node with the highest structural importance in the information layer.

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