High-speed rail toughness evaluation method based on Markov model

Abstract

The invention discloses a high-speed rail toughness evaluation method based on a Markov model, and relates to the technical field of high-speed rail evaluation, and the method comprises the steps: collecting and recording the state change data of high-speed rails in different regions under the influence of different disasters, so as to estimate an initial distribution vector, a transition probability matrix, a transition rate matrix and sojourn time distribution; two toughness evaluation indexes of inherent toughness and obtained toughness are defined, the inherent toughness is the probability that the time used by the high-speed rail system to enter the complete recovery state for the first time is earlier than the failure time, and the obtained toughness is the probability that the high-speed rail system is in the normal operation state at any moment; and according to the state change data and the Markov model type obeyed by the high-speed rail state change, calculating the corresponding inherent toughness and obtaining the toughness. According to the method, the evolution paths of the performance of the high-speed rail in different environments are comprehensively considered, and quantitative evaluation of the toughness level of the high-speed rail is realized by introducing two indexes of inherent toughness and obtained toughness and combining multiple Markov models.

Description

Technical Field

[0001] The present invention relates to the technical field of high-speed railway assessment, and in particular to a high-speed railway toughness assessment method based on a Markov model. Background Art

[0002] The high-speed rail system may be affected by disasters such as blizzards, earthquakes, heavy rains, and man-made damage, which may cause serious damage to the track structure and even cause the entire system to fail, seriously interfering with the safe and stable operation of the high-speed rail. Therefore, it is particularly important to enhance the resilience of the high-speed rail transportation system and ensure that it can quickly maintain and restore its functions and operational capabilities when encountering various external shocks and disturbances.

[0003] Resilience assessment is the basis of resilience research, which aims to evaluate the response and coping ability of existing systems to destructive events. Existing research can be divided into qualitative research and quantitative research. Qualitative research focuses on analyzing the influencing factors of complex system resilience and deriving system resilience through weighted summation, but it relies more on subjective judgment and lacks objective and accurate quantitative evaluation. Quantitative research covers methods based on performance curves, topology and characteristics, but it focuses on one aspect, and the measurement indicators are relatively single, which cannot effectively quantitatively evaluate the resilience level of high-speed rail.

[0004] In particular, the performance changes of high-speed rail may be discrete. For example, the signal system may switch from normal operation to fault state in an instant, and there is no continuous performance gradient process in the middle. This discreteness makes it difficult to accurately apply the traditional resilience assessment method based on continuous performance change curves. Using static characteristics such as node centrality or a certain resilience attribute as a single measurement indicator ignores the characteristic changes of high-speed rail resilience during disturbances, making it difficult to judge the resilience level of high-speed rail at a certain damaged stage.

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

[0006] In view of the problems existing in the prior art, a high-speed rail resilience assessment method and system based on the Markov model are proposed. The present invention comprehensively considers the evolution path of the performance of high-speed rail under different environments, introduces two indicators, inherent toughness and acquired toughness, and combines a variety of Markov models to achieve a quantitative assessment of the high-speed rail resilience level. The present invention provides strong support for the safety risk management and resilience improvement of the high-speed rail system, and effectively makes up for the shortcomings of existing research.

[0007] The technical solution of the present invention:

[0008] In one aspect, the present invention provides a high-speed rail toughness assessment method based on a Markov model, the method comprising the following steps:

[0009] Collect and record the status change data of high-speed rail in different regions under the influence of different disasters to estimate the initial distribution vector, transition probability matrix, transition rate matrix and residence time distribution;

[0010] Define two toughness evaluation indicators: inherent toughness and acquired toughness. The inherent toughness is the probability that the time taken by the high-speed rail system to enter the fully recovered state for the first time is earlier than the failure time. The acquired toughness is the probability that the high-speed rail system is in a normal operating state at any time.

[0011] According to the type of Markov model that the state change of the high-speed rail obeys, combined with the state change data, the inherent resilience and acquired resilience 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 a discrete-time Markov chain, the following steps are used to calculate the intrinsic toughness and obtain the toughness:

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

[0015] Estimate a 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, the inherent toughness and the acquired toughness are calculated.

[0018] As a preferred method, a discrete-time semi-Markov chain model of high-speed rail performance changes is constructed;

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

[0020] Using the transition probability matrix, sojourn 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, the inherent toughness and the acquired toughness are calculated.

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

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

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

[0025] Using the transfer 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, the inherent toughness and the acquired toughness are calculated.

[0027] Preferably, for a continuous-time semi-Markov process, the following steps are used to calculate the intrinsic 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 sojourn time distribution based on the collected state change data;

[0030] Using the transfer rate matrix, sojourn 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, the inherent toughness and the acquired toughness are calculated.

[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 collection module is used to collect and record the status change data of high-speed railways in different regions under the influence of different disasters;

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

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

[0036] The beneficial effects of the present invention are:

[0037] 1) Comprehensive consideration of performance evolution: The present invention adopts a variety of 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 evaluation of toughness index: By calculating inherent toughness and obtaining toughness index, the present invention realizes quantitative evaluation of high-speed rail toughness level and provides a scientific basis for high-speed rail safety risk management.

[0039] 3) Wide application scope: The method of the present invention is not only applicable to the case of continuous change of high-speed rail performance, but also to the case of discrete change, and has a wider application prospect. BRIEF DESCRIPTION OF THE DRAWINGS

[0040] In order to more clearly illustrate the specific embodiments of the present invention or the technical solutions in the prior art, the following is a brief introduction to the drawings required for the specific embodiments or the description of the prior art. 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 according to the actual scale.

[0041] Figure 1 is a flow chart of the method of the present invention;

[0042] Figure 2 Gain resilience for Markov chains;

[0043] Figure 3 Comparison chart between analytical results and simulation results of toughness obtained under Markov chain;

[0044] Figure 4 Obtain toughness for semi-Markov chains;

[0045] Figure 5 Comparison chart of analytical and simulation results of toughness obtained under semi-Markov connection;

[0046] Figure 6 Obtaining resilience under Markov processes;

[0047] Figure 7 Comparison chart between analytical results and simulation results of toughness obtained under Markov process;

[0048] Figure 8 Obtain toughness for Weibull distribution under semi-Markov process;

[0049] Fig. 9 Comparison chart between analytical and simulation results of toughness obtained for Weibull distribution under semi-Markov process;

[0050] Fig.10 Obtain toughness for the Gamma distribution under a semi-Markov process;

[0051] Fig.11 Comparison chart of analytical and simulation results of toughness obtained for Gamma distribution under semi-Markov process;

[0052] Fig.12 To obtain the sensitivity analysis of the transition probability matrix for the toughness under Markov chains;

[0053] Fig.13 Sensitivity analysis of the transition probability matrix for obtaining toughness under semi-Markov chains.

[0054] Fig.14The sensitivity analysis of toughness to the transition rate matrix is ​​obtained for Markov and semi-Markov processes. DETAILED DESCRIPTION

[0055] The following embodiments of the technical solution of the present invention are described in detail in conjunction with the accompanying drawings. The following embodiments are only used to more clearly illustrate the technical solution of the present invention, and are therefore only used as examples, and cannot be used to limit the protection scope of the present invention.

[0056] It should be noted that, unless otherwise specified, the technical terms or scientific terms used in this application should have the common meanings understood by those skilled in the art to which the invention belongs.

[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 an embodiment of the present invention, the method comprising the following steps:

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

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

[0060] In order to accurately evaluate the resilience of the high-speed rail system, the state of the high-speed rail system is divided into the following three state subsets E = {E1, E2, E3}, where E1 represents the normal operation state subset of the system, E2 represents the damaged state subset, and E3 represents the completely damaged state subset. Each subset contains at least one specific state:

[0061] Normal operation status subset E1:

[0062] The state 0 is included, which means that the high-speed rail is in normal operation. In this state, the track fasteners of the high-speed rail are intact, tightly and firmly fixing the rails on the sleepers, ensuring that the high-speed rail train can run smoothly and safely at the predetermined speed and track.

[0063] Damaged state subset E2:

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

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

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

[0067] State 3: The high-speed rail is in a fastener deformation state. The track fasteners are deformed by external force under disaster conditions and can no longer effectively fix the rails. The high-speed rail trains may shake and deviate significantly during operation, posing a potential threat to the stability and safety of the track system.

[0068] State 4: The high-speed rail is in a state of severe damage but not failure. The track fasteners and their surrounding structures (such as sleepers and track plates) are significantly damaged. The fasteners may be severely deformed, cracked or even partially broken, but the overall structure has not completely collapsed and the rails are still fixed to a certain extent.

[0069] Failure state subset E3:

[0070] Including state 5, it means that the high-speed rail is in a failed state. In this state, due to the failure of track fasteners or other serious disasters, the track system loses its function as a whole, and high-speed trains cannot run normally on any section of the track, so the entire high-speed rail system is paralyzed.

[0071] Through the detailed division and definition of the above-mentioned state subsets, the state change process of the high-speed rail system under the influence of disasters can be described more accurately, providing a solid and reliable foundation for the subsequent high-speed rail resilience assessment.

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

[0073] Step 2, define two toughness evaluation indicators: inherent toughness and acquired toughness. The inherent toughness is the probability that the time taken by the high-speed rail system to enter the full recovery state for the first time is earlier than the failure time, and the acquired toughness is the probability that the high-speed rail system is in normal operation at any time.

[0074] The inherent toughness is defined as Re I =P{T E1 <T E3}, where T E1 and T E3 Represents the moment when the high-speed rail system first enters the E1 and E3 states. The inherent resilience reflects the speed and efficiency of the system returning to normal operation. The acquired resilience is defined as Re A (t) = P{F(t)∈E1}, that is, the probability that the high-speed rail system is in normal operation at time t, t is any time, and the obtained resilience reflects the stability and reliability of the system to maintain normal operation at any time point.

[0075] Step 3, according to the Markov model type obeyed by the high-speed rail state change, combined with the state change data, calculate the inherent resilience and acquired resilience 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 the discrete-time Markov chain, the following steps are used to calculate the 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 in different states; and calculate the inherent toughness and acquired toughness based on the calculated probability distribution.

[0077] More specifically, when the state change of the HSR obeys a discrete-time Markov chain, the analytical formulas for the intrinsic toughness and the acquired toughness are derived through the following process:

[0078] Assuming that under the influence of disasters, the evolution of the state of the high-speed rail system over time follows the principle of discrete-time homogeneous Markov chain. n}, represents the state of the high-speed rail system at time n, where represents a set of natural numbers. In this process, the state of the system at the next moment is only related to the current state. The evolution of the Markov chain is represented 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 =i}, which means the probability of the high-speed rail system transferring to state j at time n+1 when it is in state i at time n. Given that the high-speed rail system contains three state sets, its transition probability matrix Will take on a specific form, specifically:

[0079]

[0080] Among them, the transition probability matrix Represents the state set Ei Transfer to state set E j The transition probability matrix (i,j=1,2,3).

[0081] According to the definition of intrinsic toughness: Given the initial performance state i∈E2 at time 0, let It represents the probability that the time taken for the high-speed rail to recover to normal operation is earlier than the failure time when it is in state i at time 0.

[0082]

[0083] All α i Written as a column vector α = [α i ], i∈E2, formula (1) is written in matrix form as follows

[0084]

[0085] in, is an |E1|-dimensional column vector with 1 as its element. So we can get

[0086]

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

[0088]

[0089] In order to study the acquired resilience of the high-speed rail system under Markov chain, some important quantities are defined first. First, let {J n ,S n}, is a Markov update chain, where {J n}, is the state of the system at the nth jump, {S n}, is the time of the nth jump. In addition, let {Y n}, Indicates that before the nth jump, in state J n-1 The residence time in 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 n time.

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

[0091]

[0092] in is the set of positive integers.

[0093] Define a probability matrix in

[0094]

[0095] It means that the system stays in the state subset E1 for a given initial state i, and after l transitions, it finally enters the state j at time n. Obviously, It is equal to 1 only when n=0, otherwise it is equal to 0. Based on the total probability formula, for We have

[0096]

[0097] Write equation (3) in matrix form where ★ is the convolution operator of the matrix, represents the probability that the system is still in the state set E1 after l-1 transitions, represents the probability that the system transfers from the state set E1 and stays in each state for n. For l ≥ 1, we have

[0098]

[0099] in, They are In addition, consider all possible values ​​of l and sum them up to get in

[0100]

[0101] Written in matrix form: Obviously,

[0102] When n = 0, we have

[0103]

[0104] It follows from this

[0105] When n>0, we have

[0106]

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

[0108]

[0109] Next, we introduce the probability matrix in

[0110]

[0111] It means that starting from state i, l transitions have been completed in E1. At time n, the system is in state j and has not yet made the (l+1)th transition. We have

[0112]

[0113] in The diagonal elements are The diagonal matrix of .

[0114] Performing Z transformation on both sides of equation (6) yields

[0115]

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

[0117]

[0118] It is worth noting that

[0119]

[0120] Among them, when i = j, δ ij =1, otherwise it is equal to 0. Write equation (8) in matrix form,

[0121]

[0122] Performing Z transformation on both sides of equation (9) yields

[0123]

[0124] Then, summing all possible values ​​of l gives 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 the system starts from state i, stays in subset E1 within the time interval [0,n), the first l-1 state transitions occur in E1, and finally leaves E1 at time n and reaches state subset E2. Obviously, Similar to formula (3), we have

[0133]

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

[0135]

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

[0137] Written in matrix form: Can get The Z transform is

[0138]

[0139] According to the definition of acquired resilience and combined with the initial state distribution of the system, acquired resilience can be expressed as

[0140]

[0141] Starting from subset E2, the system can be transferred directly from E2 to E1, or it can be composed of any number of transfer cycles E2→E1→E2, and then transferred from E2 to E1. We can get Re A The Z transform of (n) is as follows:

[0142]

[0143] In this embodiment, if it is assumed that the performance change of the high-speed rail obeys the discrete-time Markov chain, then its evolution process is controlled by the transition probability matrix P, taking

[0144]

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

[0146] According to the Monte Carlo simulation algorithm, the inherent toughness of the high-speed rail under the Markov chain model is calculated 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 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 of the semi-Markov kernel can be obtained. Through formula (12) and (13), we can get Submatrices through the semi-Markov kernel The Z transform is: pass And its Z transform is obtained; and According to formula (15), the required and Submatrices through the semi-Markov kernel and From the calculation formula (16) of the toughness, through the inverse Z transformation, we can obtain the toughness change diagram of the high-speed rail, as shown in Figure 2 shown.

[0150] At the same time, according to the Monte Carlo simulation algorithm, the obtained toughness of the high-speed rail under the Markov chain model is calculated and compared with the analytical results. It can be found that the analytical results are highly consistent with the simulation results, which verifies the correctness of the calculation method. Figure 3 shown.

[0151] Specifically, a discrete-time semi-Markov chain model of high-speed rail performance changes is constructed; the transition probability matrix and residence time distribution are estimated based on the collected state change data; the probability distribution of the high-speed rail system in different states is calculated using the transition probability matrix, residence 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 variation of the HSR obeys a discrete-time semi-Markov chain, the analytical formulas for the intrinsic toughness and the acquired toughness are derived through the following process:

[0153] Assume that the time evolution of the HSR system performance is described by a discrete-time homogeneous semi-Markov chain To describe. express The point in time when the state changes. This process is called The update point or jump point of , marks the moment when the system moves from one state to another. The stay time in each visited state is given by Indicates that generally, Record exist Time state chain Indicates that represents an embedded Markov chain. For i,j∈E and The transition probability matrix is in It represents the probability that the system is in state i at time n and in state j at time n+1. The discrete-time semi-Markov kernel of The definition is It 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 k.

[0154] According to the definition of inherent resilience Introducing inherent toughness for It means that under the condition of the state set E2, the probability that the system first enters the state set E1 is less than the first time it enters the state set E3 when the initial state i of the system at time 0 is in the state set E2. We have

[0155]

[0156] in, Formula (17) can be written in matrix form as

[0157]

[0158] in, is a |E1|-dimensional column vector with all 1s. It can be seen that

[0159]

[0160] Combined with the initial distribution of the system, we have

[0161]

[0162] Let f(k)=[f ij (k)] |E|×|E| is the conditional stay time distribution, where represents the probability that the system stays in state i for a time k under the condition that it is in state i at time n and in state j at time n+1. Obviously, for any state i, j∈E and a non-negative integer k, we have Spend at least one time unit in a state, that is, for any state i, j, there is

[0163] In order 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 The diagonal matrix of .

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

[0168]

[0169] The Z transform is

[0170]

[0171] According to the definition of gaining toughness, the expression of gaining toughness can be obtained:

[0172]

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

[0174]

[0175] In this embodiment, if the performance change of the high-speed rail obeys the discrete-time semi-Markov chain, the transition probability matrix embedded in the Markov chain is

[0176]

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

[0178] According to the Monte Carlo simulation algorithm, the inherent toughness of the high-speed rail under the semi-Markov chain model is calculated 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 2.

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

[0180]

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

[0182] Table 3 Parameters of the length of stay following the Poisson distribution

[0183]

[0184] Based on the given transition probability matrix and sojourn time distribution of the embedded Markov chain, the expression of the semi-Markov kernel can be obtained. Through formula (19) and (20), we can get Submatrices through the semi-Markov kernel The Z transform is obtained; and It is obtained by formula (21), in which and Submatrices through the semi-Markov kernel and The Z transform is obtained. From the calculation formula (22) of the toughness obtained under the semi-Markov chain model, through the inverse Z transform, we can obtain the change diagram of the toughness obtained by high-speed rail, as shown in Figure 4 shown.

[0185] The comparison between the toughness calculated by Monte Carlo simulation algorithm and the analytical result shows that the two results are highly consistent, which verifies the correctness of the analytical formula. Figure 5 shown.

[0186] Specifically, for the continuous-time Markov process, the following steps are used to calculate the intrinsic toughness and acquired toughness: construct a continuous-time Markov process model of the high-speed rail state change; 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 in different states; calculate the intrinsic toughness and acquired toughness based on the calculated probability distribution.

[0187] More specifically, when the state change of the HSR obeys a continuous-time Markov process, the analytical formulas for the intrinsic toughness and the acquired toughness are derived through the following process:

[0188] Assume that the state evolution process of the high-speed rail system under the influence of disasters is an irreducible continuous-time homogeneous Markov process {X(t)}, t≥0, whose state space is E and X(t) represents the state of the system at time t. The Markov process is represented by the transition rate matrix A = [a ij ] |E|×|E| Control, whose diagonal elements And the sum of the elements in each row is 0. ij represents the transition rate from state i to state j.

[0189] Let T1 denote 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) indicates the probability of the system transferring to state j at time T1 under the condition that the system 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 ) means the diagonal element is a ii The diagonal matrix of .

[0190] According to the definition of inherent resilience The conditional inherent toughness is introduced to facilitate the subsequent derivation and calculation, which is defined as It means that under the condition of state set E2, the probability that the system first enters state set E1 is less than the first time it enters state set E3 when the initial state i of the system at time 0 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 is a |E1|-dimensional column vector of all 1s. Then, we can get Combined with the initial distribution of the system, the inherent toughness is

[0193]

[0194] Two important quantities are introduced to derive the acquired resilience under the Markov process.

[0195] First, define a probability matrix in represents the probability that the system is in state i (i∈E1) at t=0, the system is still in the state subset E1 between (0, t) and the system is in state j (j∈E1) at time t. We can find right By doing Laplace transform on t, we can get

[0196]

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

[0198]

[0199] Writing the above formula in matrix form, we can get right By doing Laplace transform on t, we can get

[0200]

[0201] According to the definition of intrinsic resilience, we have

[0202]

[0203] The system starts at E2 and can move directly from E2 to E1 or contain any number of transition cycles E2→E1→E2 and then from E2 to E1. Because 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 the individual probability density functions, we can get Re A The Laplace transform of (t) is

[0204]

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

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

[0207]

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

[0209]

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

[0211] According to the Monte Carlo simulation algorithm, the inherent toughness of the high-speed rail under the Markov process model is calculated 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 by formula (25), where It is artificially set and known; and It is obtained by formula (26), where and is artificially set and known. From the calculation formula (27) of the obtained toughness, through the numerical inversion of Laplace transform, we can get the obtained toughness of high-speed rail as Figure 6 shown.

[0215] In addition, the obtained toughness under the Markov model is calculated according to the Monte Carlo simulation algorithm and compared with the analytical results. The results show that the curves drawn by the two are highly consistent. Therefore, the analytical results and simulation results are consistent, which verifies the correctness of the analytical formula. Figure 7 shown.

[0216] Specifically, for the 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; the transition rate matrix and residence time distribution are estimated based on the collected state change data; the probability distribution of the high-speed rail system in different states is calculated using the transition rate matrix, residence time distribution and initial distribution vector; and the inherent toughness and acquired toughness are calculated based on the calculated probability distribution.

[0217] More specifically, when the performance variation of the HSR obeys a continuous-time semi-Markov process, the analytical formulas for the intrinsic toughness and the acquired toughness are derived through the following process:

[0218] If the system's performance status evolves t≥0 is a continuous-time homogeneous semi-Markov process. yes The relevant Markov update process is Indicates the moment when the state changes. Indicates the state of access. The time interval between states is expressed by the process portrayal, among which And Y0=S0=0. In the 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] It represents the probability that the system is in state i at time n, in state j at time n+1, and the stay time in state i is less than or equal to t.

[0221] According to the definition of inherent resilience Introducing inherent toughness In order to facilitate the subsequent derivation and calculation, we express the probability that the system's initial state i at time 0 is less than the first time it enters state set E3 under the condition of state set E2. Then we have the following expression:

[0222]

[0223] in, Representative {J n}. Write equation (28) in matrix form: So we can get in They represent the transition probability matrix from state set E2 to E2 and the transition probability matrix from state set E2 to E1, respectively. Combined with the initial distribution of the system, the expression of inherent resilience is:

[0224]

[0225] Let M(t) = [M ij (t)] |E|×|E| is the conditional stay time distribution, where M ij (t) = P{Y n ≤t|J n =i,J n+1 =j}, which means the probability that the system stays in state i for less than or equal to t under the condition that the system is in state i at time n and in state j at time n+1. Obviously, for any state i, j∈E and non-negative real number t, we have

[0226] In order to derive the expression for toughness under the semi-Markov process, the Laplace-Stieltjes transform is introduced. The Laplace-Stieltjes transform of F(t) is defined as Where s is a complex variable. Based on the definition of the 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 Writing it 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 the system will eventually transfer to state j at time t after n transitions in E1 before t, under the condition that it is in state i at t = 0. Obviously, and Therefore, we have

[0230]

[0231] in, Write the above formula in matrix form,

[0232] Recursively, we have

[0233]

[0234] in, yes Further, considering all possible values ​​of n and summing them, we have in

[0235]

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

[0237]

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

[0239]

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

[0241] To calculate The probability of introduction is as follows

[0242]

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

[0244]

[0245] Performing Laplace-Stieltjes transformation on both sides of equation (33) yields

[0246]

[0247] Write the above formula in matrix form, in, equal

[0248]

[0249] It should be noted that

[0250]

[0251] Among them, δij =1(i=j) and δ ij =0(i≠j). Performing Laplace-Stieltjes transformation on both sides of equation (35), we can obtain

[0252]

[0253] Writing the above formula in matrix form, we have Then, we can get 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 the system finally transitions to state j for the nth time before time t, given that the system is in state i at t = 0, after n-1 transitions in E1 before time t. Similar to equation (30), we have

[0258]

[0259] Write it in matrix form Apparently Recursively, we can get

[0260]

[0261] Consider all possible values ​​of n and sum them up, we have in

[0262]

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

[0264]

[0265] Performing Laplace-Stieltjes transformation on equation (39) with respect to t,

[0266]

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

[0268]

[0269] The system starts from E2 and can transfer directly from E2 to E1, or it can contain any number of transformation cycles E2→E1→E2 and then transfer from E2 to E1. We can get Re A The Laplace-Stieltjes transform of (t) is

[0270]

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

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

[0273] Table 5 Parameters of the length of stay following the Weibull distribution

[0274]

[0275] Combined with the given system initial distribution vector From formula (29), we can get the intrinsic toughness of high-speed rail to be 0.627504.

[0276] According to the Monte Carlo simulation algorithm, the inherent toughness of the high-speed rail under the semi-Markov process model is calculated 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 6.

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

[0278]

[0279] Based on the given embedded Markov chain transition probability matrix and the residence time distribution M(t), we can obtain the expression of the semi-Markov kernel. It is obtained by formula (37), in which Submatrices through the semi-Markov kernel The Laplace-Stieltjes transformation is obtained; and According to formula (40), the required and Submatrices through the semi-Markov kernel and In addition, from the calculation formula (41) of the obtained toughness, through the Laplace-Stieltjes inverse transformation, we can calculate the obtained toughness of the high-speed rail as follows: Figure 8 shown.

[0280] The obtained toughness of high-speed rail was obtained by Monte Carlo simulation algorithm and compared with the analytical results. It was found that the two results were highly consistent, which verified the correctness of the analytical formula. Fig. 9 shown.

[0281] If the distribution of the HSR’s stay time in each state follows a shape parameter 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. The parameters are shown in Table 7.

[0282] Table 7 Parameters of the stay time following the Gamma distribution

[0283]

[0284] From the calculation formula (41) for toughness, using the numerical inversion of Laplace-Stieltjes transformation, we can obtain the toughness curve of high-speed rail as follows: Fig.10 shown.

[0285] The obtained toughness of the high-speed rail is obtained by Monte Carlo simulation algorithm and compared with the analytical results. It can be found that the simulation results are highly consistent with the analytical results, which further verifies the correctness of the analytical formula. Fig.11 shown.

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

[0287] By increasing the transition probability from state subset E2 to E1 and reducing 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 semi-Markov model are calculated, as shown in Table 8 and Fig.12 , Fig.13 Results shown.

[0288] Table 8 Analysis of sensitivity of inherent toughness to transition probability matrix under Markov chain and semi-Markov chain

[0289]

[0290] From Table 8, we can see that or The gradual increase, or The gradual decrease of its inherent toughness shows a significant increase. Specifically, or For every 20% increase, the intrinsic toughness of high-speed iron increases by 7%, which indicates that increasing the transition probability from E2 to E1 helps to improve the intrinsic toughness.

[0291] Depend on Fig.12 It can be seen that under the Markov chain model, before time point 2, the change in probability does not significantly affect the high-speed rail's acquisition resilience. After time point 2, as increase, As the value of decreases, the toughness gradually increases. Fig.13 As shown, under the semi-Markov chain model, as The increase and The decrease in the growth rate leads to a gradual increase in the acquired resilience. The growth rate initially decreases and then increases again, reaching a minimum at time 6. This trend shows the complex relationship between parameters and system resilience. The initial decrease in the growth rate indicates that the high-speed rail first undergoes a period of adjustment after being damaged to adapt to the changing transition state.

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

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

[0294]

[0295] From Table 9, we can see that when a 31 Increase, a 36 When decreases, the inherent toughness of HSR increases gradually under the Markov process model and semi-Markov process model. Specifically, a 31 For every 0.1 increase, the intrinsic toughness increases by about 3%.

[0296] In addition, by 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, we can also obtain the sensitivity analysis diagram of the acquired toughness to the transfer rate matrix under the Markov process and semi-Markov process system, such as Fig.14 As shown in the figure, it can be seen that when a 31 From 0.4 to 0.9, a 36 When it decreases from 0.5 to 0, the high-speed rail obtains a significant improvement in toughness. When the state's residence time distribution is Weibull distribution, the improvement in toughness is most significant; when the state's residence time distribution is exponential distribution, the improvement in toughness is not so significant; when the state's residence time follows Gamma distribution, the improvement in toughness is least obvious.

[0297] Based on the above embodiments, an embodiment of the present invention also provides a high-speed rail resilience assessment system based on a Markov model, the system comprising: a data acquisition module, used to collect and record the state change data of high-speed rails in different regions under the influence of different disasters; a resilience assessment index module, used to define two resilience assessments: inherent resilience and acquired resilience; a resilience calculation module, used to calculate the inherent resilience and acquired resilience of the high-speed rail system based on the type of Markov model to which the high-speed rail state change obeys, combined with the state change data.

[0298] It should be understood that the high-speed rail toughness assessment system based on the Markov model provided in the embodiment of the present invention and the high-speed rail toughness assessment method based on the Markov model provided in the above embodiment are based on the same inventive concept. For more specific working principles of each module in the embodiment of the present invention, please refer to the above embodiment and 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, rather than to limit it. Although the present invention has been described in detail with reference to the aforementioned embodiments, those skilled in the art should understand that they can still modify the technical solutions described in the aforementioned embodiments, or replace some or all of the technical features therein by equivalents. These modifications or replacements do not make the essence of the corresponding technical solutions deviate from the scope of the technical solutions of the embodiments of the present invention, and they should all be included in the scope of the claims and specification of the present invention.

Claims

1. A high-speed rail toughness assessment method based on a Markov model, characterized in that: The following steps are involved: Collect and record the status change data of high-speed rail in different regions under the influence of different disasters to estimate the initial distribution vector, transition probability matrix, transition rate matrix and residence time distribution; Define two toughness evaluation indicators: inherent toughness and acquired toughness. The inherent toughness is the probability that the time taken by the high-speed rail system to enter the fully recovered state for the first time is earlier than the failure time. The acquired toughness is the probability that the high-speed rail system is in a normal operating state at any time. According to the type of Markov model that the state change of the high-speed rail obeys, combined with the state change data, the inherent resilience and acquired resilience of the high-speed rail system are calculated.

2. The high-speed rail toughness assessment method based on the Markov model according to claim 2 is characterized in that: The Markov model types include discrete-time Markov chains, discrete-time semi-Markov chains, continuous-time Markov processes, and continuous-time semi-Markov processes.

3. The high-speed rail toughness assessment method based on the Markov model according to claim 3 is characterized in that: For discrete-time Markov chains, the following steps are used to calculate the intrinsic toughness and obtain the toughness: Construct a discrete-time Markov chain model of high-speed rail state changes; Estimate a 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, the inherent toughness and the acquired toughness are calculated.

4. The high-speed rail toughness assessment method based on the Markov model according to claim 3 is characterized in that: For discrete-time semi-Markov chains, the following steps are used to calculate the intrinsic toughness and obtain the toughness: Construct a discrete-time semi-Markov chain model of high-speed rail performance changes; Estimate the transition probability matrix and sojourn time distribution based on the collected state change data; Using the transition probability matrix, sojourn 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, the inherent toughness and the acquired toughness are calculated.

5. The high-speed rail toughness assessment method based on the Markov model according to claim 3 is characterized in that: For a continuous-time Markov process, the following steps are used to calculate the intrinsic toughness and obtain the toughness: Construct a continuous-time Markov process model of high-speed rail state changes; estimating a transition rate matrix based on the collected state change data; Using the transfer 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, the inherent toughness and the acquired toughness are calculated.

6. The high-speed rail toughness assessment method based on the Markov model according to claim 3 is characterized in that: For a continuous-time semi-Markov process, the following steps are used to calculate the intrinsic 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 sojourn time distribution based on the collected state change data; Using the transfer rate matrix, sojourn 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, the inherent toughness and the acquired toughness are calculated.

7. A high-speed rail toughness assessment system based on a Markov model, characterized in that: include: The data collection module is used to collect and record the status change data of high-speed railways in different regions under the influence of different disasters; The resilience assessment indicator module is used to define two resilience assessments: inherent resilience and acquired resilience; The toughness calculation module is used to calculate the corresponding inherent toughness and acquired toughness according to the Markov type obeyed by the high-speed rail state change.

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