A High-Speed ​​Railway Toughness Assessment Method Based on Markov Model

Abstract

This invention discloses a high-speed rail resilience assessment method based on Markov models, belonging to the field of high-speed rail assessment technology. The method includes collecting and recording state change data of high-speed rail in different regions under different disaster influences to estimate the initial distribution vector, transition probability matrix, transition rate matrix, and dwell time distribution; defining two resilience assessment indices: inherent resilience and acquired resilience. Inherent resilience is the probability that the time taken for the high-speed rail system to first enter a fully recovered state is earlier than the failure time, and acquired resilience is the probability that the high-speed rail system is in normal operation at any given time; based on the state change data and the type of Markov model followed by the high-speed rail state changes, the corresponding inherent resilience and acquired resilience are calculated. This invention comprehensively considers the evolution path of high-speed rail performance under different environments. By introducing two indices, inherent resilience and acquired resilience, and combining multiple Markov models, it achieves a quantitative assessment of the high-speed rail resilience level.

Description

Technical Field

[0001] This invention relates to the field of high-speed railway evaluation technology, specifically a high-speed railway resilience evaluation method based on a Markov model. Background Technology

[0002] High-speed rail systems may be affected by disasters such as blizzards, earthquakes, torrential rains, and human-caused damage. These disasters can cause serious damage to the track structure and even lead to the failure of the entire system, severely disrupting the safe and stable operation of high-speed railways. Therefore, enhancing the resilience of high-speed rail transportation systems and ensuring that they can quickly maintain and restore their functionality and operational capabilities when encountering various external shocks and disturbances is of paramount importance.

[0003] Resilience assessment is fundamental to resilience research, aiming to evaluate the response and coping capabilities of existing systems to disruptive events. Existing research can be divided into qualitative and quantitative studies. Qualitative studies focus on analyzing the influencing factors of complex system resilience and derive system resilience through weighted summation, but rely heavily on subjective judgment and lack objective and precise quantitative evaluation. Quantitative studies encompass methods based on performance curves, topology, and characteristics, but often focus on a single aspect, with relatively singular measurement indicators, and cannot effectively quantitatively assess the resilience level of high-speed rail.

[0004] In particular, the performance changes of high-speed trains can be discrete. For example, the signaling system may switch from normal operation to a fault state instantaneously, without an intermediate, continuous, gradual performance change process. This discreteness makes traditional resilience assessment methods based on continuous performance change curves difficult to apply accurately. On the other hand, using static characteristics such as nodal centrality or a certain resilience attribute as a single measure ignores the characteristic changes in the resilience of high-speed trains during disturbances, making it difficult to determine the resilience level of high-speed trains at a certain stage of damage.

[0005] Therefore, there is an urgent need for a technical method to quantitatively assess the resilience level of high-speed rail. This method should be applicable to both continuous and discrete performance variations and be able to assess the resilience level of high-speed rail at various stages of exposure to extreme events. Summary of the Invention

[0006] To address the problems existing in the current technology, this invention proposes a high-speed rail toughness assessment method and system based on Markov models. This invention comprehensively considers the evolution path of high-speed rail performance under different environments. By introducing two indicators, inherent toughness and acquired toughness, and combining multiple Markov models, it achieves a quantitative assessment of the high-speed rail toughness level. This invention provides strong support for the safety risk management and toughness improvement of high-speed rail systems and effectively makes up for the shortcomings of existing research.

[0007] The technical solution of the present invention:

[0008] On the one hand, this invention provides a method for evaluating the toughness of high-speed railways based on a Markov model, the method comprising the following steps:

[0009] Collect and record data on the state changes of high-speed railways in different regions under the influence of different disasters in order to estimate the initial distribution vector, transition probability matrix, transition rate matrix and dwell time distribution;

[0010] Two resilience assessment indices are defined: inherent resilience and acquired resilience. Inherent resilience is the probability that the time taken for the high-speed rail system to enter a fully recovered state for the first time is earlier than the failure time. Acquired resilience is the probability that the high-speed rail system is in a normal operating state at any time.

[0011] Based on the type of Markov model that the high-speed rail system follows in terms of state changes, and combined with state change data, the inherent toughness and acquired toughness of the high-speed rail system are calculated.

[0012] Preferably, the Markov model types include discrete-time Markov chains, discrete-time semi-Markov chains, continuous-time Markov processes, and continuous-time semi-Markov processes.

[0013] Preferably, for discrete-time Markov chains, the following steps are used to calculate the inherent toughness and obtain the toughness:

[0014] Construct a discrete-time Markov chain model of the state changes of high-speed rail;

[0015] Estimate the transition probability matrix based on the collected state change data;

[0016] Using the transition probability matrix and the initial distribution vector, the probability distribution of the high-speed rail system under different states is calculated;

[0017] Based on the calculated probability distribution, calculate the inherent toughness and the acquired toughness.

[0018] As a preferred approach, a discrete-time semi-Markov chain model of high-speed rail performance variation is constructed.

[0019] Estimate the transition probability matrix and dwell time distribution based on the collected state change data;

[0020] Using the transition probability matrix, dwell time distribution, and initial distribution vector, the probability distribution of the high-speed rail system under different states is calculated;

[0021] Based on the calculated probability distribution, calculate the inherent toughness and the acquired toughness.

[0022] Preferably, for a continuous-time Markov process, the following steps are used to calculate the inherent toughness and obtain the toughness:

[0023] Construct a continuous-time Markov process model of the state changes of high-speed rail;

[0024] Estimate the transition rate matrix based on the collected state change data;

[0025] Using the transition rate matrix and the initial distribution vector, the probability distribution of the high-speed rail system under different states is calculated;

[0026] Based on the calculated probability distribution, calculate the inherent toughness and the acquired toughness.

[0027] Preferably, for a continuous-time semi-Markov process, the following steps are used to calculate the inherent toughness and obtain the toughness:

[0028] Construct a continuous-time semi-Markov process model of high-speed rail performance changes;

[0029] Estimate the transition rate matrix and dwell time distribution based on the collected state change data;

[0030] Using the transition rate matrix, dwell time distribution, and initial distribution vector, the probability distribution of the high-speed rail system under different states is calculated;

[0031] Based on the calculated probability distribution, calculate the inherent toughness and the acquired toughness.

[0032] On the other hand, the present invention provides a high-speed rail toughness assessment system based on a Markov model, the system comprising:

[0033] The data acquisition module is used to collect and record data on the status changes of high-speed railways in different regions under the influence of different disasters;

[0034] The resilience assessment index module is used to define two resilience assessments: inherent resilience and acquired resilience.

[0035] The toughness calculation module is used to calculate the inherent toughness and acquired toughness of the high-speed rail system based on the type of Markov model that the high-speed rail's state changes follow, combined with state change data.

[0036] The beneficial effects of this invention are as follows:

[0037] 1) Comprehensive consideration of performance evolution: This invention uses multiple Markov models to characterize the performance evolution process of high-speed rail under different environments, which can more accurately reflect the dynamic characteristics of high-speed rail toughness.

[0038] 2) Quantitative assessment of toughness indicators: By calculating inherent toughness and obtaining toughness indicators, this invention achieves a quantitative assessment of the toughness level of high-speed rail, providing a scientific basis for high-speed rail safety risk management.

[0039] 3) Wide range of applications: The method of the present invention is applicable not only to situations where the performance of high-speed rail changes continuously, but also to situations where it changes discretely, and has a wider range of application prospects. Attached Figure Description

[0040] To more clearly illustrate the specific embodiments of the present invention or the technical solutions in the prior art, the accompanying drawings used in the description of the specific embodiments or the prior art will be briefly introduced below. In all the drawings, similar elements or parts are generally identified by similar reference numerals. In the drawings, the elements or parts are not necessarily drawn to scale.

[0041] Figure 1 This is a flowchart of the method described in this invention;

[0042] Figure 2 To achieve toughness under Markov chains;

[0043] Figure 3 A comparison of analytical and simulation results of toughness obtained under Markov chains;

[0044] Figure 4 To achieve toughness under a semi-Markov chain;

[0045] Figure 5 A comparison chart of toughness analysis results and simulation results obtained under semi-Markov continuous stress;

[0046] Figure 6 To obtain resilience under Markov processes;

[0047] Figure 7 A comparison chart showing the toughness analysis results and simulation results obtained under the Markov process;

[0048] Figure 8 To obtain resilience for the Weibull distribution under a semi-Markov process;

[0049] Figure 9 A comparison of analytical and simulation results of the toughness of the Weibull distribution under a semi-Markov process;

[0050] Figure 10 To obtain resilience for the Gamma distribution under a semi-Markov process;

[0051] Figure 11 A comparison of analytical and simulation results of the toughness obtained from the Gamma distribution under a semi-Markov process;

[0052] Figure 12 Sensitivity analysis of the transition probability matrix for obtaining resilience under Markov chains;

[0053] Figure 13 Sensitivity analysis of the transition probability matrix for obtaining resilience in a semi-Markov chain.

[0054] Figure 14Sensitivity analysis of toughness to the transfer rate matrix is ​​performed for Markov and semi-Markov process systems. Detailed Implementation

[0055] The embodiments of the technical solution of the present invention will now be described in detail with reference to the accompanying drawings. These embodiments are merely illustrative of the technical solution of the present invention and are therefore intended to limit the scope of protection of the present invention.

[0056] It should be noted that, unless otherwise stated, the technical or scientific terms used in this application should have the ordinary meaning as understood by those skilled in the art to which this invention pertains.

[0057] Please refer to Figure 1 , Figure 1 A flowchart of a high-speed rail toughness assessment method based on a Markov model provided in this invention is included, comprising the following steps:

[0058] Step 1: Collect and record data on the state changes of high-speed railways in different regions under the influence of different disasters in order to estimate the initial distribution vector, transition probability matrix, transition rate matrix and stay time distribution.

[0059] The first step in high-speed rail resilience assessment is to extensively collect and record detailed state change data from high-speed rail systems in different regions under the influence of various disasters (including but not limited to natural disasters such as earthquakes, blizzards, and mudslides, as well as structural damage that may be caused by human-caused damage or daily wear and tear). This data is extremely important for subsequent accurate estimation of the initial distribution vector, transition probability matrix, transition rate matrix, and dwell time distribution of the high-speed rail system.

[0060] To accurately assess the resilience of the high-speed rail system, the system's state is divided into three subsets E = {E1, E2, E3}, where E1 represents the subset of normally operating states, E2 represents the subset of damaged states, and E3 represents the subset of completely damaged states. Each subset contains at least one specific state.

[0061] Normal operating state subset E1:

[0062] State 0 indicates that the high-speed rail is in normal operation. In this state, the track fasteners are intact, tightly and firmly fixing the rails to the sleepers, ensuring that the high-speed train can run smoothly and safely at the predetermined speed and trajectory.

[0063] Damaged state subset E2:

[0064] It includes four specific states, from state 1 to state 4, describing the high-speed rail as being damaged to varying degrees but not completely failed.

[0065] Status 1: The high-speed rail is in a state of slight wear. Although the track fasteners have undergone a period of use and wear, the degree of wear is still light and has no significant impact on the fasteners' fixing function and the overall stability of the track system. The high-speed train can still operate normally.

[0066] Status 2: The high-speed rail is in a state of loose fasteners. The track fasteners have become loose due to long-term external force or wear, which has led to a decrease in the stability of the rails. The high-speed train may experience slight shaking or deviation during operation.

[0067] State 3: The high-speed rail is in a state of fastener deformation. The track fasteners are deformed due to external impact under disaster conditions and can no longer effectively fix the rails. The high-speed train may experience significant shaking and deviation during operation, posing a potential threat to the stability and safety of the track system.

[0068] Status 4: The high-speed rail is in a state of severe damage but not yet failure. The track fasteners and their surrounding structures (such as sleepers and track slabs) have suffered significant damage. The fasteners may have severe deformation, cracks or even partial breakage, but the overall structure has not completely collapsed and still maintains the fixation of the rails to a certain extent.

[0069] Failure State Subset E3:

[0070] State 5 indicates that the high-speed rail is in a failure state. In this state, due to the failure of track fasteners or the impact of other serious disasters, the entire track system loses its function, high-speed trains cannot operate normally on any section of track, and the entire high-speed rail system is thus paralyzed.

[0071] By dividing and defining the above-mentioned state subsets in detail, we can more accurately describe the state change process of the high-speed rail system under the influence of disasters, providing a solid and reliable foundation for subsequent high-speed rail resilience assessment.

[0072] In this embodiment, the initial state of the high-speed rail system is defined as the damaged state after the disaster, i.e., E2, and its initial vector distribution is η. E2 =[η i .i∈E2], where η i Let F(t) represent the probability that the system is in state i, and let F(t) represent the state of the high-speed rail system at time t.

[0073] Step 2: Define two resilience assessment indicators: inherent resilience and acquired resilience. Inherent resilience is the probability that the time taken for the high-speed rail system to enter a fully recovered state for the first time is earlier than the failure time. Acquired resilience is the probability that the high-speed rail system is in a normal operating state at any time.

[0074] It possesses inherent toughness, defined as Re. I =P{T E1 <T E3}, where T E1 and T E3 These represent the moments when the high-speed rail system first enters states E1 and E3, respectively. Inherent resilience reflects the speed and efficiency with which the system recovers to normal operating conditions. Resilience is defined as Re... A F(t) = P{F(t)∈E1}, which is the probability that the high-speed rail system is in normal operation at time t, where t is any time. The resilience reflects the stability and reliability of the system in maintaining normal operation at any point in time.

[0075] Step 3: Based on the type of Markov model that the high-speed rail's state changes follow, and combined with the state change data, calculate the inherent toughness and acquired toughness of the high-speed rail system; wherein, the Markov model type includes discrete-time Markov chain, discrete-time semi-Markov chain, continuous-time Markov process, and continuous-time semi-Markov process.

[0076] Specifically, for discrete-time Markov chains, the following steps are used to calculate inherent toughness and acquired toughness: construct a discrete-time Markov chain model of high-speed rail state changes; estimate the transition probability matrix based on the collected state change data; use the transition probability matrix and the initial distribution vector to calculate the probability distribution of the high-speed rail system under different states; and calculate the inherent toughness and acquired toughness based on the calculated probability distribution.

[0077] More specifically, when the state changes of a high-speed train follow a discrete-time Markov chain, the analytical formulas for inherent toughness and acquired toughness are derived through the following process:

[0078] Assuming that under the influence of a disaster, the state evolution of the high-speed rail system over time follows the discrete-time homogeneous Markov chain principle. Let {Z} n}, This represents the state of the high-speed rail system at time n, where Let P represent the set of natural numbers. In this process, the system's state at the next step depends only on the current state. The evolution of a Markov chain is determined by the transition probability matrix P = [p...]. ij ] |E|×|E| Control, where |E| represents the number of state subsets, p ij =P{Z n+1 =j|Z n Let $\mathbf{i}$ represent the probability that the high-speed rail system, currently in state $i$ at time $n$, will transition to state $j$ at time $n+1$. Given that the high-speed rail system comprises three state sets, its transition probability matrix is... It will be presented in a specific form, specifically as follows:

[0079]

[0080] Wherein, the transition probability matrix Indicates from state set Ei Transition to state set E j The transition probability matrix (i,j=1,2,3).

[0081] According to the definition of inherent toughness: Given the initial performance state i∈E2 at time 0, let This represents the probability that, given a high-speed train is in state i at time 0, the time taken to return to normal operation is earlier than the failure time.

[0082]

[0083] All α i Write it as a column vector α = [α i ], i∈E2, Equation (1) can be written in matrix form as follows

[0084]

[0085] in, Let E1 be a |E1|-dimensional column vector with elements of 1. Therefore, we can obtain...

[0086]

[0087] Combined with the initial distribution of the system The expression for the inherent toughness measure of a high-speed rail system is:

[0088]

[0089] To study the acquired toughness of the high-speed rail system under a Markov chain, some important quantities are first defined. First, let {J} n ,S n}, It is a Markov update chain, where {J n}, It is the state of the system at the nth jump, {S n}, This is the time of the nth jump. Additionally, let {Y} n}, This indicates that the state J is in the state before the nth jump. n-1 The time spent in Y n =S n -S n-1 And Y0 = S0 = 0. The relevant semi-Markov kernel V(n) = [ν ij (n)] |E|×|E| Defined as ν ij (n)=P{J k+1 =j,S k+1 -S k =n|J k =i}=P{Jk+1 =j,Y k =n∣J k =i}, which represents the probability that the system is in state i at time k, in state j at time k+1, and stays in state i for a duration of n.

[0090] It is worth noting that a Markov chain with a state transition probability matrix can be viewed as a special case of a Markov update chain with a semi-Markov kernel. The semi-Markov kernel of a Markov chain is...

[0091]

[0092] in It is a set of positive integers.

[0093] Define a probability matrix in

[0094]

[0095] This indicates that, given an initial state i, the system remains in the subset E1 for a time interval [0, n), undergoes l transitions, and finally enters state j at time n. Clearly, It equals 1 only when n = 0, otherwise it equals 0. Based on the law of total probability, for We have

[0096]

[0097] Write equation (3) in matrix form Where ★ represents the matrix convolution operator. This represents the probability that the system remains in state set E1 after l-1 transitions. Let represent the probability that the system transitions from state set E1 and stays in each state for a duration of n. For l ≥ 1, we have

[0098]

[0099] in, They are The l-1 and l-fold convolutions. Furthermore, considering all possible values ​​of l and summing them, we obtain... in

[0100]

[0101] Written in matrix form as Obviously,

[0102] When n = 0, we have

[0103]

[0104] Therefore, we can conclude that...

[0105] When n > 0, we have

[0106]

[0107] Performing a Z-transform on both sides of equation (4) yields...

[0108]

[0109] Next, the probability matrix is ​​introduced. in

[0110]

[0111] Let represent the probability that, starting from state i, l transitions have been completed within E1, and at time n, the system is in state j and has not yet undergone the (l+1)th transition. We have

[0112]

[0113] in Indicates that the diagonal elements are A diagonal matrix.

[0114] Taking the Z-transform of both sides of equation (6), we can obtain

[0115]

[0116] Writing equation (7) in matrix form, we have

[0117]

[0118] It is worth noting

[0119]

[0120] Where, when i = j, δ ij =1, otherwise equal to 0. Rewrite equation (8) in matrix form.

[0121]

[0122] Taking the Z-transform of both sides of equation (9), we can obtain

[0123]

[0124] Then, summing all possible values ​​of l yields the result. Its Z-transform is as follows:

[0125]

[0126] When n = 0,

[0127]

[0128] When n > 0

[0129]

[0130] Define a probability matrix in

[0131] g ij,l (n)=P{S l =n;J l =j; J1,J2,...,J l-1 ∈E1|J0=i},i∈E1,j∈E2.

[0132] It describes the probability that a system starts from state i, stays in subset E1 for the time interval [0, n), the first l-1 state transitions occur in E1, and finally leaves E1 at time n to reach state subset E2. Clearly, Similar to formula (3), we have

[0133]

[0134] Rewrite the above equation in matrix form. Based on equation (14), for l≥1, we have

[0135]

[0136] Furthermore, considering all possible values ​​of l and summing them, we have in,

[0137] Written in matrix form as It can be obtained The Z-transform is

[0138]

[0139] Based on the definition of acquired resilience and considering the initial state distribution of the system, acquired resilience can be expressed as follows:

[0140]

[0141] Starting from subset E2, the system can either directly transition from E2 to E1, or consist of any number of transition loops from E2→E1→E2, followed by a transition from E2 to E1. We can obtain Re. A The Z-transform of (n) is as follows:

[0142]

[0143] In this embodiment, if we assume that the performance changes of high-speed rail follow a discrete-time Markov chain, then its evolution process is controlled by the transition probability matrix P.

[0144]

[0145] The initial state distribution vector is Based on formula (2), we can calculate the inherent toughness of the high-speed rail as 0.729605.

[0146] The inherent toughness of high-speed rail under the Markov chain model was calculated using the Monte Carlo simulation algorithm and compared with the analytical results. The results show that the error between the simulation and analytical results is less than one-thousandth, verifying the correctness of the analytical formula, as shown in Table 1.

[0147] Table 1 Comparison of analytical and simulation results under the Markov chain model.

[0148]

[0149] Based on the given transition probability matrix P, the expression for the semi-Markov kernel can be obtained. The formulas (12) and (13) are used to obtain the formula. Submatrices through a semi-Markov kernel The Z-transform is obtained as follows: pass It is obtained from its Z-transform; and The calculation process requires the following information, obtained through formula (15). and Submatrices through a semi-Markov kernel and The Z-transform is obtained. From the formula for calculating toughness (16), through the inverse Z-transform, we can obtain the toughness variation diagram of the high-speed rail, as shown below. Figure 2 As shown.

[0150] Simultaneously, based on the Monte Carlo simulation algorithm, the obtained toughness of the high-speed rail under the Markov chain model was calculated, and compared with the analytical results. It can be found that the analytical results and simulation results are highly consistent, verifying the correctness of the calculation method. Figure 3 As shown.

[0151] Specifically, a discrete-time semi-Markov chain model of high-speed rail performance changes is constructed; the transition probability matrix and dwell time distribution are estimated based on the collected state change data; the probability distribution of the high-speed rail system under different states is calculated using the transition probability matrix, dwell time distribution, and initial distribution vector; and the inherent toughness and acquired toughness are calculated based on the calculated probability distribution.

[0152] More specifically, when the performance changes of high-speed rail follow a discrete-time semi-Markov chain, the analytical formulas for inherent toughness and acquired toughness are derived through the following process:

[0153] Assume that the performance of the high-speed rail system evolves over time according to a discrete-time homogeneous semi-Markov chain. To describe. Let express The point in time when the state changes. This process is called... The update point or jump point marks the moment when the system transitions from one state to another. The dwell time in each access state is determined by... It means that among them generally, Record exist Chain of states It means that among them This represents an embedded Markov chain. For i,j∈E and The transition probability matrix is in This represents the probability that the system is in state i at time n and in state j at time n+1. Discrete-time semi-Markov kernel The definition of This represents the probability that the system is in state i at time n, in state j at time n+1, and stays in state i for a time of k.

[0154] According to the definition of inherent toughness Introducing inherent resilience for Let represent the probability that, given the initial state i of the system at time 0 is in state set E2, the time it takes for the system to first enter state set E1 is less than the time it takes to first enter state set E3. We have...

[0155]

[0156] in, Equation (17) can be written in matrix form as follows:

[0157]

[0158] in, It is an E1-dimensional column vector consisting entirely of 1s. Therefore, it can be seen that...

[0159]

[0160] Based on the initial distribution of the system, we have

[0161]

[0162] Let f(k) = [f ij (k)] |E|×|E| For the conditional stay time distribution, where Let represent the probability that the system stays in state i for a duration of k, given that the system is in state i at time n and in state j at time n+1. Clearly, for any state i, j ∈ E and a non-negative integer k, we have Spend at least one unit of time in a certain state, that is, for any states i and j, we have

[0163] To derive the system's resilience, two key quantities are calculated. and Its derivation is similar to that in the Markov chain model. and

[0164] When n = 0, The Z-transform is

[0165]

[0166] in, The diagonal elements are A diagonal matrix.

[0167] When n > 0 The Z-transform is

[0168]

[0169] The Z-transform is

[0170]

[0171] Based on the definition of acquiring toughness, we can obtain the expression for acquiring toughness.

[0172]

[0173] Starting from state set E2, the system can transition directly from E2 to E1, or it can consist of any number of transition loops from E2→E1→E2, and then transition from E2 to E1. Therefore, Re A The Z-transform of (n) is

[0174]

[0175] In this embodiment, if the performance changes of high-speed rail follow a discrete-time semi-Markov chain, then the transition probability matrix of the embedded Markov chain is:

[0176]

[0177] Combined with the initial distribution vector of high-speed rail Based on formula (18), the inherent toughness of the high-speed rail is calculated to be 0.729605.

[0178] The inherent toughness of high-speed rail under a semi-Markov chain model was calculated using the Monte Carlo simulation algorithm and compared with the analytical results. The results showed that the error between the simulation and analytical results was less than one-thousandth, which verified the correctness of the analytical formula, as shown in Table 2.

[0179] Table 2 Comparison of analytical and simulation results under the semi-Markov chain model.

[0180]

[0181] Assume that the dwell time of the high-speed train follows a Poisson distribution with parameter λ, i.e. Where n is a non-negative integer. Let the residence time in state i be distributed under the condition of transitioning from state i to state j. All parameters are shown in Table 3.

[0182] Table 3. Parameters for the Poisson Distribution of Stay Time

[0183]

[0184] Based on the given transition probability matrix and residence time distribution of the embedded Markov chain, the expression for the semi-Markov kernel can be obtained. We obtain this from formulas (19) and (20), in which formula... Submatrices through a semi-Markov kernel The Z-transform is obtained; and The formula (21) is used to obtain the result. and Submatrices through a semi-Markov kernel and The Z-transform is obtained. From the formula (22) for calculating toughness under the semi-Markov chain model, we can obtain the toughness variation diagram of high-speed rail through the inverse Z-transform, as shown in... Figure 4 As shown.

[0185] A comparison between the toughness calculated using the Monte Carlo simulation algorithm and the analytical results shows a high degree of consistency between the two results, verifying the correctness of the analytical formula. Figure 5 As shown.

[0186] Specifically, for continuous-time Markov processes, the following steps are used to calculate inherent toughness and acquired toughness: construct a continuous-time Markov process model of the high-speed rail's state changes; estimate the transition rate matrix based on the collected state change data; use the transition rate matrix and the initial distribution vector to calculate the probability distribution of the high-speed rail system under different states; and calculate the inherent toughness and acquired toughness based on the calculated probability distribution.

[0187] More specifically, when the state changes of a high-speed train follow a continuous-time Markov process, the analytical formulas for inherent toughness and acquired toughness are derived through the following process:

[0188] Assume the state evolution of a high-speed rail system under disaster is an irreducible continuous-time homogeneous Markov process {X(t)}, t≥0, with a state space of E, and X(t) representing the system's state at time t. The Markov process is defined by the transition rate matrix A = [a...]. ij ] |E|×|E| Control, its diagonal elements And the sum of the elements in each row is 0. ij This represents the transition rate from state i to state j.

[0189] Let T1 represent the time of the first visit to state X(T1), then Define a probability matrix Q = [q ij ] |E|×|E| , where q ij =P{X(T1)=j|X(0)=i}=-a ij / a ii (i≠j) represents the probability that the system transitions to state j at time T1, given that it is in state i at time 0, and q ii =P{X(T1)=i|X(0)=i}=0. Therefore, we have A=diag(a ii [QI],diag(a ii ) indicates that the diagonal element is a ii A diagonal matrix.

[0190] According to the definition of inherent toughness Conditional inherent toughness is introduced to facilitate subsequent derivation and calculation; it is defined as follows: Let represent the probability that, given the initial state i of the system at time 0, the time it takes for the system to first enter state set E1 is less than the time it takes to first enter state set E3, given that the system's initial state i is in state set E2. Then we have the following expression:

[0191]

[0192] in, By defining a column vector β = [β i ,i∈E2], we can write equation (23) in matrix form, that is in It is a |E1|-dimensional column vector consisting entirely of 1s. Next, we can obtain... Based on the initial distribution of the system, the inherent toughness is

[0193]

[0194] Two important quantities are introduced to derive the acquisition of resilience under Markov processes.

[0195] First, define a probability matrix. in Let represent the probability that the system is in state i (i∈E1) at t=0, remains in state subset E1 between (0,t), and is in state j (j∈E1) at time t. We can observe that... right Taking the Laplace transform of t, we can obtain...

[0196]

[0197] Secondly, define a probability matrix. in i∈E1, j∈E2 represents the probability that the system is in state i (i∈E1) between (0,t) and still in state subset E1, and is in state j (j∈E2) at time t. Considering each possible intermediate state r from i to j, we have

[0198]

[0199] Rewriting the above equation in matrix form, we can obtain... right Taking the Laplace transform of t, we can obtain...

[0200]

[0201] According to the definition of inherent toughness, we have

[0202]

[0203] The system starts at E2 and can transition directly from E2 to E1, or it can contain any number of transition loops E2→E1→E2, and then transition back to E1. Since the probability density function of the sum of random variables is the convolution of their individual probability density functions, and the Laplace transform of the desired probability density function is the product of the Laplace transforms of their individual probability density functions, we can obtain Re. A The Laplace transform of (t) is

[0204]

[0205] Re is obtained by numerical inversion using the Laplace transform. A (t).

[0206] In this embodiment, if the performance change of the high-speed rail follows a continuous-time homogeneous Markov process, it is controlled by a Markov process with a transition rate matrix, wherein the transition rate matrix...

[0207]

[0208] According to A = diag(a) ii [QI], we can obtain

[0209]

[0210] At this point, the initial distribution vector of the high-speed rail is According to formula (24), the inherent toughness of high-speed rail is 0.627504.

[0211] The inherent toughness of high-speed rail under the Markov process model was calculated using the Monte Carlo simulation algorithm and compared with the analytical results. It can be found that the error between the simulation results and the analytical results is less than one-thousandth, which verifies the correctness of the analytical formula, as shown in Table 4.

[0212] Table 4 Comparison of analytical and simulation results under the Markov process model

[0213]

[0214] Based on the given state-rate matrix A It is obtained through formula (25), where It is a known fact set by humans; and It is obtained through formula (26), where and It is a known, artificially set condition. From the formula for obtaining toughness (27), through numerical inversion using the Laplace transform, we can obtain the obtained toughness of high-speed rail as follows: Figure 6 As shown.

[0215] Furthermore, the acquired toughness under the Markov model was calculated using the Monte Carlo simulation algorithm and compared with the analytical results. The results show that the curves plotted by both are highly consistent, thus confirming the consistency between the analytical and simulation results and verifying the correctness of the analytical formula. Figure 7 As shown.

[0216] Specifically, for a continuous-time semi-Markov process, the following steps are used to calculate the inherent toughness and acquired toughness to construct a continuous-time semi-Markov process model of high-speed rail performance changes; estimate the transition rate matrix and dwell time distribution based on the collected state change data; use the transition rate matrix, dwell time distribution, and initial distribution vector to calculate the probability distribution of the high-speed rail system under different states; and calculate the inherent toughness and acquired toughness based on the calculated probability distribution.

[0217] More specifically, when the performance changes of high-speed rail follow a continuous-time semi-Markov process, the analytical formulas for inherent toughness and acquired toughness are derived through the following process:

[0218] If the system's performance state evolves The condition t≥0 is a continuous-time homogeneous semi-Markov process. Let... yes The relevant Markov update process, Indicates the moment when the state changes. This indicates the state being accessed. The time interval between state transitions is represented by a procedure. Depiction, among which And Y0 = S0 = 0. In a semi-Markov process, the semi-Markov kernel V(t) = [v ij (t)] |E|×|E| Defined as

[0219] ν ij (t)=P(J n+1 =j,S n+1 -S n ≤t|J n =i),

[0220] This represents the probability that the system is in state i at time n, in state j at time n+1, and that the time spent in state i is less than or equal to t.

[0221] According to the definition of inherent toughness Introducing inherent resilience For ease of subsequent derivation and calculation, let's express the probability that the system's initial state i at time 0, given state set E2, occurs at time E1 less than at time E3. Then we have the following expression:

[0222]

[0223] in, Represents {J n The transition probability matrix of}. Equation (28) can be written in matrix form. Therefore we can obtain in Let E1 and E2 represent the transition probability matrices of the system from state set E2 to E2 and from state set E2 to E1, respectively. Combining this with the system's initial distribution, the expression for the inherent toughness is:

[0224]

[0225] Let M(t) = [M ij (t)] |E|×|E| Let M be the conditional stay time distribution, where M ij (t)=P{Y n ≤t|J n =i,J n+1 Let =j} represent the probability that the time spent in state i is less than or equal to t, given that the system is in state i at time n and in state j at time n+1. Clearly, for any state i, j ∈ E and a nonnegative real number t, we have

[0226] To derive the expression for obtaining toughness under a semi-Markov process, the Laplace-Stieltjes transform is introduced. The Laplace-Stieltjes transform of F(t) is defined as follows: Where s is a complex variable. Based on the definition of a semi-Markov kernel, it is clear that v ii (t) = 0. When i ≠ j, v ij The Laplace-Stieltjes transform of (t) with respect to t is defined as Representing this in matrix form, we have Ψ(s) = [ψ ij (s)] |E|×|E| .

[0227] Define a probability matrix, in

[0228]

[0229] It describes the probability that, given a state i at t=0, after n transitions within E1 before t, the system will eventually transition to state j at time t. Clearly, and Therefore, we have

[0230]

[0231] in, Rewrite the above equation in matrix form.

[0232] Recursive, has

[0233]

[0234] in, yes The n-fold convolution. Further, considering all possible values ​​of n and summing them, we have... in

[0235]

[0236] Based on formula (31), we can obtain

[0237]

[0238] Perform a Laplace-Stieltjes transformation on both sides of equation (32),

[0239]

[0240] Note that when t = 0, for i ≠ j, we have and Obviously

[0241] In order to calculate The probability of introduction is as follows

[0242]

[0243] This means that the system starts from state i, is in state j at time t, and has completed n transitions within E1 but has not yet made the (n+1)th transition. We have

[0244]

[0245] By performing the Laplace-Stieltjes transformation on both sides of equation (33), we can obtain

[0246]

[0247] Rewrite the above equation in matrix form. in, equal

[0248]

[0249] It is important to note that

[0250]

[0251] Where, δij =1 (i=j) and δ ij =0 (i≠j). Applying the Laplace-Stieltjes transformation to both sides of equation (35), we can obtain...

[0252]

[0253] Rewriting the above equation in matrix form, we have: Next, we can obtain The Laplace-Stieltjes transform is as follows:

[0254]

[0255] Define a probability matrix in

[0256] n g ij (t)=P(S n ≤t,J n =j,J1,J2,...,J n-1 ∈E1|J0=i), i∈E1,j∈E2.

[0257] It describes the probability that, given a state i at t=0, after n-1 transitions within E1 before time t, the system will eventually transition to state j for the nth time before time t. Similar to equation (30), we have...

[0258]

[0259] Write it in matrix form Obviously Recursively, we can obtain

[0260]

[0261] Considering all possible values ​​of n and summing them, we have in

[0262]

[0263] Based on equation (38), we have

[0264]

[0265] Perform a Laplace-Stieltjes transformation on equation (39) with respect to t.

[0266]

[0267] According to the definition of achieving resilience, we have

[0268]

[0269] The system starts at E2 and can transition directly from E2 to E1, or it can contain any number of transition loops E2→E1→E2, and then transition back from E2 to E1. We can obtain Re. A The Laplace-Stieltjes transformation of (t) is:

[0270]

[0271] Re is obtained by numerical inversion using the Laplace-Stieltjes transform. A (t).

[0272] In this embodiment, if the high-speed rail performance changes follow a continuous-time homogeneous semi-Markov process, and the state dwell time follows a shape parameter α... ij The scale parameter is β ij The Weibull distribution has a cumulative distribution function as follows: The residence time distribution parameters for different state transitions are shown in Table 5.

[0273] Table 5. Parameters for the Weibull Distribution of Stay Time

[0274]

[0275] Combined with the given initial distribution vector of the system From formula (29), we can obtain the inherent toughness of high-speed rail as 0.627504.

[0276] The inherent toughness of high-speed rail under a semi-Markov process model was calculated using the Monte Carlo simulation algorithm and compared with the analytical results. The results showed that the error between the simulation and analytical results was less than one-thousandth, which verified the correctness of the analytical formula, as shown in Table 6.

[0277] Table 6 Comparison of analytical and simulation results under the semi-Markov process model.

[0278]

[0279] Based on the given transition probability matrix of the embedded Markov chain From the residence time distribution M(t), we can obtain the expression for the semi-Markov kernel. The formula (37) is used to obtain the result. Submatrices through a semi-Markov kernel The Laplace-Stieltjes transform is used to obtain it; and The calculation process requires the following information, obtained through formula (40). and Submatrices through a semi-Markov kernel and The Laplace-Stieltjes transform is used to obtain it. Furthermore, from the formula for obtaining toughness (41), through the inverse Laplace-Stieltjes transform, we can calculate the obtained toughness of the high-speed rail as follows: Figure 8 As shown.

[0280] The high-speed rail's resilience was obtained using the Monte Carlo simulation algorithm and compared with analytical results. The results showed a high degree of consistency, verifying the correctness of the analytical formula. Figure 9 As shown.

[0281] If the dwell time distribution of the high-speed train in each state follows a shape parameter of a ij And the scale parameter is b ij The Gamma distribution has a cumulative distribution function of: Where γ(a) ij ,b ij t) is an incomplete gamma function, Γ(a) ij ) is the gamma function. Parameters are shown in Table 7.

[0282] Table 7. Parameters for Dwell Time Following a Gamma Distribution

[0283]

[0284] From the formula for calculating toughness (41), and using the numerical inversion of the Laplace-Stieltjes transform, we can obtain the toughness curve of the high-speed rail as shown in Figure 41. Figure 10 As shown.

[0285] The high-speed rail's resilience was obtained through Monte Carlo simulation and compared with analytical results. The results show a high degree of consistency between the simulation and analytical approaches, further validating the correctness of the analytical formula. Figure 11 As shown.

[0286] Finally, this embodiment performs a sensitivity analysis to observe the changes in the inherent toughness and acquired toughness of the high-speed rail.

[0287] By increasing the transition probability from state subset E2 to E1 while decreasing the transition probability from E2 to E3, the changes in the inherent toughness and acquired toughness of high-speed rail under the Markov chain model and the semi-Markov model are calculated, and the results are shown in Table 8. Figure 12 , Figure 13 The results are shown.

[0288] Table 8. Sensitivity analysis of the inherent toughness of Markov chains and semi-Markov chains to the transition probability matrix.

[0289]

[0290] As shown in Table 8, with or The gradual increase, at the same time or As the concentration of pollutants gradually decreases, its inherent resilience shows a significant increase. Specifically, or For every 20% increase, the inherent toughness of high-speed rail increases by 7%. This indicates that increasing the transition probability from E2 to E1 helps improve inherent toughness.

[0291] Depend on Figure 12 It can be seen that, under the Markov chain model, before time point 2, the change in probability does not significantly affect the resilience of the high-speed rail. However, after time point 2, as... The increase, As the value decreases, toughness gradually increases. Similarly, as... Figure 13 As shown, in the semi-Markov chain model, with The increase and The decrease in the coefficient of performance leads to a gradual increase in resilience. Specifically, the growth rate initially decreases, then increases again, reaching its minimum at time 6. This trend indicates a complex relationship between the parameter and system resilience; the initial decrease in the growth rate suggests that the high-speed rail underwent an adjustment period after the damage to adapt to the constantly changing transitional state.

[0292] By increasing the transition rate from state 2 to 1 while decreasing the transition rate from state 2 to 5, the changes in the inherent toughness and acquired toughness of high-speed rail under the Markov process model and the semi-Markov process model were calculated, and the results are shown in Table 9.

[0293] Table 9 Sensitivity analysis of inherent toughness to the transfer rate matrix

[0294]

[0295] As shown in Table 9, when a 31 Increase, a 36 As the speed decreases, the inherent toughness of high-speed rail gradually increases under both the Markov process model and the semi-Markov process model. Specifically, a 31 For every 0.1 increase, the inherent toughness increases by approximately 3%.

[0296] Furthermore, calculating the changes in the inherent toughness and acquired toughness of high-speed rail under the Markov process model and the semi-Markov process model can also yield sensitivity analysis diagrams of the acquired toughness to the transfer rate matrix under the Markov process and semi-Markov process systems, such as... Figure 14 As shown in the figure, when a 31 From 0.4 to 0.9, a 36 When the value is reduced from 0.5 to 0, the resilience of high-speed rail is significantly improved. The improvement in resilience is most significant when the dwell time distribution of the state follows a Weibull distribution; the improvement in resilience is less significant when the dwell time distribution of the state follows an exponential distribution; and the improvement in resilience is least obvious when the dwell time of the state follows a Gamma distribution.

[0297] Based on the above embodiments, this invention also provides a high-speed rail resilience assessment system based on a Markov model. The system includes: a data acquisition module for collecting and recording state change data of high-speed rail in different regions under the influence of different disasters; a resilience assessment index module for defining two resilience assessments: inherent resilience and acquired resilience; and a resilience calculation module for calculating the inherent resilience and acquired resilience of the high-speed rail system based on the Markov model type followed by the state change data.

[0298] It should be understood that the high-speed rail toughness assessment system based on Markov model provided in this embodiment of the invention and the high-speed rail toughness assessment method based on Markov model provided in the above embodiment are based on the same inventive concept. For more specific working principles of each module in this embodiment of the invention, please refer to the above embodiment, which will not be repeated in this embodiment.

[0299] 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 them. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some or all of the technical features therein. Such modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention, and they should all be covered within the scope of the claims and specification of the present invention.

Claims

1. A method for evaluating the toughness of high-speed railways based on a Markov model, characterized in that, Includes the following steps: Collect and record data on the state changes of high-speed railways in different regions under the influence of different disasters in order to estimate the initial distribution vector, transition probability matrix, transition rate matrix and dwell time distribution; The state of the high-speed rail system is divided into the following three state subsets. , This represents a subset of the system's normal operating states. Represents a subset of damaged states. Represents a subset of completely damaged states, and each subset contains at least one concrete state: Two resilience assessment indices are defined: inherent resilience and acquired resilience. Inherent resilience is the probability that the time taken for the high-speed rail system to enter a fully recovered state for the first time is earlier than the failure time. Acquired resilience is the probability that the high-speed rail system is in a normal operating state at any time. Based on the type of Markov model that the high-speed rail system follows in terms of state changes, and combined with state change data, the inherent toughness and acquired toughness of the high-speed rail system are calculated. The Markov model types include discrete-time Markov chains, discrete-time semi-Markov chains, continuous-time Markov processes, and continuous-time semi-Markov processes. When the state changes of a high-speed train follow a discrete-time Markov chain, its inherent toughness is: in For the initial distribution, Due to inherent resilience, It is the identity matrix. To start from the state set Transition to state set The transition probability matrix, To start from the state set Transition to state set The transition probability matrix, For elements with a value of 1 3D column vector; When the state changes of the high-speed rail follow a discrete-time Markov chain, the Z-transform for obtaining resilience is: in, for z-transform, To start from the state set Transition to state set The transition probability matrix; The probability matrix is ​​set. z-transform, The probability matrix is ​​set. z-transform; When the performance of high-speed rail follows a discrete-time semi-Markov chain, the inherent toughness is: in For the system At all times in the state set , At all times in the state set The probability matrix, For the system At all times in the state set , At all times in the state set The probability matrix; When the performance changes of high-speed rail follow a discrete-time semi-Markov chain, the Z-transform for obtaining toughness is: in, for z-transform; The probability matrix is ​​set. z-transform, The probability matrix is ​​set. z-transform; When the state changes of a high-speed train follow a continuous-time Markov process, its inherent toughness is: in This indicates that the system is in the state set at time 0. conditions, Transition to state set at time The probability matrix, This indicates that the system is in the state set at time 0. conditions, Transition to state set at time The probability matrix; When the state changes of a high-speed train follow a continuous-time Markov process, the Laplace transform for obtaining resilience is: in for about The Laplace transform matrix, This indicates that the system is in a state set. , The system is still in a state set. And in Always in the state set The probability matrix; for about The Laplace transform matrix, This indicates that the system is in a state set. , The system is still in a state set. And in Always in the state set The probability matrix; for about The Laplace transform matrix, Indicates in The system is in a state set , The system is still in a state set. And in At any given moment, the system is in the state set. The probability matrix; When the performance of high-speed rail follows a continuous-time semi-Markov process, its inherent toughness is: in This indicates that the system starts from the state set. arrive Transition probability matrix, This indicates that the system starts from the state set. arrive The transition probability matrix; When the performance changes of high-speed rail follow a continuous-time semi-Markov process, the Laplace-Stieltjes transformation for obtaining toughness is: in for about The Laplace-Stieltjes transformation matrix, for about The Laplace-Stieltjes transformation matrix, for about The Laplace-Stieltjes transformation matrix.

2. The high-speed rail toughness assessment method based on the Markov model according to claim 1, characterized in that, For discrete-time Markov chains, the following steps are used to calculate the inherent toughness and obtain the toughness: Construct a discrete-time Markov chain model of the state changes of high-speed rail; Estimate the transition probability matrix based on the collected state change data; Using the transition probability matrix and the initial distribution vector, the probability distribution of the high-speed rail system under different states is calculated; Based on the calculated probability distribution, calculate the inherent toughness and the acquired toughness.

3. The high-speed rail toughness assessment method based on the Markov model according to claim 1, characterized in that, For discrete-time semi-Markov chains, the following steps are used to calculate the inherent toughness and obtain the toughness. Construct a discrete-time semi-Markov chain model of high-speed rail performance variation; Estimate the transition probability matrix and dwell time distribution based on the collected state change data; Using the transition probability matrix, dwell time distribution, and initial distribution vector, the probability distribution of the high-speed rail system under different states is calculated; Based on the calculated probability distribution, calculate the inherent toughness and the acquired toughness.

4. The high-speed rail toughness assessment method based on the Markov model according to claim 1, characterized in that, For a continuous-time Markov process, the following steps are used to calculate the inherent toughness and obtain the toughness: Construct a continuous-time Markov process model of the state changes of high-speed rail; Estimate the transition rate matrix based on the collected state change data; Using the transition rate matrix and the initial distribution vector, the probability distribution of the high-speed rail system under different states is calculated; Based on the calculated probability distribution, calculate the inherent toughness and the acquired toughness.

5. The high-speed rail toughness assessment method based on the Markov model according to claim 1, characterized in that, For a continuous-time semi-Markov process, the following steps are used to calculate the inherent toughness and obtain the toughness: Construct a continuous-time semi-Markov process model of high-speed rail performance changes; Estimate the transition rate matrix and dwell time distribution based on the collected state change data; Using the transition rate matrix, dwell time distribution, and initial distribution vector, the probability distribution of the high-speed rail system under different states is calculated; Based on the calculated probability distribution, calculate the inherent toughness and the acquired toughness.

6. A high-speed rail toughness assessment system according to any one of claims 1 to 5, characterized in that, include: The data acquisition module is used to collect and record data on the status changes of high-speed railways in different regions under the influence of different disasters; The resilience assessment index module is used to define two resilience assessments: inherent resilience and acquired resilience. The toughness calculation module is used to calculate the inherent toughness and acquired toughness according to the Markov type followed by the changes in the state of the high-speed rail.

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