Reliability modeling and evaluation method for repairable system with imperfect repair

Abstract

The application discloses a kind of repairable equipment reliability modeling and evaluation method considering incomplete repair.By using three-parameter boundary intensity process model (3-BIP) as benchmark failure intensity function to correct proportional hazard intensity (LPIM) model, the reliability of repairable equipment is modeled;The model parameter estimation problem is converted into the maximum value solving problem of nonlinear objective function, and the particle swarm optimization algorithm is used to solve the model parameters;Thirdly, the inverse Fisher information matrix method is used to estimate the confidence interval of model parameters, and the point estimation and interval estimation method based on Delta method of key reliability indicators are given;The application is applied to the fault truncated time data of a wind generator, and the Akaike information criterion, Bayesian information criterion and goodness-of-fit index are used to test the goodness of fit of the model.The goodness of fit of the model is better than BIP and LPIM model, and the reliability evaluation result obtained is more in line with engineering practice.

Description

Technical Field

[0001] This invention relates to a method for reliability modeling and evaluation of repairable equipment that takes into account incomplete repair, belonging to the field of reliability evaluation of mechanical equipment or other repairable equipment. Background Technology

[0002] Repairable equipment differs from simple electromechanical equipment that is scrapped upon failure; maintenance activities affect its reliability. In practice, incomplete maintenance is more common for repairable equipment. The effect of incomplete maintenance lies between complete and minimal maintenance, restoring components to a state between "like new" and "like old," which better reflects engineering realities. As the number of maintenance operations increases and service time accumulates, the time it can operate normally after maintenance becomes shorter, and the probability of failure increases.

[0003] Repairable equipment refers to equipment whose operational capability can be restored by any means other than replacing the entire system. In this sense, reliability is interpreted as the probability of no failure occurring within a specific period. The reliability of repairable equipment is modeled and analyzed using stochastic point processes. Stochastic point processes focus on the equipment's failure interval time, studying both the operational and failure states of the system, with different stochastic point processes representing different maintenance outcomes. Numerous research results exist on the reliability modeling and evaluation of repairable equipment under incomplete maintenance outcomes. Currently, research on reliability modeling methods based on failure time data mainly focuses on the Generalized Renewal Process (GRP) and the Proportional Intensity Model (PIM).

[0004] Kijima et al. proposed two generalized renewal process models, GRP-I and GRP-II. These models are based on the concepts of a "renewal process" and a "non-homogeneous Poisson process," introducing a service life regression factor to describe the maintenance effect by the degree of system service life reduction. However, solving for the model parameters is complex. Methods such as Monte Carlo (MC) simulation, Maximum Likelihood Estimation (MLE), Bayesian methods, likelihood ratio tests, the Expectation-Maximum (EM) algorithm, and nonlinear constraint programming have been proposed to address this issue. However, due to the discontinuity of the model itself, these methods all struggle to obtain closed-form solutions for the reliability index.

[0005] The Proportional Intensity Model (PIM) proposed by Cox et al. considers the impact of repair activities on the reliability of equipment during service and emphasizes the assessment of the maintenance level of equipment through the intensity reduction coefficient. Based on this, the article "Practical methods for modeling repairable systems with time trends and repair effects" published in Reliability and Maintainability Symposium, California: IEEE 2006, 182-188 comprehensively considers time trends, proportional failure intensity, and cumulative maintenance effects, and introduces the cumulative failure intensity function as a covariate into the proportional intensity model to conduct reliability assessment of repairable equipment under the minimum maintenance assumption. The article "Reliability Assessment of CNC Machine Tools for Incomplete Maintenance" published in the Journal of Mechanical Engineering 2013, 49(23):136-141 proposes the Log-linear Proportional Intensity Model (LPIM) to evaluate the reliability of CNC machine tools under incomplete maintenance conditions. However, LPIM cannot describe the bathtub-shaped failure process with early failures and wear failures. In light of this, the paper "Reliability Modeling and Evaluation of CNC Machine Tools for Incomplete Maintenance," published in the *Journal of Tsinghua University (Natural Science Edition)* in 2022, proposes a superposed logarithmic proportional strength model (S-LPIM) to model the reliability of CNC machine tools under incomplete maintenance conditions. This model fits well the bathtub-shaped failure rate characteristic of CNC machine tools. The above PIM model and its improved models mainly use power law processes (PLP) and log linear processes (LLP) as the benchmark functions for system failure strength. A common drawback of these two process models is that the failure strength increases infinitely with the accumulation of service time, meaning the model does not have a closed-form solution, which is inconsistent with engineering reality. In fact, after maintainable equipment enters the wear and tear failure period, the maintenance frequency increases accordingly with the decrease in reliability. Frequent maintenance makes the failure strength a bounded increasing function, not an infinitely increasing one.Therefore, the paper "A bounded intensity process for the reliability of repairable equipment" published in the Journal of Quality Technology, 2001, 33(4): 480-492, argues that for some repairable equipment, repeated maintenance measures will produce a finite bound of continuously increasing fault intensity, that is, the fault intensity asymptotically approaches a certain constant as the system operating time increases. Based on this, a Bounded Intensity Process (BIP) is proposed, which is suitable for reliability assessment of repairable equipment in the wear-out period. However, this model has a zero fault intensity at the initial moment of data acquisition, which is inconsistent with engineering reality. Before the actual wear-out period is the period of random failure, and the fault intensity during the period of random failure is not zero. The paper "Three-parameter Bounded Intensity Process Model and Its Application in Reliability Assessment of CNC Machine Tools" published in the Journal of Xi'an Jiaotong University, 2014, 48(05): 107-112, introduces position parameters on the basis of the BIP model to establish a three-parameter boundary intensity process model (3-BIP), which realizes an effective assessment of the reliability of CNC machine tools under the minimum maintenance assumption. The paper "Continuous Proportional Strength Model and Application for Incomplete Maintenance of Repairable Equipment," published in the *Journal of Central South University (Natural Science Edition)*, 2015, 46(07):2467-2473, considers the impact of operating time and maintenance behavior on equipment reliability. It proposes a continuous proportional strength model with a closed-form solution to achieve reliability modeling and assessment under incomplete maintenance. However, this model assumes zero equipment failure intensity at the initial data acquisition moment. Furthermore, since unit reliability is affected by multiple random factors, the accuracy of single-line fitting has significant uncertainty. The fitting accuracy depends on the dispersion of failure time data; the worse the clustering of failure time data, the worse the fitting accuracy. Using this reliability assessment result as the theoretical basis for maintenance decisions may be either conservative or aggressive.

[0006] In solving reliability model parameters, commonly used methods include least squares, Minimum Lexical Estimation (MLE), and Bayesian methods. The first two are statistical methods developed based on the principle of minimizing empirical risk, but their accuracy is poor when the sample size is small. Bayesian methods show advantages with small sample data, but their estimation accuracy depends on the chosen prior distribution. MLE solves the system of nonlinear transcendental equations by directly taking the partial derivative of the log-likelihood function, followed by analytical, numerical, and parameter optimization methods. Analytical methods are difficult to solve the nonlinear transcendental equations of complex multi-parameter distribution models. Numerical iterative methods such as Newton's method and the EM algorithm can lead to divergence in the iteration process and easy getting trapped in local optima if the initial values ​​are not chosen properly. These drawbacks of traditional MLE algorithms limit their application in multi-parameter reliability model parameter estimation. Multi-parameter reliability model parameter estimation is essentially a nonlinear model parameter optimization problem. Particle Swarm Optimization (PSO) and Genetic Algorithm (GA) are intelligent optimization algorithms that can be used to solve optimization problems of nonlinear complex functions, representing a feasible approach. PSO algorithms require fewer parameter adjustments, have a fast convergence speed, are conceptually simple, and are easy to implement, avoiding the complex operations of GA algorithms. PSO algorithms do not rely on the analytical properties of the objective function; based on the fundamental idea of ​​Model-Like Optimization (MLE), they determine the fitness function, establish a constrained nonlinear optimization model, and solve for the reliability model parameters by searching for global and local extrema in parallel, thus reducing the difficulty of the solution.

[0007] Therefore, in view of the above-mentioned problems in reliability modeling and evaluation of repairable equipment under incomplete maintenance based on failure time data, there is an urgent need to invent a solution that conforms to the actual engineering situation. Summary of the Invention

[0008] The technical problem to be solved by the present invention is to provide a reliability modeling and evaluation method for repairable equipment that takes into account incomplete maintenance, thereby overcoming the shortcomings of the existing methods.

[0009] To solve the above-mentioned technical problems, the technical solution adopted by the present invention is as follows:

[0010] A method for reliability modeling and evaluation of repairable equipment that considers incomplete repair, comprising the following steps:

[0011] By collecting and sorting out the equipment's alarm logs and maintenance records, as well as conducting on-site investigations and verifications, the equipment's failure time data was compiled.

[0012] Perform a trend test on the failure time data. If the test results show that the equipment failure intensity is increasing and the equipment is in the "wear-out period" stage, then proceed to step S3.

[0013] The fault intensity function of LPIM is

[0014] (2)

[0015] In equation (2), This indicates the number of failures per unit of time. It is the baseline fault intensity function. To repair the efficacy factors, The average cumulative number of failures; LPIM believes that each maintenance activity affects equipment reliability, and even if the severity of the failure changes, the cumulative repair effect is... To reflect; when When, it indicates that maintenance activities have an effect on improving equipment reliability; when When the maintenance activity has no significant impact on equipment reliability, it indicates that the maintenance activity has no significant impact on equipment reliability; when This indicates that maintenance activities have worsened the reliability of the equipment;

[0016] The average cumulative number of failures is expressed as

[0017] (3)

[0018] It can be deduced that

[0019] (4)

[0020] A three-parameter boundary strength process model is used as the benchmark fault strength function for LPIM to describe the fault process. Defined as

[0021] (5)

[0022] In equation (5), 𝛼 is the maximum fault intensity of the system; 𝛽 is the time taken for the fault intensity to increase from the initial intensity to 𝛼; 𝜀 represents the fault intensity function along time. t The position of the axis changes; when t=0, This value represents the actual fault intensity at the initial moment. hour, ;

[0023] Will Substituting into equation (2), we can obtain

[0024] (6)

[0025] The closed-form solution can be obtained by solving the differential equation, which can be obtained from equations (4) and (6).

[0026] (7)

[0027] By transforming both sides of equation (7) and taking the indefinite integral, we obtain

[0028] (8)

[0029] Solving equation (8), we can obtain

[0030] (9)

[0031] According to boundary conditions It can be found that Substituting into equation (9), we get

[0032] (10)

[0033] Taking the first derivative of equation (10), the fault intensity function can be obtained as follows:

[0034] (11)

[0035] S41: Maximum likelihood estimation is used to estimate model parameters: assuming a single device is within the fault observation interval... The time of the internal failure is ,when The time is the fault truncation Experiment; is a time-truncated experiment; to obtain the maximum likelihood estimate of the model parameters, the conditional distribution function is defined as follows:

[0036] (12)

[0037] No. i The conditional reliability probability density function of the equipment before the next failure is:

[0038] (13)

[0039] No. i Secondary failure time The conditional probability density function is expressed as:

[0040] (14)

[0041] If the fault time data is truncated, then the likelihood function is:

[0042] (15)

[0043] In equation (15), T is the test end time; if the fault time data is a fault truncation, then ,and For a single device, when the failure time data is truncated, its likelihood function is:

[0044] (16)

[0045] Taking the logarithm of both sides of equation (16) yields its log-likelihood function expression, which is:

[0046] (17)

[0047] When the fault time data is fault-trunculated, its likelihood function is: (18)

[0048] Taking the logarithm of both sides of equation (18) yields its log-likelihood function expression, which is:

[0049] (19)

[0050] Then take the partial derivatives of equation (19) with respect to , , , and .

[0051] (20)

[0052] (twenty one)

[0053] (twenty two)

[0054] (twenty three)

[0055] Setting equations (20) to (23) equal to 0, we obtain the likelihood equation system.

[0056] (twenty four)

[0057] S42: Solve equation system (24) using the PSO algorithm to obtain the maximum likelihood point estimates of , , , . Let the equation system (24) contain... The system of equations (24) is rewritten as follows:

[0058] (25)

[0059] Solving the system of equations (25) can be transformed into solving the following optimization problem: (26)

[0060] Since the objective function consists of four transcendental equations, the search for the optimal solution first considers the maximum value of any possible solution within the objective function. Based on this, search the solution space to minimize the value of the maximum value in the solution space. If the fitness of the corresponding model parameters is the highest, then the solution is close to the true value.

[0061] Based on the solved model parameters, the maximum likelihood point estimates of the equipment reliability index are expressed as follows:

[0062] (1) Instantaneous fault intensity function The point estimate is (27)

[0063] (2) The longer the Instantaneous Mean Time Between Failures (IMTBF), the more reliable the equipment, expressed as: (28)

[0064] Then its point estimate is

[0065] (29)

[0066] (3) Average cumulative number of failures The point estimate is

[0067] (30)

[0068] (4) The cumulative mean time between failures is expressed as: (31)

[0069] Then its point estimate is (32)

[0070] (5) The cumulative fault intensity function is expressed as follows: (33)

[0071] Then its point estimate is (34)

[0072] Confidence intervals for estimating model parameters using the asymptotic log-normal distribution property of maximum likelihood estimates. (35)

[0073] In equation (35), For parameters to be estimated or reliability indicators; For confidence level The quantiles of the standard normal distribution; The standard deviation of the estimated parameter or reliability index is given. ; The variance of the parameter or reliability index to be estimated is given; by finding the variance of the parameter or reliability index, the confidence interval can be obtained.

[0074] Model parameters of a single device using the inverse Fisher information matrix method and The variance var and covariance cov are respectively:

[0075] (36)

[0076] By taking the second-order partial derivatives of each parameter in equation (19), we get (37)

[0077] (38)

[0078] (39)

[0079] (40)

[0080] (41)

[0081] (42)

[0082] (43)

[0083] (44)

[0084] (45)

[0085] (46)

[0086] Substituting the values ​​of equations (37)-(46) into the right side of equation (36), and then finding the inverse matrix on the right side, we can obtain the result. Substituting these values ​​into equation (35) will yield the confidence interval for the parameter to be estimated.

[0087] The confidence intervals for reliability indices are estimated using the Delta method: Let and be the reliability indices respectively. The variance of is (47)

[0088] Using equipment reliability indicators respectively To replace 𝜑, find the corresponding partial derivative in equation (47), i.e. equation (48)-(67), and then substitute the variance var and covariance cov calculated by equation (36) into equation (47) to obtain the confidence interval of the reliability index from equation (35).

[0089] (48)

[0090] (49)

[0091] (50)

[0092] (51)

[0093] (52)

[0094] (53)

[0095] (54)

[0096] (55)

[0097] (56)

[0098] (57)

[0099] (58)

[0100] (59)

[0101] (60)

[0102] (61)

[0103] (62)

[0104] (63)

[0105] (64)

[0106] (65)

[0107] (67)

[0108] S81: The Akaike Information Criterion (AIC) is based on the concept of entropy to balance the complexity of the estimated model with the goodness of fit to the data. It further introduces the Bayesian Information Criterion (BIC) with a relatively larger penalty term to prevent excessive model complexity caused by high model accuracy. Its expression is defined as follows:

[0109] (68)

[0110] (69)

[0111] In the formula, m The number of model parameters. n Let be the number of fault time data points, and maxln φ be the maximum log-likelihood function of the fault time data.

[0112] S82: To further verify that the established model is the best model for describing the equipment failure process, and to test the model's fit, a goodness-of-fit test is performed; the goodness-of-fit evaluation index is defined as follows:

[0113] (70)

[0114] In the formula, for i The actual cumulative number of faults at any given time. R is an estimate of the expected number of faults. The larger the R value, the better the model fit.

[0115] As a preferred embodiment, step S2, the failure time data trend verification, is as follows:

[0116] The TTT chart method was used to test the trend of failure time data.

[0117] (1) Let n This represents the total number of equipment failures. S 0=0, calculate TTT value S i ;

[0118] (1)

[0119] (2) Regularized TTT value S i ,but ;

[0120] (3) with i / n The x-axis is... Plot a TTT chart with the vertical axis as the ordinate;

[0121] If the scattered points in the TTT graph are located above the diagonal of the unit square, it indicates that the equipment failure intensity is increasing and the equipment is in the "wear-out period" stage. Then, proceed to step S3.

[0122] As a preferred embodiment, step S2, the fault time data trend verification, is specifically as follows: using the cumulative fault number versus time graph verification method, a cumulative fault number versus time graph is plotted. When the graph is concave, it indicates that the system fault intensity has an increasing trend and the equipment is in the "wear-out period" stage, then step S3 is executed.

[0123] As a preferred approach, the cumulative failure count versus time plot test is used to plot the cumulative failure count versus time plot. When the sample size exceeds 90, the total observation time is divided into several equal time intervals. The cumulative failure count versus time plot test is applied to the data in each time interval to plot the cumulative failure count versus time plot. Simultaneously, Equation (A) is used to calculate the average failure rate for each interval. (A)

[0124] In formula (A), From time 0 to the i The total number of faults observed within a time interval. It is the length of each time interval; if the graph is concave and each calculated If the value increases successively, it indicates that the device performance is deteriorating, then proceed to step S3.

[0125] The beneficial effects of this invention are:

[0126] This invention proposes a reliability modeling and evaluation method for repairable equipment considering incomplete maintenance. The invention uses 3-BIP as the baseline fault intensity function to modify the LPIM model, models the reliability of repairable equipment with incomplete maintenance, and uses the PSO algorithm to solve the model parameters. This reliability model has clear physical meaning and quantifies the impact of maintenance activities on reliability, making it applicable to solving practical engineering problems. The invention also considers practical engineering realities such as the initial fault intensity being non-zero, the fault intensity increasing with service time, and asymptotically approaching a certain constant. The reliability modeling and evaluation method for repairable equipment considering incomplete maintenance proposed in this invention allows for both clear boundary characteristics and a certain range of flexibility when using reliability evaluation results for maintenance decisions, avoiding both conservative and aggressive maintenance decisions. This invention is a beneficial supplement to existing reliability modeling and evaluation methods. Attached Figure Description

[0127] Figure 1 This is a flowchart of the method.

[0128] Figure 2 This is a TTT (Time-to-Time) graph of the fault time data of 12 wind turbine generators in the embodiment.

[0129] Figure 3 This is a fitted curve of the cumulative number of faults of the 12 wind turbine generators in the embodiment.

[0130] Figure 4 The curves are fitting curves of the instantaneous fault intensity function of 12 wind turbine generator sets in the embodiment. Detailed Implementation

[0131] The following is in conjunction with the appendix Figures 1 to 4 Tables 1 to 4 and specific embodiments provide a detailed description of the present invention. Obviously, the described embodiments are only a portion, not all, of the embodiments of the present invention. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without inventive effort should fall within the scope of protection of the present invention.

[0132] A method for reliability modeling and evaluation of repairable equipment that considers incomplete repair, comprising the following steps:

[0133] Taking 12 wind turbine units of the same model from a certain company that have been in operation for 10 years as an example, with a designed lifespan of 20 years, we collected and analyzed alarm logs, maintenance records, and other information, as well as conducted on-site investigations and verifications. We compiled fault time data for the 12 units from 00:00 on January 1, 2019 to 24:00 on December 31, 2021, for a total observation time T=26280h. The collected fault times are shown in Table 1. Because the fault time data in this example is truncated, the following analysis focuses on the truncated fault data. The analysis approach for time-truncated fault data remains consistent and will not be repeated here.

[0134] Table 1. Failure time data for 12 generating units

[0135]

[0136] The TTT (Total Time on Test) chart method was used to test the trend of failure time data. Let n be the total number of failures in the unit. S 0=0, calculate TTT value S i Regularized TTT value S i ,but . i / n The x-axis is... A TTT chart was plotted with the vertical axis as the ordinate. The results are shown in the attached figure. Figure 2 As shown in the figure. i / n For the first i The ratio of individual faults to the total number of faults. (From Appendix) Figure 2 As can be seen, except for a few units with individual fault points in the early stages, the remaining scattered points are all distributed above the diagonal of the unit square, indicating that the fault intensity of the units is increasing and they are in the "wear-out period". Therefore, the model proposed in this invention can be used for reliability modeling and analysis.

[0137] (1) Substitute the failure time data in Table 1 into the reliability model established in step S3;

[0138] (2) The maximum likelihood estimates of the model parameters of the 12 units are obtained by using step S4;

[0139] (3) Calculate the confidence interval of the model parameters with a confidence level of 95% using step S6.

[0140] The maximum likelihood estimates and confidence intervals of the model parameters are shown in Table 2.

[0141] Table 2 Point and Interval Estimates of Model Parameters

[0142]

[0143] (1) Substitute the point estimation results of the model parameters in Table 2 into equations (27)-(34) in step S5 to calculate the point estimation results of the key reliability index at the time of the last failure.

[0144] (2) Use step S4 to solve for the confidence interval of the reliability index with a confidence level of 95%.

[0145] The estimated values ​​of the reliability index and the calculation results of the confidence interval are shown in Table 3.

[0146] Table 3 Point and Interval Estimations of Reliability Indicators

[0147]

[0148] S7: Goodness-of-fit test

[0149] The model proposed in this invention was compared with other models (LPIM, 3-BIP), and the AIC and BIC calculation results of different models are shown in Table 4. Clearly, the model proposed in this invention has the lowest AIC and BIC, preliminarily indicating that the model proposed in this invention is the optimal model.

[0150] To further verify that the established model is the best model for describing the unit failure process, and to test the model's fitting effect, formula (70) was used to perform a goodness-of-fit test. The calculation results are shown in Table 4. The goodness-of-fit evaluation index R of the model proposed in this invention is the highest, further demonstrating that the model proposed in this invention is the best model.

[0151] Table 4. Comparison of goodness-of-fit test results for different model parameters

[0152]

[0153] The maximum likelihood estimation fitting curve for the cumulative number of failures of the 12 units is attached. Figure 3 Appendix Figure 3 The image shows a scatter plot of the observed failure time data, the model's predicted failure number, and its 95% confidence interval. Clearly, the estimated cumulative failure number curve fits the failure time data well.

[0154] The fitting results of the instantaneous fault intensity function for the 12 generating units and their 95% confidence intervals are attached. Figure 4 It can be seen that:

[0155] (1) The initial fault intensity of all units is not zero because the period before the actual wear-out period is the period of random faults, and the fault intensity during the period of random faults is not zero. The fault intensity of units in the wear-out period increases, but it does not increase indefinitely; instead, it approaches a certain constant. This shows that the method proposed in this invention, which uses 3-BIP as the benchmark fault intensity function for LPIM for reliability modeling, is consistent with engineering practice.

[0156] (2) Except for Units 5 and 12, the fault intensity of the other units showed a clear trend of first increasing and then decreasing. Compared with the actual situation, the gearboxes of these 10 units had experienced serious faults such as planetary gear breakage, housing bolt breakage, and gear tooth breakage, resulting in a high failure rate of the gearboxes. It was verified that this was caused by a design problem of the wind turbine. Subsequently, the wind farm replaced the gearboxes. With continuous break-in and maintenance, the fault intensity decreased significantly, indicating that incomplete maintenance activities have an effect on improving the reliability of the units.

[0157] The above embodiments are merely illustrative of the principles and effects of the present invention, as well as some examples of its application, and are not intended to limit the present invention. It should be noted that those skilled in the art can make various modifications and improvements without departing from the inventive concept of the present invention, and these modifications and improvements are all within the scope of protection of the present invention.

Claims

1. A method for reliability modeling and evaluation of repairable equipment considering incomplete repair, comprising the following steps: S1: Analysis of Failure Time Data By collecting and sorting out the equipment's alarm logs and maintenance records, as well as conducting on-site investigations and verifications, the equipment's failure time data was compiled. S2: Failure Time Data Trend Verification Perform a trend test on the failure time data. If the test results show that the equipment failure intensity is increasing and the equipment is in the "wear-out period" stage, then proceed to step S3. S3: Reliability Modeling The fault intensity function of LPIM is （2） In equation (2), This indicates the number of failures per unit of time. It is the baseline fault intensity function. To repair the efficacy factors, The average cumulative number of failures; LPIM believes that each maintenance activity affects equipment reliability, and even if the severity of the failure changes, the cumulative repair effect is... To reflect; when When, it indicates that maintenance activities have an effect on improving equipment reliability; when When the maintenance activity has no significant impact on equipment reliability, it indicates that the maintenance activity has no significant impact on equipment reliability; when This indicates that maintenance activities have worsened the reliability of the equipment; The average cumulative number of failures is expressed as （3） It can be deduced that （4） A three-parameter boundary strength process model is used as the benchmark fault strength function for LPIM to describe the fault process. Defined as （5） In equation (5), 𝛼 is the maximum fault strength of the system; 𝛽 is the time taken for the fault strength to increase from the initial strength to 𝛼; 𝜀 represents the fault intensity function along time t Changes in the position of the axis; When t=0, This value represents the actual fault intensity at the initial moment. hour, ; Will Substituting into equation (2), we can obtain （6） The closed-form solution can be obtained by solving the differential equation, which can be obtained from equations (4) and (6). （7） By transforming both sides of equation (7) and taking the indefinite integral, we obtain （8） Solving equation (8), we can obtain （9） According to boundary conditions It can be found Substituting into equation (9), we get （10） Taking the first derivative of equation (10), the fault intensity function can be obtained as follows: （11） S4: Point Estimation of Reliability Model Parameters S41: Maximum likelihood estimation is used to estimate model parameters: assuming a single device is within the fault observation interval... The time of the internal failure is ,when The time is the fault truncation Experiment; is a time-truncated experiment; to obtain the maximum likelihood estimate of the model parameters, the conditional distribution function is defined as follows: （12） No. i The conditional reliability probability density function of the equipment before the next failure is: （13） No. i Secondary failure time The conditional probability density function is expressed as: （14） If the fault time data is truncated, then the likelihood function is: （15） In equation (15), T is the test end time; if the fault time data is a fault truncation, then ,and For a single device, when the failure time data is truncated, its likelihood function is: （16） Taking the logarithm of both sides of equation (16) yields its log-likelihood function expression, which is: （17） When the fault time data is fault-trunculated, its likelihood function is: （18） Taking the logarithm of both sides of equation (18) yields its log-likelihood function expression, which is: （19） Then take the partial derivatives of equation (19) with respect to , , , and . （20） （21） （22） （23） Setting equations (20) to (23) equal to 0, we obtain the likelihood equation system. （24） S42: Solve equation system (24) using the PSO algorithm to obtain the maximum likelihood point estimates of , , , . Let the equation system (24) contain... The system of equations (24) is rewritten as follows: （25） Solving the system of equations (25) can be transformed into solving the following optimization problem: (26) Since the objective function consists of four transcendental equations, the search for the optimal solution first considers the maximum value of any possible solution within the objective function. Based on this, search the solution space to minimize the value of the maximum value in the solution space. If the fitness of the corresponding model parameters is the highest, then the solution is close to the true value. S5: Reliability Index Point Estimation Based on the solved model parameters, the maximum likelihood point estimates of the equipment reliability index are expressed as follows: (1) Instantaneous fault intensity function The point estimate is (27) (2) The longer the Instantaneous Mean Time Between Failures (IMTBF), the more reliable the equipment, expressed as: (28) Then its point estimate is （29） (3) Average cumulative number of failures The point estimate is （30） (4) The cumulative mean time between failures is expressed as: (31) Then its point estimate is (32) (5) The cumulative fault intensity function is expressed as follows: (33) Then its point estimate is (34) S6: Interval Estimation of Model Parameters Confidence intervals for estimating model parameters using the asymptotic log-normal distribution property of maximum likelihood estimates. (35) In equation (35), For parameters to be estimated or reliability indicators; For confidence level The quantiles of the standard normal distribution; The standard deviation of the estimated parameter or reliability index is given. ; The variance of the parameter or reliability index to be estimated is given; by finding the variance of the parameter or reliability index, the confidence interval can be obtained. Model parameters of a single device using the inverse Fisher information matrix method and The variance var and covariance cov are respectively: （36） By taking the second-order partial derivatives of each parameter in equation (19), we get (37) （38） （39） （40） （41） （42） （43） （44） （45） （46） Substituting the values ​​of equations (37)-(46) into the right side of equation (36), and then finding the inverse matrix on the right side, we can obtain the result. Substituting these values ​​into equation (35) will yield the confidence interval for the parameter to be estimated. S7: Reliability Index Range Estimation The confidence intervals for reliability indices are estimated using the Delta method: Let and be the reliability indices respectively. The variance of is (47) Using equipment reliability indicators respectively To replace 𝜑, find the corresponding partial derivative in equation (47), i.e. equation (48)-(67), and then substitute the variance var and covariance cov calculated by equation (36) into equation (47) to obtain the confidence interval of the reliability index from equation (35). （48） （49） （50） （51） （52） （53） （54） （55） （56） （57） （58） （59） （60） （61） （62） （63） （64） （65） （67） S8: Goodness-of-fit test S81: The Akaike Information Criterion (AIC) is based on the concept of entropy to balance the complexity of the estimated model with the goodness of fit to the data. It further introduces the Bayesian Information Criterion (BIC) with a relatively larger penalty term to prevent excessive model complexity caused by high model accuracy. Its expression is defined as follows: （68） （69） In the formula, m The number of model parameters. n Let be the number of fault time data points, and maxln φ be the maximum log-likelihood function of the fault time data. S82: To further verify that the established model is the best model for describing the equipment failure process, and to test the model's fit, a goodness-of-fit test is performed; the goodness-of-fit evaluation index is defined as follows: （70） In the formula, for i The actual cumulative number of faults at any given time. R is an estimate of the expected number of faults. The larger the R value, the better the model fit.

2. The reliability modeling and evaluation method for repairable equipment considering incomplete maintenance as described in claim 1, characterized in that: Step S2, the failure time data trend verification, is as follows: The TTT chart method was used to test the trend of failure time data. (1) Let n This represents the total number of equipment failures. S 0=0, calculate TTT value S i ; （1） (2) Regularized TTT value S i ,but ; (3) with i / n The x-axis is... Plot a TTT chart with the vertical axis as the ordinate; If the scattered points in the TTT graph are distributed above the diagonal of the unit square, it indicates that the equipment failure intensity is increasing and it is in the "wear-out period" stage, then proceed to step S3.

3. The reliability modeling and evaluation method for repairable equipment considering incomplete maintenance as described in claim 1, characterized in that: Step S2, the fault time data trend test, is as follows: the cumulative fault number versus time graph test method is used to draw the cumulative fault number versus time graph. When the graph is concave, it indicates that the system fault intensity has an increasing trend and the equipment is in the "wear-out period" stage. Then, step S3 is executed.

4. The reliability modeling and evaluation method for repairable equipment considering incomplete maintenance as described in claim 3, characterized in that: The cumulative failure count versus time plot test method was used to plot the cumulative failure count versus time plot. When the sample size exceeded 90, the total observation time was divided into several time intervals of equal size. The cumulative failure count versus time plot test method was used for the data in each time interval to plot the cumulative failure count versus time plot. At the same time, the average failure rate of each interval was calculated using equation (A). (A) In formula (A), From time 0 to the i The total number of faults observed within a time interval. It is the length of each time interval; If the graph is concave and each calculated If the value increases successively, it indicates that the device performance is deteriorating, then proceed to step S3.

Images