Lifetime Prognostics for Multi-Phase Degradation with Random Jump at the Change Point

Owing to operating condition changing, physical mutation, and sudden shocks, degradation trajectories usually exhibit multi-phase features, and the abrupt jump often appears at the changing time, which makes the traditional methods of lifetime estimation unavailable. In this paper, we mainly focus on how to estimate the lifetime of the multi-phase degradation process with abrupt jumps at the change points under the concept of the first passage time (FPT). Firstly, a multi-phase degradation model with jumps based on the Wiener process is formulated to describe the multi-phase degradation pattern. Then, we attain the lifetime’s closed-form expression for the two-phase model with fixed jump relying on the distribution of the degradation state at the change point. Furthermore, we continue to investigate the lifetime estimation of the degradation process with random effect caused by unit-to-unit variability and the multi-phase degradation process. We extend the results of the two-phase case with fixed parameters to these two cases. For better implementation, a model identification method with off-line and on-line parts based on Expectation Maximization (EM) algorithm and Bayesian rule is proposed. Finally, a numerical case study and a practical example of gyro are provided for illustration.

Measurement error and random failure threshold widely exist in the real degradation process. Aiming at the current methods about the remaining useful lifetime (RUL) estimation without considering the influence of measurement error and random failure threshold, an approximate analytical RUL distribution in a closed-form of a linear Wiener based degradation process with measurement errors is proposed. The expectation maximization (EM) algorithm is used to estimate the unknown parameters in the model, and the Bayesian method is used to update the model parameters. Then, the probability density function (PDF) of RUL is derived based on the random failure threshold. The simulation results show that considering measurement errors and random failure threshold can significantly improve the accuracy of RUL estimation.

Richtmyer–Meshkov instability (RMI) in gas-particle mixtures is investigated both numerically and analytically. The linear amplitude growth rate for a RMI in a two-phase mixture is derived by using a dusty gas formulation for small Stokes number (St⪡1.0), and it is shown that the problem can be characterized by mass loading and St. The model predictions are compared with numerical results under two conditions, i.e., a shock wave hitting (1) a perturbed species interface of air and SF6 surrounded by uniformly distributed particles, and (2) a perturbed shape particle cloud in uniform air. In the first case, the interaction between the instability of the species perturbation and the particles is investigated. The multiphase growth model accurately predicts the growth rates when St⪡1.0, and the amplitude growth normalized by the two-phase RMI velocity shows good agreement with the single-phase RMI growth rate as well. It is also shown that the two-phase model results are in accordance with the growth rates obtained from the simulations even for cases corresponding to St≈10. However, for St⪢10, particles do not follow the RMI motion, and the RMI growth rate agrees with the original Richtmyer’s model [R. D. Richtmyer, “Taylor instability in shock acceleration of compressible fluids,” Commun. Pure Appl. Math. 13, 297 (1960)]. Preferential concentration of particles are observed around the RMI roll-ups at late times when St is of order unity, whereas when St⪡1.0, the particles respond rapidly to the flow, causing them to distribute within the roll-ups. In the second problem, the two-phase RMI growth model is extended to study whether a perturbed dusty gas front shows RMI-like growth due to the impact of a shock wave. When St⪡1.0, good agreement with the multiphase model is again seen. Moreover, the normalized growth rates are very close to the single-phase RMI growth rates even at late times, which suggest that the two-phase growth model is applicable to this type of perturbed shape particle clouds as well. However, when St is close to unity or larger (St>1.0), the particles do not experience impulsive acceleration but rather a continuous one, which results in exponential growth rates as seen in a Rayleigh–Taylor instability.

Machinery condition prognosis systems use long-term historical data to predict the remaining useful life (RUL). One of the critical steps to reach this purpose is to segment long-term data into two or several degradation stages (healthy, unhealthy, critical stage). Finding a changing point between stages may be a crucial preliminary task for further prediction of degradation process. However, finding the accurate partition into two or more stages is a challenging task in actual application when noise inherent in the observed process exhibits non-Gaussian characteristics. In this paper, a framework for data-driven segmentation is presented for prognosis of machinery long-term data in presence of heavy-tailed distributed noise with finite variance. It is assumed that three different stages are inherent in degradation process and each segment of data follows a specific trend (constant, linear, exponential or polynomial). At first, data is divided into three parts. Trend functions are fitted to the data by using robust regression method, and cumulative error is calculated. This process is done iteratively for all possible partitions into three intervals to find the segmentation which minimizes the error. The framework has been tested via empirical analysis of estimators of the changing points obtained in Monte Carlo simulations. Also, discussed approaches are applied to the real data. In such measurement, data that are commonly available (in condition monitoring systems) is aggregated from the raw signal and sampled at long intervals. Finally, effectiveness of the segmentation results is assessed by comparing them with envelope frequency analysis of raw signal to confirm the fact that detected changing points coincide with start time of the fault in the machine or not.

Reliability analysis for a Wiener degradation process model under changing failure thresholds

In this paper, a novel condition‐based maintenance (CBM) policy for systems with mutually dependent competing failure processes (DCFPs) is studied. The cumulative random shocks contribute to the increased rate of degradation, meanwhile, the hard failure threshold is reduced when the amount of degradation reaches a threshold. The degradation process is simulated by a non‐stationary Gamma process and random shocks are modeled as a homogeneous Poisson process. Preventive replacement (PR) and corrective replacement (CR) are accounted for in the developed maintenance strategy. The optimal periodic inspection interval, PR threshold, and the number of cumulative random shocks can be derived by minimizing the average long‐term maintenance cost rate. In the end, a numerical case is applied to demonstrate the availability of the developed models.

Joint optimization of preventive maintenance and inventory management for standby systems with hybrid-deteriorating spare parts

Cost-effectiveness criteria are easier to state in theory than to apply in practice. The cumulative cost-per-foot (CCF) criterion for ending the use of a rock bit is an example. Ref. 1 simply advises, "When the cost per foot begins to increase, the bit should be pulled." This prescription then is illustrated by a numerical case in prescription then is illustrated by a numerical case in which the calculated CCF increased for the first time by less than 0.5% over a 10-ft depth increment. At this point there are two major sources of uncertainty in the point there are two major sources of uncertainty in the decision either to stop or to continue drilling:inaccuracies in the measurement and prediction of variable input quantities (footage, drilling time, and trip time) to the CCF equation andlack of precise knowledge of possible formation changes affecting future penetration possible formation changes affecting future penetration rates. The first source suggests that a 0.5% increase may be more numerical than real; the second admits the chance that the CCF may "turn around" i.e., decrease to a new overall minimum after once increasing. A well-known drilling assistance program, COSTCAL, eliminates much of the first type of uncertainty. An incremental cost-per-foot (ICF) calculation is performed for each, say, 10-ft depth increase. performed for each, say, 10-ft depth increase. COSTCAL argues that bit and trip costs need not be prorated to the ICF's. Whether they are or not, the prorated to the ICF's. Whether they are or not, the principle of the program remains the sameas depth principle of the program remains the sameas depth increases, the ICF tends to overtake and eventually to exceed the CCF, and it becomes uneconomical to continue drilling with the used bit. We then would have a sharp criterion for bit pull-out when ICF first exceeds CCF if the second source of uncertainty were not present. The numerical example and discussion in Ref. 2 illustrate the resulting ambiguity by the statements: ". . . the bit should have been pulled at 11,210 feet" and "when ‘$‘ is to the right of ‘C' [ICF exceeds CCF] a change should be considered." The italics are added. The COSTCAL data are given in Table 1 for depths above and below the transition depth of 11,210 ft, at which the ICF first became greater than the CCF. These data show that pulling the bit at 11,210 ft is indeed premature; after an additional 40 ft of drilling, the CCF reached a new absolute minimum! What is needed at the depth at which ICF first exceeds CCF is a probabilistic test of the sequence of data points for the probabilistic test of the sequence of data points for the subsequent increments. The test should proceed from the question, "Now that the difference D = ICF-CCF first has become positive, what is the probability that there is a trend toward increasingly larger values of D?" If this probability is sufficiently large when combined with an probability is sufficiently large when combined with an inexact prediction of deeper strata changes and consequent penetration-rate effects, the driller could decide to penetration-rate effects, the driller could decide to pull the bit with full recognition of the complexities of pull the bit with full recognition of the complexities of the problem and with a nice combination of analysis and judgment. Fortunately, a very suitable test for the trend in D appears in the statistical literature. Called the Cox and Stuart Test for Trend, it tests any sequence of D's for an upward trend, beginning with the first D that is greater than zero. Denoting the number of D's generally by n, the number of comparisons by c, and the first D in a sequence by D1, the next D2, etc., we begin the test by grouping the D's into pairs (D1, Dc+1), (D2, Dc+2) . . . (Dn-c, Dn), where c = n/2 if n is even, and c = (n+1)/2 if n is odd. If Di less than Dc +i for any i, we replace the pair with a "+"; if Di greater than Dc +i, we write a "--" . In substance, we are comparing the first half of the sequence of n D's with the latter half. There are c distinct comparisons. The presence of more +'s for a given sequence length is stronger evidence for an upward trend in the D's. The Cox and Stuart Test for Trend is performed in this application in "real time," i.e., after several increments following our first indication that continued drilling is becoming uneconomical, we make the test. If the data show insufficient evidence to stop drilling, we drill another increment and reperform the test with the greater set of data. At no point are data discarded. p. 2903

The estimation of PV-modules lifetime facilitates the further development and helps to lower risks for producers and investors. One base for this extensive testing and simulation work is the knowledge of the chemical degradation processes and their kinetics, as well as of the permeation of water and oxygen into the module, especially of the encapsulant. Besides ethylen-vinylacetate copolymer (EVA), which is the dominant material for encapsulation, new materials become available and need the assessment of their properties and the durability impact. Accelerated durability tests were performed on different EVA materials. The paper reports on several measurement methods for analysis of the polymers that were used, FT-IR with attenuated total reflection (ATR), and Raman microscopy, e.g. It is very important to identify degradation products and intermediates in order to identify the leading degradation processes and their kinetics as well as potential interactions between different processes. Another important factor for the degradation of the PV-modules and the concerned polymers in particular is the permeation of reactive substances, especially of water vapor, into and inside the modules. The paper shows results of permeation measurements of the new materials, as well as FEM-based numerical simulations of the humidity diffusion within a PV-module what is an important step towards the calculation of the chemical degradation using numerical simulation tools in the future.

Despite its ever-increasing computing power, dynamic origin–destination (OD) estimation in congested networks remains troublesome. In previous research, we have shown that an unbiased estimation requires the calculation of the sensitivity of the link flows to all origin–destination flows, in order to incorporate the effects of congestion spillback. This is, however, computationally infeasible for large-scale networks. To overcome this issue, we propose a hierarchical approach for off-line application that decomposes the dynamic OD estimation procedure in space. The main idea is to perform a more accurate dynamic OD estimation only on subareas where there is congestion spillback. The output of this estimation is then used as input for the OD estimation on the whole network. This hierarchical approach solves many practical and theoretical limitations of traditional OD estimation methods. The main advantage is that different OD estimation method can be used for different parts of the network as necessary. This allows applying more advanced and accurate, but more time-consuming, methods only where necessary. The hierarchical approach is tested on a study network and on a real network. In both cases the proposed methodology performs better than traditional OD estimation approaches, indicating its merit.

A program is described for fitting a regression model in which the relationship between the dependent and the independent variables is described by two regression equations, one for each of two mutually exclusive ranges of the independent variable. The point at which the change from one equation to the other occurs is often unknown, and thus must be estimated. In cognitive psychology, such models are relevant for studying the phenomenon of strategy shifts. The program uses a (weighted) least squares algorithm to estimate the regression parameters and the change point. The algorithm always finds the global minimum of the error sum of squares. The model is applied to data from a mental-rotation experiment. The program’s estimates of the point at which the strategy shift occurs are compared with estimates obtained from a nonlinear least squares minimization procedure in SPSSX.

One classic problem in air traffic management (ATM) has been the problem of detection and resolution of conflicts between aircraft. Traditionally, a conflict between two aircraft is detected whenever the two protective cylinders surrounding the aircraft intersect. In Trajectory-based Air Traffic Management, a baseline for the next generation of air traffic management system, we suggest that these protective cylinders be deformable volumes induced by variations in weather information such as wind speed and directions subjected to uncertainties of future states of trajectory controls. Using contact constraints on deforming parametric surfaces of these protective volumes, a constrained minimization algorithm is proposed to compute collision between two deformable bodies, and a differential optimization scheme is proposed to resolve detected conflicts. Given the covariance matrix representing the state of aircraft trajectory and its control and objective functions, we consider the problem of maximizing the variance explained by a particular linear combination of the input variables where the coefficients in this combination are required to be non-negative, and the number of non-zero coefficients is constrained (e.g. state of trajectory and estimated time of arrival over one change point). Using convex relaxation and re-weighted l 1 technique, we reduce the problem to solving some semi-definite programming ones, and reinforce the non-negative principal components that satisfy the sparsity constraints. Numerical results show that the method presented in this paper is efficient and reliable in practice. Since the proposed method can be applied to a wide range of dynamic modeling problems such as collision avoidance in dynamic autonomous robots environments, dynamic interactions with 4D computer animation scenes, financial asset trading, or autonomous intelligent vehicles, we also attempt to keep all descriptions as general as possible.

Bus dwelling time at stops is affected by the number of boarding and alighting passengers. Due to the uncertainty of traffic state, dwelling time will become unstable, resulting in unreliable bus operations. A short-turn strategy shares passenger demand in certain segments of the full-length vehicles; therefore, this strategy can reduce the dwell time of the full-length vehicles and have potential to improve reliability of bus operations. Firstly, this paper builds a bus operation state model considering short-turn vehicles. Secondly, the process of bus operation is analyzed to investigate the impact of passenger demand on reliability. Then, we propose a short-turn dispatching strategy, which can accurately determine the operation area and departure time. Finally, a numerical simulation is used to evaluate different strategies. As the result, the analysis reveals the following: (1) the bus operational reliability of full-length vehicles under high passenger demand could be improved by a well-designed short-turn strategy; (2) it is beneficial to improve the reliability of bus operation when short-turn vehicles departure at the change point; and (3) an undesigned short-turn dispatching strategy has a negative effect on full-length vehicles operational reliability. This study can help design a short-turn strategy and make valuable contributions to the bus management system.

Depth-averaged systems of equations describing the motion of fluid–sediment mixtures have been widely adopted by scientists in pursuit of models that can predict the paths of dangerous overland flows of debris. As models have become increasingly sophisticated, many have been developed from a multi-phase perspective in which separate, but mutually coupled sets of equations govern the evolution of different components of the mixture. However, this creates the opportunity for the existence of pathological instabilities stemming from resonant interactions between the phases. With reference to the most popular approaches, analyses of two- and three-phase models are performed, which demonstrate that they are more often than not ill posed as initial-value problems over physically relevant parameter regimes – an issue which renders them unsuitable for scientific applications. Additionally, a general framework for detecting ill posedness in models with any number of phases is developed. This is used to show that small diffusive terms in the equations for momentum transport, which are sometimes neglected, can reliably eliminate this issue. Conditions are derived for the regularisation of models in this way, but they are typically not met by multi-phase models that feature diffusive terms.

Degradation process has been successfully applied in modeling the deterioration failure modes like wearout and corrosion for reliability assessment. However, deterioration failure modes are different from each other due to different physical mechanisms with different products operated under different environments, and specific attentions should be paid for non-standard circumstances. As it is well known, the failure with fatigue crack is composed of two sequential stages: the crack initiation and the crack propagation. The lifetime distribution models have been applied for the crack initiation, and the degradation process is well modeled by degradation process. Intuitively, a lifetime delayed degradation process (LDDP) would provide a general framework for such typical mixture failure mode composed of hard and soft ones sequentially. In this study, a general reliability inference approach has been proposed for the LDDP based on the maximum likelihood method, with numerical simulations and practical application with real testing data. Furthermore, the inspection planning issue is also studied for LDDP, with the first inspection time addressed which can increase the efficiency of inspection.