Power system toughness maintenance method based on multi-stage state under ice disaster

Abstract

The invention belongs to the technical field of power system maintenance, and particularly relates to a power system toughness maintenance method based on a multi-stage state under an ice disaster, and the method comprises the following steps: S1, building a time-varying continuous Markov state transition model for source-network-load side equipment in a power system; s2, dividing to-be-overhauled equipment in the system into two types: pre-maintenance equipment a and non-pre-maintenance equipment b; defining three state evolution paths possibly experienced by the equipment; respectively constructing expected availability functions of the pre-maintenance equipment a and the non-pre-maintenance equipment b; s3, constructing a toughness maintenance optimization model with the goal of minimizing the total cost; the decision variables of the optimization model comprise the emergency maintenance trigger time TD (a) of the pre-maintenance equipment a, and the pre-maintenance time Tpre (b) and the emergency maintenance trigger time TD (b) of the non-pre-maintenance equipment b; and S4, solving the toughness maintenance optimization model, and generating an offline toughness maintenance plan. According to the method, refined and foresight maintenance scheduling of source-network-load side key equipment can be realized.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of power system maintenance, and particularly relates to a power system resilience maintenance method based on multi-stage states under ice disaster. BACKGROUND

[0002] In recent years, with the continuous improvement of the penetration rate of renewable energy, the sensitivity of power systems to extreme weather events has significantly increased. Among them, ice disaster, as a natural disaster with long duration and wide influence, frequently causes problems such as transmission line icing, tower collapse, and equipment insulation failure, seriously threatening the safe and stable operation of the power grid, and even leading to large-scale and long-time power outages. In the face of such disasters, system operators need to develop scientific and efficient maintenance strategies to improve the resilience of power systems in ice disaster environments and ensure the continuous power supply capacity of critical loads.

[0003] However, ice disaster is essentially different from instantaneous disasters such as typhoon and earthquake: its duration often lasts for several days or even weeks, during which the weather conditions (such as temperature, humidity, and wind speed) and ice thickness change constantly, leading to obvious time-varying and cumulative effects in the degradation process of power equipment. Traditional maintenance methods are difficult to effectively respond to this complex dynamic scenario. Existing strategies mainly include two categories: one is pre-disaster regular preventive maintenance, which usually establishes a static or quasi-static equipment outage model based on historical data, predicts the failure probability of equipment within a certain fixed period in the future, and arranges maintenance plans accordingly; the other is post-disaster failure response maintenance, which schedules repairs according to resource constraints and priorities after equipment failure. Both methods have obvious limitations: regular maintenance does not consider the characteristics of accelerated equipment degradation rate over time during the ice disaster process, which easily causes maintenance timing lag or resource waste; while failure maintenance is a passive response, which cannot intervene before the equipment enters the serious degradation stage but has not failed, making it difficult to avoid load loss, and may exacerbate system collapse risk under resource constraints.

[0004] Therefore, how to achieve fine and forward-looking maintenance scheduling of key equipment on the source-grid-load side has become a problem to be solved. SUMMARY

[0005] In view of the above deficiencies of the prior art, the application provides a power system resilience maintenance method based on multi-stage states under ice disaster, which can realize fine and forward-looking maintenance scheduling of key equipment on the source-grid-load side.

[0006] To solve the above technical problems, the application adopts the following technical solutions:

[0007] A power system resilience maintenance method based on multi-stage states under ice disaster, comprising the following steps:

[0008] S1. Based on the cumulative effect of ice storms, a time-varying continuous Markov state transition model is established for the source-grid-load side equipment in the power system to characterize the multi-stage evolution process of the equipment from normal state through moderately deteriorated state, severely deteriorated state to fault state during ice storms; wherein, the state transition intensity of the Markov model is a function of time, used to reflect the dynamic changes of equipment failure rate and repair rate with the duration of ice storms; the Markov model also introduces an emergency maintenance state to characterize the controlled outage state that the equipment enters due to emergency maintenance when it is in a severely deteriorated state.

[0009] S2. Divide the equipment to be repaired in the system into two categories: those with a preset fixed pre-maintenance time T. pre (a) Pre-maintenance equipment a, and non-pre-maintenance equipment b for which no fixed pre-maintenance time is set;

[0010] For each device, three possible state evolution paths are defined during the ice storm: Path 1: The device undergoes emergency maintenance at time T. D The device is determined to be in a severely degraded state, triggering emergency maintenance, and enters emergency maintenance mode; Path Two: The device does not trigger Path One, but enters emergency maintenance mode within the pre-maintenance time T. pre A sudden malfunction occurred before arrival; Path 3: The equipment did not trigger Path 1, and was in T pre No malfunctions occurred before arrival, thus the pre-maintenance task was performed as planned;

[0011] Based on the Markov state transition model constructed by S1, the occurrence probabilities of the three paths for each device are calculated. Based on the three paths and their corresponding occurrence probabilities, the expected availability function of the pre-maintenance device a and the expected availability function of the non-pre-maintenance device b are constructed respectively.

[0012] S3. Construct a resilience maintenance optimization model with the goal of minimizing total cost; total cost includes load loss cost, pre-maintenance cost, and emergency maintenance cost; among which, load loss cost is calculated based on the equipment unavailability probability determined by the expected availability function constructed in S2;

[0013] The decision variables for the optimization model include the emergency maintenance trigger time T of the pre-maintenance device a. D (a) Pre-maintenance time T of non-pre-maintenance equipment b pre (b) and emergency maintenance trigger time T D (b);

[0014] S4. Solve the toughness maintenance optimization model to obtain the optimal T. D *(a), T pre *(b) and T D *(b) Generate an offline resilience maintenance plan;

[0015] S5. During the actual evolution of the ice storm, monitor the equipment status in real time and dynamically determine the path the equipment is on based on the three state evolution paths defined in S2 and their corresponding triggering conditions. When the equipment status is detected to meet the triggering conditions of path one or path two, immediately execute the corresponding emergency maintenance operation.

[0016] Compared with the prior art, the present invention has the following beneficial effects:

[0017] 1. This approach fully characterizes the time-varying characteristics of equipment degradation and innovatively introduces three mutually exclusive and complete state evolution paths. Traditional maintenance methods typically make decisions based on static failure rates or simple binary states (normal / failure), which are insufficient to reflect the dynamic law of gradual degradation of equipment performance during the ice storm. Based on the cumulative effect of the ice storm, this solution establishes a continuous Markov model where the state transition intensity is a function of time, accurately describing the natural evolution process of equipment from normal → moderate degradation → severe degradation → failure. Simultaneously, it clearly delineates three state evolution paths covering all possible scenarios: Path 1 (triggered emergency maintenance), Path 2 (sudden failure), and Path 3 (planned pre-maintenance). This path-based modeling not only reflects the time-varying risks of equipment status but also provides a probabilistic basis for differentiated response strategies, significantly outperforming the passive "one-size-fits-all" or reactive modes of existing technologies.

[0018] 2. By optimizing the T of pre-maintenance equipment a D (a) T with non-pre-maintenance equipment b D (b) Achieving risk-driven proactive intervention. Existing methods often rely on fixed thresholds or empirical rules to trigger emergency repairs, which can easily lead to delayed response or wasted resources. This solution will use T D (a) and T D (b) Incorporating T as a key decision variable into the optimization model: For equipment a with scheduled pre-maintenance, T D (a) This determines whether to intervene before scheduled maintenance—if monitoring shows accelerated deterioration, emergency maintenance can be initiated at the optimal time to avoid cascading failures caused by operation with defects; for equipment b without scheduled pre-maintenance, T D The introduction of (b) enables it to possess "dynamic emergency response capabilities," meaning that although there is no fixed plan, precise intervention can be achieved by setting personalized trigger points. Together, these two constitute a "prevention-emergency" collaborative mechanism, significantly enhancing the system's proactive defense capabilities against high-risk equipment.

[0019] 3. Introduce the virtual pre-maintenance time T for non-pre-maintenance equipment b. pre (b) Enhance the model's versatility and scheduling flexibility. Although non-pre-maintenance equipment b (such as some distribution lines and load nodes) does not undergo pre-maintenance in practice, T will be used to improve the model's versatility and scheduling flexibility. pre(b) Incorporating it as a virtual decision variable into the model has significant value: on the one hand, it maintains structural consistency in the availability function expression between pre-maintenance and non-pre-maintenance equipment, facilitating overall optimization; on the other hand, the optimized T... D *(b) can serve as a reference indicator for the "potential optimal maintenance window," used to assess whether high-risk equipment should be upgraded to pre-maintenance targets. This design breaks through the limitations of the rigid "maintenance / non-maintenance" classification in traditional methods, providing quantitative support for prioritization under limited resources.

[0020] 4. Clarify the mechanism for transitioning equipment from a severely deteriorated state to an emergency maintenance state in "Path One," enabling precise intervention before failure. The core of "Path One" lies in: when the equipment is in T... D When a device is determined to be in a severely degraded state, emergency maintenance is immediately triggered, and the device enters an independent "emergency maintenance state." This design differs from existing technologies that vaguely equate "severe degraded" with "imminent failure." By explicitly defining a "controlled shutdown state," it achieves: modeling of the maintenance action itself (such as repair time and resource consumption); precise quantification of equipment availability during maintenance (through conditional availability functions); and a strict distinction between "intervention-friendly high-risk states" and "irreversible failures."

[0021] In summary, this method can achieve refined and forward-looking maintenance and scheduling of key equipment on the source-grid-load side.

[0022] Preferably, the probability of path one occurring is equal to the probability of the device triggering emergency maintenance at time T. D The probability of the device entering a severely degraded state for the first time; the probability of path two occurring is that the device did not trigger path one, but occurred within the pre-maintenance time T. pre The probability of entering a fault state for the first time; the probability of path three occurring is 1 minus the sum of the probabilities of path one and path two occurring.

[0023] Preferably, in S1, the state of electrical equipment is represented as a set. Where the subscript k represents device k, Indicates a normal state. This indicates mild degradation. This indicates severe degradation. Indicates a fault; Indicates an emergency maintenance status;

[0024] Time-varying state probability of device k during an ice storm

[0025] This setup allows for different types of equipment (such as generators, transmission lines, and load nodes) to exhibit varying degradation patterns under ice storms. This solution employs a general state set approach, facilitating unified modeling of various equipment types while allowing adjustment of transfer strength parameters based on specific equipment characteristics. This modular design enables the method to be extended to complex power systems and is applicable to resilience analysis under diverse disaster scenarios.

[0026] Preferably, the expected availability function of the pre-maintenance device a constructed in S2 is:

[0027]

[0028] In the formula, A m (a,t) represents the expected availability of pre-maintenance equipment a; The expected availability of the three paths for pre-maintenance device a is as follows; The probabilities of occurrence for the three paths of pre-maintenance device a are respectively;

[0029] This represents the probability that the device first enters a severely degraded state before time t. Let be the cumulative distribution function of the device when it first enters a severely degraded state before time u; This is a conditional availability function for equipment under emergency maintenance. This is a conditional availability function for devices in a faulty state. Let be the conditional availability function of the equipment under normal conditions; u is the integral variable, representing the specific moment when the equipment first enters a severely degraded state; Indicates the device is in tT D (a) The probability of first entering a fault state before time; d represents the cumulative distribution function of the device when it first enters a fault state before time u; m The duration of the pre-maintenance task.

[0030] This setup, by introducing convolutional integrals, accurately captures the impact of historical degradation events on current availability. Traditional methods often use static probability or simple exponential decay models to estimate availability, making it difficult to reflect the impact of "when degradation occurs" on subsequent performance. This solution couples the probability of a device entering a severely degraded state at different times with its subsequent operability through an integral term. This design allows the model to not only predict "whether a failure will occur," but also quantify "at what point in time degradation will affect the power supply capacity for how long," thereby achieving a more refined risk assessment.

[0031] 2. Establish a complete mapping relationship from state transition strength to availability function to support the global optimization objective. This model transforms the state transition strength (such as failure rate and repair rate) in a Markov process into a specific availability function A.m (a,t) is integrated into total availability through path probability weighting. This not only completes the transformation from "physical process to mathematical expression" but also provides direct input for load loss cost calculation in S3, realizing the integrated modeling from equipment-level degradation to system-level resilience indicators.

[0032] 3. Clearly distinguish the timing logic between emergency maintenance and planned maintenance to improve the dynamic adaptability of maintenance decisions. For pre-maintenance equipment a, existing strategies often rely solely on a fixed T... pre (a) Execution and maintenance lack the ability to respond to real-time status. This solution addresses this by setting T... D (a) As an emergency maintenance trigger threshold, it allows T pre (a) Initiate maintenance in advance. This two-layer control mechanism enables the system to maintain its planned operation while also having the flexibility to cope with accelerated degradation, avoiding the risk of cascading failures caused by delayed maintenance.

[0033] Preferably, in S2, the expected availability function of the constructed non-pre-maintenance equipment b is:

[0034]

[0035] In the formula, A nm (b,t) represents the expected availability of non-pre-maintenance equipment b; and The expected availability of the three paths for non-pre-maintenance equipment (b) is as follows; The probabilities of the three paths for non-pre-maintenance equipment b are respectively;

[0036] This represents the probability that the device first enters a severely degraded state before time t. The cumulative distribution function represents the device's first entry into a severely degraded state before time u; A conditional availability function representing equipment under emergency maintenance conditions; This function represents the conditional availability of a device when it is in a normal state.

[0037] This setup has two main drawbacks: 1. Traditional methods cannot proactively schedule equipment without fixed maintenance cycles. This solution addresses this by using T... pre (b) A "virtual but executable" planned maintenance time point was constructed as an optimization variable. This means that even if the equipment does not have a preset maintenance time, the system can dynamically determine the best maintenance time based on its degradation trend and system impact weight, thereby transforming passive response into proactive prevention and control, and significantly improving the overall system resilience.

[0038] 2. Maintain consistent modeling logic with pre-maintenance equipment to ensure consistent evaluation of all equipment within the system. Although equipment b has no preset maintenance time, its availability function structure is completely consistent with that of equipment a: both adopt a three-path partitioning, probability weighting, and convolution integral form. This unified mathematical framework avoids model inconsistencies caused by different equipment classifications, enabling various equipment on the source-grid-load side to be compared and optimized under the same evaluation system, enhancing the overall coordination and scalability of the method.

[0039] Preferably, in S3, the objective function of the toughness maintenance optimization model is:

[0040]

[0041] In the formula, N m N represents the number of pre-maintenance devices. nm The number of non-maintenance equipment; R m (a) R nm (b) Repair costs for pre-maintenance equipment and non-pre-maintenance equipment, respectively; R C Cost of load loss.

[0042] Preferably,

[0043]

[0044] In the formula, These are the expected pre-maintenance costs, expected emergency maintenance costs, and expected failure maintenance costs for the pre-maintenance equipment, respectively. These represent the expected pre-maintenance costs, expected emergency maintenance costs, and expected failure maintenance costs for non-pre-maintenance equipment, respectively; Ω m (t) represents the maintenance scenario set at time t, including pre-maintenance, emergency maintenance, and no maintenance scenarios; π ω (t) is the probability that maintenance scenario ω will occur at time t; and R ω (t) is the load loss associated with time t and maintenance scenario ω.

[0045] This approach addresses two key issues: 1) Existing maintenance strategies typically focus only on maintenance costs or the reliability of individual devices, neglecting the impact of load loss at the system level. This innovative approach integrates pre-maintenance costs, emergency maintenance costs, fault maintenance costs, and expected load loss costs into the objective function, forming a multi-objective cost structure. This allows the optimization process to consider not only the economic costs of "whether to repair" and "when to repair," but also the socio-economic losses from "power outages due to non-repair," thereby achieving optimal overall benefits with limited resources.

[0046] 2. Differentiate the cost structure of pre-maintenance and non-pre-maintenance equipment to enhance model adaptability. Traditional optimization models often assume all equipment has the same maintenance pattern, making it difficult to handle situations where both "planned maintenance" and "unplanned maintenance" coexist in actual operation and maintenance. This solution defines R... m (a) and R nm (b) allows for independent modeling of the two cost items, respecting existing operation and maintenance procedures while giving the system the ability to dynamically schedule unplanned equipment, thus improving the model's practical applicability.

[0047] 3. A scenario-based probability-weighted load loss calculation method is introduced to improve the accuracy of risk assessment. Most existing methods use static load loss estimation, which cannot reflect the uncertainties brought about by different maintenance strategies during the evolution of ice storms. This solution uses π... ω (t) represents the probability of maintenance scenario ω occurring at time t, combined with R ω (t) Calculate the expected load loss. This probabilistic scenario-based modeling approach can effectively capture the randomness of the equipment state evolution path, avoid underestimating the risk due to ignoring extreme cases, and make the optimization results more robust and resistant to disturbances.

[0048] Preferably, in S3, the constraints of the toughness maintenance optimization model include:

[0049] Maintenance time constraints and maintenance resource constraints can be expressed as follows:

[0050]

[0051] In the formula, u k,t Let u be the maintenance state variable of device k at time t. k,t =0 indicates that device k requires maintenance, u k,t =1 indicates that device k is in operation; α k and β k Let N be the start and end times of maintenance for device k, respectively; N is the total number of devices; and ε(t) is the number of maintenance teams available to perform actions at time t.

[0052] Current constraints:

[0053]

[0054] Among them, Ω B Ω L and Ω G Represent the power system node set, transmission line set, and generator set set, respectively; P i,t Q i,t Let P be the active and reactive power of node i at time t; g,t N represents the active power output of generator g at node i at time t.B P represents the total number of busbar nodes. id,t Let ΔP be the active power demand of node i at time t. id,t It is the active power loss of node i at time t; U i,t θ is the voltage at node i at time t; i,t It is the voltage phase angle of node i at time t; G ij and B ij These are the conductance and magnetic reluctance of the transmission line (i, j), respectively.

[0055] u l,t and u g,t These are binary variables, representing whether the node and generator are in normal operating condition, respectively, with a value of 1 when they are in normal condition. and P g,t P respectively g,t The upper and lower limits of Q; g,t Let be the reactive power output of generator g at node i at time t; and Q g,t Q g,t The upper and lower limits of Q; id,t Let ΔQ be the reactive power demand of node i at time t. id,t P represents the reactive power loss of node i at time t. ij,t Let be the active power transmission of line (i, j) at time t; The thermal limit capacity of the line; θ i,t Let be the voltage phase angle of node i at time t; and θ i,t θ i,t The upper and lower limits; and U i,t U i,t The upper and lower limits.

[0056] This setup, encompassing multiple dimensions such as time, resources, and electrical operation, forms a well-structured and logically rigorous mixed-integer nonlinear programming problem. Despite its complexity, each part can be analytically expressed, making it suitable for efficient solving using advanced solvers. Compared to heuristic methods lacking physical constraints, this approach generates solutions with greater global optimum.

[0057] Preferably, in S3, the constraints of the toughness maintenance optimization model also include:

[0058]

[0059] In the formula, N1 represents the number of lines in the power grid; A(t) is the temperature of the j-th line connector at time t; the time constant ΔT A,jT represents the duration for which the temperature of the j-th line connector exceeds the limit; max,j It is the upper limit of the temperature of the line connector; t0 is More than T max,j The first time period;

[0060] After the power supply is transferred, when the line joint temperature no longer exceeds the limit, the line joint temperature rise constraint in the power supply optimization model is:

[0061]

[0062] In the formula, The temperature of the j-th line connector before the temperature rise due to load change at time t; This represents the temperature rise of the j-th line connector at time t due to load changes.

[0063] This approach addresses several issues. Firstly, existing power system resilience assessments often focus on macroscopic indicators like equipment failure rates and power flow exceedances, neglecting physical nodes like line joints that are susceptible to icing and current surges. Secondly, this solution introduces temperature rise duration and recovery constraints to quantitatively model the thermal stress process at joints, significantly improving the ability to identify failure modes with "small components, big impact" characteristics and enhancing overall system safety.

[0064] 2. The "duration threshold" mechanism effectively distinguishes between transient disturbances and actual overheating risks. During ice storms, load fluctuations may cause a short-term increase in joint temperature, but this does not necessarily constitute permanent damage. Traditional methods might treat such fluctuations as a danger signal and trigger an emergency response, resulting in wasted resources. This solution sets a ΔT threshold... A,j As a minimum duration threshold, the constraint is only applied when the joint continuously exceeds the temperature for a specified duration, thereby avoiding overreaction to short-term fluctuations and improving the robustness and rationality of decision-making.

[0065] Preferably, in S4, a gradient-type nonlinear programming solver is invoked to solve the resilience maintenance optimization model. Attached Figure Description

[0066] To make the objectives, technical solutions, and advantages of the invention clearer, the invention will now be described in further detail with reference to the accompanying drawings, wherein:

[0067] Figure 1 This is a flowchart of the method;

[0068] Figure 2 This is a schematic diagram of the availability variation curves of G1 and L1-5 in Example 2;

[0069] Figure 3 This is a schematic diagram of the system load change curve in Example 2. Detailed Implementation

[0070] The following detailed explanation illustrates the specific implementation methods:

[0071] Example 1

[0072] like Figure 1 As shown in the figure, this embodiment discloses a power system resilience maintenance method based on multi-stage states under ice storms, including the following steps:

[0073] S1. Based on the cumulative effect of ice storms, a time-varying continuous Markov state transition model is established for the source-grid-load side equipment in the power system to characterize the multi-stage evolution process of the equipment from normal state through moderately deteriorated state, severely deteriorated state to fault state during ice storms. The state transition intensity of the Markov model is a function of time, used to reflect the dynamic changes of equipment failure rate and repair rate with the duration of ice storms. The Markov model also introduces an emergency maintenance state to characterize the controlled outage state that the equipment enters due to emergency maintenance when it is in a severely deteriorated state.

[0074] In practice, the state of electrical equipment is represented as a set. Where the subscript k represents device k, Indicates a normal state. This indicates mild degradation. This indicates severe degradation. Indicates a fault; Indicates an emergency maintenance status;

[0075] Time-varying state probability of device k during an ice storm

[0076] During the study, the continuous Markov process was discretized into small intervals Δt. Given the device state... The probability of occurrence at time t can be calculated as described above, with a time step of 15 minutes. The state probabilities of these devices remain stable within 15-minute intervals. After 15 minutes, the state probabilities of the devices are updated once and need to be recalculated based on a Markov process. Over time, the probability of the device being in a faulty state increases.

[0077] Different types of equipment (such as generators, transmission lines, and load nodes) exhibit different degradation patterns under ice storms. This solution adopts a general state set approach, facilitating unified modeling of various equipment types while allowing adjustment of transfer strength parameters based on specific equipment characteristics. This modular design enables the method to be extended to complex power systems and is applicable to resilience analysis under various disaster scenarios.

[0078] S2. Divide the equipment to be repaired in the system into two categories: those with a preset fixed pre-maintenance time T. pre(a) Pre-maintenance equipment a, and non-pre-maintenance equipment b for which no fixed pre-maintenance time is set;

[0079] For each device, three possible state evolution paths are defined during the ice storm: Path 1: The device undergoes emergency maintenance at time T. D The device is determined to be in a severely degraded state, triggering emergency maintenance, and enters emergency maintenance mode; Path Two: The device does not trigger Path One, but enters emergency maintenance mode within the pre-maintenance time T. pre A sudden malfunction occurred before arrival; Path 3: The equipment did not trigger Path 1, and was in T pre No malfunctions occurred before arrival, thus the pre-maintenance task was performed as planned;

[0080] Based on the Markov state transition model constructed by S1, the occurrence probabilities of the three paths for each device are calculated. Based on the three paths and their corresponding occurrence probabilities, the expected availability function of the pre-maintenance device a and the expected availability function of the non-pre-maintenance device b are constructed respectively.

[0081] Among them, the probability of path one occurring is the probability of the device triggering emergency maintenance at time T. D The probability of the device entering a severely degraded state for the first time; the probability of path two occurring is that the device did not trigger path one, but occurred within the pre-maintenance time T. pre The probability of entering a fault state for the first time; the probability of path three occurring is 1 minus the sum of the probabilities of path one and path two occurring.

[0082] In practical implementation, the expected availability function of the pre-maintenance device a is:

[0083]

[0084]

[0085] In the formula, A m (a,t) represents the expected availability of pre-maintenance equipment a; The expected availability of the three paths for pre-maintenance device a is as follows; The probabilities of occurrence for the three paths of pre-maintenance device a are respectively;

[0086] This represents the probability that the device first enters a severely degraded state before time t. Let be the cumulative distribution function of the device when it first enters a severely degraded state before time u; This is a conditional availability function for equipment under emergency maintenance. This is a conditional availability function for devices in a faulty state. Let be the conditional availability function of the equipment under normal conditions; u is the integral variable, representing the specific moment when the equipment first enters a severely degraded state; Indicates the device is in tT D (a) The probability of first entering a fault state before time; d represents the cumulative distribution function of the device when it first enters a fault state before time u; m The duration of the pre-maintenance task.

[0087] Traditional methods often employ static probability or simple exponential decay models to estimate availability, failing to capture the impact of "when degradation occurs" on subsequent performance. This approach couples the probability of equipment entering a severely degraded state at different times with its subsequent operability through an integral term. This design allows the model to not only predict "whether a failure will occur" but also quantify "when degradation will affect power supply capacity and for how long," thus achieving a more refined risk assessment. Furthermore, the model transforms the state transition strengths (such as failure rate and repair rate) in the Markov process into a specific availability function A. m (a,t) is then integrated into total availability through path probability weighting. This not only completes the transformation from "physical process to mathematical expression" but also provides direct input for load loss cost calculation in S3, achieving seamless modeling from equipment-level degradation to system-level resilience indicators. Furthermore, for pre-maintenance equipment a, existing strategies often rely solely on a fixed T. pre (a) Execution and maintenance lack the ability to respond to real-time status. This solution addresses this by setting T... D (a) As an emergency maintenance trigger threshold, it allows T pre (a) Initiate maintenance in advance. This two-layer control mechanism enables the system to maintain its planned operation while also having the flexibility to cope with accelerated degradation, avoiding the risk of cascading failures caused by delayed maintenance.

[0088] The expected availability function of the constructed non-pre-maintenance device b is:

[0089]

[0090]

[0091] In the formula, A nm (b,t) represents the expected availability of non-pre-maintenance equipment b; and The expected availability of the three paths for non-pre-maintenance equipment (b) is as follows; The probabilities of the three paths for non-pre-maintenance equipment b are respectively;

[0092] This represents the probability that the device first enters a severely degraded state before time t. The cumulative distribution function represents the device's first entry into a severely degraded state before time u; A conditional availability function representing equipment under emergency maintenance conditions; This function represents the conditional availability of a device when it is in a normal state.

[0093] This scheme will use T pre (b) A "virtual but executable" planned maintenance time point was constructed as the optimization variable. This means that even if the equipment did not originally have a preset maintenance time, the system can dynamically determine the optimal maintenance time based on its degradation trend and system impact weight, thereby transforming passive response into proactive prevention and control, significantly improving the overall system resilience. Furthermore, although equipment b has no preset maintenance time, its availability function structure is completely consistent with that of equipment a: both adopt a three-path partitioning, probability weighting, and convolution integral form. This unified mathematical framework avoids model inconsistencies caused by different equipment classifications, enabling various types of equipment on the source-grid-load side to be compared and optimized under the same evaluation system, enhancing the overall coordination and scalability of the method.

[0094] S3. Construct a resilience maintenance optimization model with the goal of minimizing total cost; total cost includes load loss cost, pre-maintenance cost, and emergency maintenance cost; among which, load loss cost is calculated based on the equipment unavailability probability determined by the expected availability function constructed in S2;

[0095] The decision variables for the optimization model include the emergency maintenance trigger time T of the pre-maintenance device a. D (a) Pre-maintenance time T of non-pre-maintenance equipment b pre (b) and emergency maintenance trigger time T D (b)

[0096] In practical implementation, the objective function of the resilience maintenance optimization model is:

[0097]

[0098] In the formula, N m N represents the number of pre-maintenance devices. nm The number of non-maintenance equipment; R m (a) R nm (b) Repair costs for pre-maintenance equipment and non-pre-maintenance equipment, respectively; R C Cost of load loss.

[0099]

[0100] In the formula, These are the expected pre-maintenance costs, expected emergency maintenance costs, and expected failure maintenance costs for the pre-maintenance equipment, respectively. These represent the expected pre-maintenance costs, expected emergency maintenance costs, and expected failure maintenance costs for non-pre-maintenance equipment, respectively; Ωm (t) represents the maintenance scenario set at time t, including pre-maintenance, emergency maintenance, and no maintenance scenarios; π ω (t) is the probability of maintenance scenario ω occurring at time t (i.e., 1 minus the corresponding expected availability function); and R ω (t) is the load loss associated with time t and maintenance scenario ω.

[0101] Existing maintenance strategies typically focus only on maintenance costs or the reliability of individual devices, neglecting the impact of load loss at the system level. This solution innovatively incorporates pre-maintenance costs, emergency maintenance costs, fault maintenance costs, and expected load loss costs into the objective function, forming a multi-objective integrated cost structure. This allows the optimization process to consider not only the economic costs of "whether to repair" and "when to repair," but also the socio-economic losses caused by "power outages due to non-repair," thus achieving optimal overall benefits with limited resources. Furthermore, traditional optimization models often assume all equipment has the same maintenance pattern, making it difficult to handle situations where "planned maintenance" and "unplanned maintenance" coexist in actual operation and maintenance. This solution defines R... m (a) and R nm (b) This approach allows for independent modeling of both components in terms of cost, respecting existing operational procedures while also empowering the system with the ability to dynamically schedule unplanned equipment, thus enhancing the model's practical applicability. Furthermore, most existing methods employ static load loss estimation, failing to reflect the uncertainties arising from different maintenance strategies during the ice storm evolution process. This solution utilizes π... ω (t) represents the probability of maintenance scenario ω occurring at time t, combined with R ω (t) Calculate the expected load loss. This probabilistic scenario-based modeling approach can effectively capture the randomness of the equipment state evolution path, avoid underestimating the risk due to ignoring extreme cases, and make the optimization results more robust and resistant to disturbances.

[0102] The constraints of the toughness maintenance optimization model include:

[0103] Maintenance time constraints and maintenance resource constraints can be expressed as follows:

[0104]

[0105]

[0106] In the formula, u k,t Let u be the maintenance state variable of device k at time t. k,t =0 indicates that device k requires maintenance, u k,t =1 indicates that device k is in operation; α k and β kLet N be the start and end times of maintenance for device k, respectively; N is the total number of devices; and ε(t) is the number of maintenance teams available to perform actions at time t.

[0107] Current constraints:

[0108]

[0109] Among them, Ω B Ω L and Ω G Represent the power system node set, transmission line set, and generator set set, respectively; P i,t Q i,t Let P be the active and reactive power of node i at time t; g,t N represents the active power output of generator g at node i at time t. B P represents the total number of busbar nodes. id,t Let ΔP be the active power demand of node i at time t. id,t It is the active power loss of node i at time t; U i,t θ is the voltage at node i at time t; i,t It is the voltage phase angle of node i at time t; G ij and B ij These are the conductance and magnetic reluctance of the transmission line (i, j), respectively.

[0110] u l,t and u g,t These are binary variables, representing whether the node and generator are in normal operating condition, respectively, with a value of 1 when they are in normal condition. and P g,t P respectively g,t The upper and lower limits of Q; g,t Let be the reactive power output of generator g at node i at time t; and Q g,t Q g,t The upper and lower limits of Q; id,t Let ΔQ be the reactive power demand of node i at time t. id,t P represents the reactive power loss of node i at time t. ij,t Let be the active power transmission of line (i, j) at time t; The thermal limit capacity of the line; θ i,t Let be the voltage phase angle of node i at time t; and θ i,t θ i,t The upper and lower limits; and U i,t U i,t The upper and lower limits.

[0111] The constraints of the toughness maintenance optimization model also include:

[0112]

[0113] In the formula, N1 represents the number of lines in the power grid; A(t) is the temperature of the j-th line connector at time t; the time constant ΔT A,j T represents the duration for which the temperature of the j-th line connector exceeds the limit; max,j It is the upper limit of the temperature of the line connector; t0 is More than T max,j The first time period;

[0114] After the power supply is transferred, when the line joint temperature no longer exceeds the limit, the line joint temperature rise constraint in the power supply optimization model is:

[0115]

[0116] In the formula, The temperature of the j-th line connector before the temperature rise due to load change at time t; This represents the temperature rise of the j-th line connector at time t due to load changes.

[0117] Current power system resilience assessments primarily focus on macroscopic indicators such as equipment failure rates and power flow exceedances, neglecting physical nodes like line joints that are susceptible to icing and current surges. This solution, by introducing temperature rise duration and recovery constraints, achieves quantitative modeling of the thermal stress process at joints, significantly improving the ability to identify failure modes with "small components, big impacts" and enhancing overall system safety. Furthermore, during ice storms, load fluctuations may cause short-term increases in joint temperature, but this does not necessarily constitute permanent damage. Traditional methods might interpret such fluctuations as danger signals and trigger emergency responses, resulting in wasted resources. This solution sets ΔT... A,j As a minimum duration threshold, the constraint is only applied when the joint continuously exceeds the temperature for a specified duration, thereby avoiding overreaction to short-term fluctuations and improving the robustness and rationality of decision-making.

[0118] S4. Use a gradient-based nonlinear programming solver to solve the resilience maintenance optimization model and obtain the optimal T. D *(a), T pre *(b) and T D *(b) Generate an offline resilience maintenance plan;

[0119] S5. During the actual evolution of the ice storm, monitor the equipment status in real time and dynamically determine the path the equipment is on based on the three state evolution paths defined in S2 and their corresponding triggering conditions. When the equipment status is detected to meet the triggering conditions of path one or path two, immediately execute the corresponding emergency maintenance operation.

[0120] Traditional maintenance methods typically rely on static failure rates or simple binary states (normal / failed) for decision-making, failing to reflect the dynamic pattern of gradual equipment performance degradation during prolonged ice storms. This solution, based on the cumulative effect of ice storms, establishes a continuous Markov model where state transition intensity is a function of time, accurately describing the natural evolution of equipment from normal → moderate degradation → severe degradation → failure. Simultaneously, it clearly delineates three state evolution paths covering all possible scenarios: Path 1 (triggered emergency maintenance), Path 2 (sudden failure), and Path 3 (planned pre-maintenance). This path-based modeling not only reflects the time-varying risk of equipment status but also provides a probabilistic basis for differentiated response strategies, significantly outperforming the passive "one-size-fits-all" or reactive modes of existing technologies. Furthermore, existing methods often rely on fixed thresholds or empirical rules to trigger emergency repairs, easily leading to response delays or resource waste. This solution will... D (a) and T D (b) Incorporating T as a key decision variable into the optimization model: For equipment a with scheduled pre-maintenance, T D (a) This determines whether to intervene before scheduled maintenance—if monitoring shows accelerated deterioration, emergency maintenance can be initiated at the optimal time to avoid cascading failures caused by operation with defects; for equipment b without scheduled pre-maintenance, T D The introduction of (b) enables it to possess "dynamic emergency response capabilities," meaning that although there is no fixed plan, precise intervention can be achieved by setting personalized trigger points. Together, these two constitute a "prevention-emergency" collaborative mechanism, significantly enhancing the system's proactive defense capabilities against high-risk equipment.

[0121] Although non-pre-maintenance equipment b (such as some power distribution lines and load nodes) does not undergo pre-maintenance in practice, T will be used for pre-maintenance. pre (b) Incorporating it as a virtual decision variable into the model has significant value: on the one hand, it maintains structural consistency in the availability function expression between pre-maintenance and non-pre-maintenance equipment, facilitating overall optimization; on the other hand, the optimized T... D *(b) can serve as a reference indicator for the "potential optimal maintenance window," used to assess whether high-risk equipment should be upgraded to pre-maintenance status. This design breaks through the limitations of the rigid "maintenance / non-maintenance" classification in traditional methods, providing quantitative support for prioritization under limited resources. Furthermore, by clarifying the mechanism by which equipment transitions from a severely deteriorated state to an emergency maintenance state in "Path One," precise intervention before failure is achieved. The core of "Path One" lies in: when the equipment is in T... DWhen a device is determined to be in a severely degraded state, emergency maintenance is immediately triggered, and the device enters an independent "emergency maintenance state." This design differs from existing technologies that vaguely equate "severe degraded" with "imminent failure." By explicitly defining a "controlled shutdown state," it achieves: modeling of the maintenance action itself (such as repair time and resource consumption); precise quantification of equipment availability during maintenance (through conditional availability functions); and a strict distinction between "intervention-friendly high-risk states" and "irreversible failures."

[0122] This method enables refined and proactive maintenance and scheduling of key equipment on the source-grid-load side.

[0123] Example 2

[0124] To better understand this method, the following comparative examples are provided.

[0125] In this example, three methods are used for repair.

[0126] Method 1 (Case 1): The maintenance plan is fixed, and the pre-defined maintenance plan does not change with the equipment status.

[0127] Method 2 (Case 2): Using a maintenance scheduling model, but P D The probability of triggering emergency maintenance is set to a fixed value of 0.6.

[0128] Method 3 (Case 3): This is the method described above.

[0129] like Figure 2 As shown below, the changes in device availability under these three scenarios are compared.

[0130] 1) Equipment availability varies significantly at different times, with L1-5 having a higher failure rate than G1, resulting in greater availability fluctuations during the scheduling period.

[0131] 2) Compared to Case 1, the availability in cases 2 and 3 at time T pre The equipment is in a higher condition because it requires emergency maintenance in the early stages, and the equipment is in better condition when pre-maintenance is performed.

[0132] 3) Compared to Case 1, Case 2 and Case 3 showed rapid increases around 26 days and 36 days, respectively, indicating that T... D This is because the equipment condition improved after emergency maintenance was implemented, thus significantly increasing equipment availability.

[0133] Overall, Case 3 has higher device availability compared to Case 2, suggesting that optimization of T... DThis further improves availability. Whether using pre-maintenance or non-pre-maintenance equipment, performing emergency repairs within an appropriate timeframe based on changes in equipment condition can effectively improve equipment availability.

[0134] Table 1. Estimated Total Cost of System Failure and Equipment Repair

[0135]

[0136] Table 2 Maintenance and Optimization Plan

[0137]

[0138] Tables 1 and 2 show the total power system load loss cost and equipment maintenance cost, along with the maintenance optimization results, for three different scenarios. It should be noted that the pre-maintenance time shown in Table 2 is not fixed but is set according to the operation and maintenance strategies adopted in different cases. For example, Case 1 uses an earlier pre-maintenance time to reduce the risk of failure, Case 2 adopts a more conservative time arrangement, while Case 3 further optimizes the emergency maintenance triggering timing while maintaining the original pre-maintenance plan. By comparing the total cost under different pre-maintenance times, it can be verified that the present invention can effectively improve system resilience under various initial conditions.

[0139] "M" indicates maintenance, and "F" indicates no maintenance. The optimized repair time T is shown in case 3. D As shown in Table 3. The system load curve is as follows. Figure 3 As shown.

[0140] Table 3 Optimal Emergency Maintenance Time

[0141]

[0142] The results for Cases 1-2 in Table 1 show a significant reduction in total maintenance costs and system load loss, indicating that real-time correlation between maintenance scheduling and equipment status can effectively improve equipment reliability and system resilience. Since a small portion of pre-maintenance is canceled due to equipment failure, redundant maintenance is reduced, resulting in lower pre-maintenance costs for Case 2 compared to Case 1. Furthermore, because maintenance tasks for pre-pending equipment can be proactively advanced in Case 2, equipment failure maintenance costs are significantly lower than in Case 1. Regarding non-prepaid maintenance costs, although Case 2 involves additional maintenance costs, failure maintenance costs are significantly reduced. In terms of system resilience, the load loss cost of Case 2 is significantly lower than in Case 1 because equipment availability is greatly improved in Case 2. Figure 3 In Case 3, the system resilience was improved compared to Case 2. This indicates that optimizing the emergency maintenance time T... D Improving system resilience from this perspective is an effective strategy.

[0143] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and not to limit the technical solutions. Those skilled in the art should understand that any modifications or equivalent substitutions to the technical solutions of the present invention without departing from the spirit and scope of the present invention should be covered within the scope of the claims of the present invention.

Claims

1. A method for resilience maintenance of power systems under ice storms based on multi-stage states, characterized in that, Includes the following steps: S1. Based on the cumulative effect of ice storms, a time-varying continuous Markov state transition model is established for the source-grid-load side equipment in the power system to characterize the multi-stage evolution process of the equipment from normal state through moderately deteriorated state, severely deteriorated state to fault state during ice storms; wherein, the state transition intensity of the Markov model is a function of time, used to reflect the dynamic changes of equipment failure rate and repair rate with the duration of ice storms; the Markov model also introduces an emergency maintenance state to characterize the controlled outage state that the equipment enters due to emergency maintenance when it is in a severely deteriorated state. S2. Divide the equipment to be repaired in the system into two categories: those with a preset fixed pre-maintenance time T. pre (a) Pre-maintenance equipment a, and non-pre-maintenance equipment b for which no fixed pre-maintenance time is set; For each device, three possible state evolution paths are defined during the ice storm: Path 1: The device undergoes emergency maintenance at time T. D The device is determined to be in a severely degraded state, triggering emergency maintenance, and enters emergency maintenance mode; Path Two: The device does not trigger Path One, but enters emergency maintenance mode within the pre-maintenance time T. pre A sudden malfunction occurred before arrival; Path 3: The equipment did not trigger Path 1, and was in T pre No malfunctions occurred before arrival, thus the pre-maintenance task was performed as planned; Based on the Markov state transition model constructed by S1, the occurrence probabilities of the three paths for each device are calculated. Based on the three paths and their corresponding occurrence probabilities, the expected availability function of the pre-maintenance device a and the expected availability function of the non-pre-maintenance device b are constructed respectively. S3. Construct a resilience maintenance optimization model with the goal of minimizing total cost; total cost includes load loss cost, pre-maintenance cost, and emergency maintenance cost; among which, load loss cost is calculated based on the equipment unavailability probability determined by the expected availability function constructed in S2; The decision variables for the optimization model include the emergency maintenance trigger time T of the pre-maintenance device a. D (a) Pre-maintenance time T of non-pre-maintenance equipment b pre (b) and emergency maintenance trigger time T D (b); S4. Solve the toughness maintenance optimization model to obtain the optimal T. D *(a), T pre *(b) and T D *(b) Generate an offline resilience maintenance plan; S5. During the actual evolution of the ice storm, monitor the equipment status in real time and dynamically determine the path the equipment is on based on the three state evolution paths defined in S2 and their corresponding triggering conditions. When the equipment status is detected to meet the triggering conditions of path one or path two, immediately execute the corresponding emergency maintenance operation.

2. The power system resilience maintenance method based on multi-stage states under ice storms as described in claim 1, characterized in that: The probability of path one occurring is the time T during emergency maintenance triggering the device. D The probability of the device entering a severely degraded state for the first time; the probability of path two occurring is that the device did not trigger path one, but occurred within the pre-maintenance time T. pre The probability of entering a fault state for the first time; the probability of path three occurring is 1 minus the sum of the probabilities of path one and path two occurring.

3. The power system resilience maintenance method based on multi-stage states under ice storms as described in claim 1, characterized in that: In S1, the state of electrical equipment is represented as a set. Where the subscript k represents device k, Indicates a normal state. This indicates mild degradation. This indicates severe degradation. Indicates a fault; Indicates an emergency maintenance status; Time-varying state probability of device k during an ice storm 4. The power system resilience maintenance method based on multi-stage states under ice storms as described in claim 1, characterized in that: The expected availability function of the pre-maintenance device a constructed in S2 is: In the formula, A m (a,t) represents the expected availability of pre-maintenance equipment a; The expected availability of the three paths for pre-maintenance device a is as follows; The probabilities of occurrence for the three paths of pre-maintenance device a are respectively; This represents the probability that the device first enters a severely degraded state before time t. Let be the cumulative distribution function of the device when it first enters a severely degraded state before time u; This is a conditional availability function for equipment under emergency maintenance. This is a conditional availability function for devices in a faulty state. Let be the conditional availability function of the equipment under normal conditions; u is the integral variable, representing the specific moment when the equipment first enters a severely degraded state; Indicates the device is in tT D (a) The probability of first entering a fault state before time; d represents the cumulative distribution function of the device when it first enters a fault state before time u; m The duration of the pre-maintenance task.

5. The power system resilience maintenance method based on multi-stage states under ice storms as described in claim 4, characterized in that: In S2, the expected availability function of the non-pre-maintenance device b is: In the formula, A nm (b,t) represents the expected availability of non-pre-maintenance equipment b; and The expected availability of the three paths for non-pre-maintenance equipment (b) is as follows; The probabilities of the three paths for non-pre-maintenance equipment b are respectively; This represents the probability that the device first enters a severely degraded state before time t. The cumulative distribution function represents the device's first entry into a severely degraded state before time u; A conditional availability function representing equipment under emergency maintenance conditions; This function represents the conditional availability of a device when it is in a normal state.

6. The power system resilience maintenance method based on multi-stage states under ice storms as described in claim 5, characterized in that: In S3, the objective function of the toughness maintenance optimization model is: In the formula, N m N represents the number of pre-maintenance devices. nm The number of non-maintenance equipment; R m (a) R nm (b) Repair costs for pre-maintenance equipment and non-pre-maintenance equipment, respectively; R C Cost of load loss.

7. The power system resilience maintenance method based on multi-stage states under ice storms as described in claim 6, characterized in that: In the formula, These are the expected pre-maintenance costs, expected emergency maintenance costs, and expected failure maintenance costs for the pre-maintenance equipment, respectively. These represent the expected pre-maintenance costs, expected emergency maintenance costs, and expected failure maintenance costs for non-pre-maintenance equipment, respectively; Ω m (t) represents the maintenance scenario set at time t, including pre-maintenance, emergency maintenance, and no maintenance scenarios; π ω (t) is the probability that maintenance scenario ω will occur at time t; and R ω (t) is the load loss associated with time t and maintenance scenario ω.

8. The power system resilience maintenance method based on multi-stage states under ice storms as described in claim 1, characterized in that: In S3, the constraints of the toughness maintenance optimization model include: Maintenance time constraints and maintenance resource constraints can be expressed as follows: In the formula, u k,t Let u be the maintenance state variable of device k at time t. k,t =0 indicates that device k requires maintenance, u k,t =1 indicates that device k is in operation; α k and β k Let N be the start and end times of maintenance for device k, respectively; N is the total number of devices; and ε(t) is the number of maintenance teams available to perform actions at time t. Current constraints: Among them, Ω B Ω L and Ω G Represent the power system node set, transmission line set, and generator set set, respectively; P i,t Q i,t Let P be the active and reactive power of node i at time t; g,t N represents the active power output of generator g at node i at time t. B P represents the total number of busbar nodes. id,t Let ΔP be the active power demand of node i at time t. id,t It is the active power loss of node i at time t; U i,t θ is the voltage at node i at time t; i,t It is the voltage phase angle of node i at time t; G ij and B ij These are the conductance and magnetic reluctance of the transmission line (i, j), respectively. u l,t and u g,t These are binary variables, representing whether the node and generator are in normal operating condition, respectively, with a value of 1 when they are in normal condition. and P g,t P respectively g,t The upper and lower limits of Q; g,t Let be the reactive power output of generator g at node i at time t; and Q g,t Q g,t The upper and lower limits of Q; id,t Let ΔQ be the reactive power demand of node i at time t. id,t P represents the reactive power loss of node i at time t. ij,t Let be the active power transmission of line (i, j) at time t; The thermal limit capacity of the line; θ i,t Let be the voltage phase angle of node i at time t; and i i,t θ i,t The upper and lower limits; and U i,t U i,t The upper and lower limits.

9. The power system resilience maintenance method based on multi-stage states under ice storms as described in claim 8, characterized in that: In S3, the constraints of the toughness maintenance optimization model also include: In the formula, N1 represents the number of lines in the power grid; A(t) is the temperature of the j-th line connector at time t; the time constant ΔT A,j T represents the duration for which the temperature of the j-th line connector exceeds the limit; max,j It is the upper limit of the temperature of the line connector; t0 is More than T max,j The first time period; After the power supply is transferred, when the line joint temperature no longer exceeds the limit, the line joint temperature rise constraint in the power supply optimization model is: In the formula, The temperature of the j-th line connector before the temperature rise due to load change at time t; This represents the temperature rise of the j-th line connector at time t due to load changes.

10. The power system resilience maintenance method based on multi-stage states under ice storms as described in claim 1, characterized in that: In S4, the gradient-type nonlinear programming solver is called to solve the resilience maintenance optimization model.

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