Report on RAMS&LCC integrated modelsinfralert.eu/wp-content/multiverso-files/21_572b4a01a6b... ·...

This project has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No 636496

H2020-MG-8.1a-2014

INFRALERT: Linear Infrastructure Efficiency Improvement by Automated Learning and Optimised Predictive Maintenance Techniques

Grant Agreement number: 636496

Work Package WP5. RAMS & LCC models and analysis

Task T5.3. RAMS simulations at system level T5.4. LCC assessment at system level

Revision 4 Due date 31/01/2017

Revision date 02/02/2017 Submission date 03/02/2017

Deliverable type R

Deliverable leader

CEMOSA

Contributing partners

Fraunhofer IVI, Universidad de Sevilla, Infraestruturas de Portugal, Lulea Tekniska Universitet

Dissemination Level

PU Public X

CO Confidential, only for members of the consortium (including the Commission Services)

CI Classified, as referred to in Commission Decision 2001/844/EC.

Deliverable D5.2

Report on RAMS&LCC integrated models

D5.2 – RAMS and LCC integrated models. Revision 4 - 02/02/2017

INFRALERT - 636496 PU

Document status

Revision Date Description

1 15/07/2016 First draft (internal). General overview and outline of the work.

2 21/12/2016 Second draft. Significant changes included. Models incorporated. Case study analysis incorporated.

3 30/01/2017 Third draft. Grammar, spelling, typos has been included. Suggestions/recommendations from LTU included.

4 02/02/2017 Last improvements included in the final version

Status Final



Executive Summary

The aim of WP5 is to assess the RAMS and Life-Cycle Costs (LCC) of the whole infrastructure, related with the operation phase. RAMS parameters give measures of the reliability of our system, and the effectiveness of the maintenance interventions to keep targeted availability and safety levels. These availability and safety levels are achieved at the expense of a maintenance (operational) cost.

The deliverable D5.2 belongs to tasks 5.3 “RAMS Simulations at system level” and task 5.4 “LCC Assessment at system level”, from WP5, “RAMS & LCC models and analysis”. These tasks are a continuation of the work carried out in tasks 5.1 and 5.2 for failure data and RAMS analysis at component level which was reported in deliverable D5.1 (Ref. [1]). The main difference of this deliverable is that the reliability models applied here consider the system as a whole, composed by a set of components that may behave differently. Moreover, the description of how to implement RAMS including LCC analyses is described in this deliverable in detail.

This deliverable is intended to provide the basis for a system level analysis of Reliability, Availability, Maintainability and Safety, as well as the Life-Cycle Cost Analysis. Specifically, the present document reports the methodology implemented in WP5 for the integrated RAMS and LCC analysis, which is the basis for the next task 5.5, development of the algorithms, and the cornerstone of WP5.

Most of the analysis is intended to be as general as possible, although as an example, the study of S&Cs in the railway use case has been chosen to guide in the implementation and explanation of the methodologies. This has two objectives: in one side it allows for the analysis of such an important component of the railway infrastructure, and at the same time it implements part of the algorithms that will be finally provided in task 5.5 “Development of algorithms and tools for the use of RAMS&LCC analysis in the eIMS”. Therefore, what is presented in this deliverable will be part of the actual algorithms provided to the eIMS.

The document presents a thorough description of the RAMS models at system level in Section 3 and the LCC methodology that follows the assessment of the RAMS in Section 0. In Section 5 the methodology to integrate RAMS into LCC is explained. This methodology is demonstrated in Section 6 for a single case study, which is, as stated above, the S&Cs of the railway use case. This case study helps to understand how the models are implemented in practice. Supporting material and additional information is provided in the Appendices.

The methodologies and methods presented in this document might be subject to revision during the progress of INFRALERT in order to find an optimal fit with the other parts of the project.



Table of content

1 Background ......................................................................................................................................... 10

2 Objectives ........................................................................................................................................... 11

3 RAMS at System Level ........................................................................................................................ 13

3.1 Reliability model .......................................................................................................................... 13

3.1.1 Parametric Non-Homogeneous Poisson Process (NHPP) ..................................................... 16

3.1.2 Non-parametric estimators .................................................................................................. 17

3.1.3 Proportional hazard (PH) model ........................................................................................... 18

3.2 Availability and Maintainability Analysis ..................................................................................... 18

3.2.1 Parametric maintainability models ...................................................................................... 20

3.2.2 Proportional repair model (PRM) ......................................................................................... 20

3.2.3 Extended PRM with time-dependent covariates ................................................................. 21

3.3 Safety Analysis ............................................................................................................................. 21

4 LCC Assessment at System Level ........................................................................................................ 22

4.1 LCC methodology overview ......................................................................................................... 22

4.2 Cost model for cost drivers .......................................................................................................... 24

4.2.1 Acquisition Cost (LCCA) ........................................................................................................ 25

4.2.2 Termination Cost (LCCT) ....................................................................................................... 25

4.2.3 Ownership Cost (LCCO) ........................................................................................................ 25

4.2.3.1 Support Cost (LSC) ......................................................................................................... 25

4.2.3.2 Consequential or Unavailability Cost (LUC) ................................................................... 27

4.3 Asset degradation and maintenance policy ................................................................................ 27

4.4 Propagating uncertainties: sensitivity analysis ............................................................................ 28

5 Combined RAMS and LCC Analysis ..................................................................................................... 29

6 Case Study: Railway S&Cs Analysis ..................................................................................................... 30

6.1 The Case Study database ............................................................................................................. 30

6.2 Costs of maintenance .................................................................................................................. 32

6.3 Basic concepts about S&Cs .......................................................................................................... 33

6.4 Reliability model for S&Cs ........................................................................................................... 35

6.5 Cost analysis of maintenance actions on S&Cs ............................................................................ 37

7 Conclusions ......................................................................................................................................... 42

Appendices ............................................................................................................................................ 43

A. Useful relations and notation used in reliability formulas ........................................................ 43

B. Kaplan-Meier (KM) estimator of the survival function ............................................................. 44

C. Nelson-Aalen (NA) estimator of the cumulative hazard ........................................................... 44

D. Counting processes ................................................................................................................... 45



D.1 Definitions ............................................................................................................................... 45

D.2 Types of counting processes ................................................................................................... 46

D.3 Homogeneous Poisson process (HPP) .................................................................................... 47

D.4 Non-homogeneous Poisson process (NHPP) .......................................................................... 47

D.5 Parametric NHPP models ........................................................................................................ 48

D.6 NHPP models with time dependent covariates ...................................................................... 49

D.7 non-parametric estimators ..................................................................................................... 49

E. Symbols and abbreviations in LCC formulas ............................................................................. 51

F. Sample of raw data (for S&Cs) .................................................................................................. 52

F.1 Data from asset register database (BIS) .................................................................................. 52

F.2 Data from maintenance database (Ofelia) .............................................................................. 52

F.3 Data from inspection report database (Bessy) ........................................................................ 53

F.4 Data from track recording car (Optram) ................................................................................. 53

G. Sample of r code for ROCOF determination.............................................................................. 55

Glossary of terms................................................................................................................................... 57

References ............................................................................................................................................. 58



List of tables

Table 1: Models for Recurrent Events Data .......................................................................................... 16 Table 2: Availability and Maintainability parameters ........................................................................... 19 Table 3: Relation between asset phases and owner costs .................................................................... 22 Table 4: Example of cost in LCC analysis ............................................................................................... 23 Table 5: Operational volumes for rail maintenance in M€ (taken from ref. [27]) ................................ 32 Table 6: Subsystem components of an S&C .......................................................................................... 34 Table 7: S&C Components in the Case Study Database. ....................................................................... 35 Table 8: Estimated parameters for unit S&C No.23 .............................................................................. 36 Table 9: Data for S&C according to asset register ................................................................................. 38 Table 10: Types of S&Cs in the railway use case ................................................................................... 38 Table 11: Randomly generated maintenance cost/action in Euro ........................................................ 39

List of figures

Figure 1: Work flow in the RAMS/LCC analysis ..................................................................................... 11 Figure 2: Statistical models ................................................................................................................... 14 Figure 3: RAMS from Corrective and Preventive Maintenance ............................................................ 19 Figure 4: Cost element concept (IEC-60300-3-3) .................................................................................. 23 Figure 5: Example of Cost Breakdown Structure in LCC analysis .......................................................... 24 Figure 6: Schema of the combined RAMS/LCC calculation framework ................................................ 29 Figure 7: Railway system asset hierarchy tree ...................................................................................... 31 Figure 8: Diagram of a simple S&C (taken from http://www.railway-technical.com/) ........................ 34 Figure 9: System breakdown of an S&C ................................................................................................ 34 Figure 10: Event plot for the system of S&Cs ........................................................................................ 35 Figure 11: Non-parametric estimators for S&Cs reliability ................................................................... 36 Figure 12: Rate Of Occurrence of Failures modelled to unit 23 data ................................................... 37 Figure 13: Cumulative density of failures of unit 23 ............................................................................. 37 Figure 14: Distributions used in Monte Carlo simulations of LCC ......................................................... 40 Figure 15: Maintenance actions on S&Cs of track section 1 ................................................................. 40 Figure 16: Maintenance actions on S&Cs of track section 2 ................................................................. 40 Figure 17: Maintenance actions by type on section 1 .......................................................................... 41 Figure 18: Maintenance actions by type on section 2 .......................................................................... 41 Figure 19: Relation between number of events, interoccurrence times and calendar times .............. 45 Figure 20: Types of repairs and stochastic processes ........................................................................... 46 Figure 21: Sample of basic data from asset register (S&Cs) .................................................................. 52 Figure 22: Sample of corrective maintenance data (S&Cs) ................................................................... 53 Figure 23: Sample data of track recording car ...................................................................................... 54 Figure 24: Block diagram for ROCOF determination ............................................................................. 56



Abbreviations

Abbreviation Description

AFT Accelerated Failure Time

CBS Cost Breakdown Structure

DB Database

eIMS Expert-based Infrastructure Management System

GRP General Renewal Process

HPP Homogeneous Poisson Process

IM Infrastructure Manager

KM Kaplan-Meier estimator

LCC Life-Cycle Cost

MDT Mean Down Time

MGT Millions Gross Tones

MLE Maximum Likelihood Estimation

MTBF Mean Time Between Failures

MTTR Mean Time To Repair/Restore

MUT Mean Up Time

NA Nelson-Aalen estimator

NHPP Non-Homogeneous Poisson Process

NPV Net Present Value

PDF Probability Density Function

PH Proportional Hazard

PRM Proportional Repair Model

RAMS Reliability, Availability, Maintainability and Safety

ROCOF Rate Of Occurrence Of Failure

RP Renewal Process

RRM Reliability Regression Model

TRV Trafikverket

WO Work Order



Mathematical symbols

Symbol Description

Intensity function or ROCOF

Expected or cumulative number of failures

Reliability function

Scale parameter in the ROCOF

Shape parameter in the ROCOF

Likelihood function

Log-likelihood function

N Number of Failures of a single unit

m Number of units of a system

Total operation time of unit j

Failure occurrence of unit j at time i

Probability density function with covariates

Cumulative density function with covariates

z(t) Hazard function (also called hazard rate or failure rate)

Z(t) Cumulative hazard function

Set of covariates or regressors

Set of features

Baseline function in PH models

Expected value

M(t) Maintainability function

Repair rate

LCCA Life-Cycle Acquisition Cost

LCCT Life-Cycle Termination cost

LCCO Life-Cycle Ownership Cost

LSC Life-Cycle Support Cost

LUC Life-Cycle consequential Unavailability Cost

CO Operational Cost



Symbol Description

EC Cost of Energy

CC Clearance Cost

OSC Operational Staff Cost

CYCM Yearly Corrective Maintenance Cost

CYPM Yearly Preventive Maintenance Cost

CI Initial Cash Inflow

Ck Cash Inflow in period k

r Interest rate

λij / fij / ηij Failure / Maintenance / Periodical PM Frequency of action i and unit j

CL Labour Cost

CP Spare Part Cost

CE Maintenance Equipment Cost

MRT Mean Repair Time

MLT Mean Logistic Time

MAT Mean (Preventive Maintenance) Action Time

CYIN Asset Installation Cost

CPPM Periodical Preventive Maintenance Cost

pij Probability of delay of action i and unit j

CDelay Cost of delay

MDT Mean Delay Time



1 Background

Railway and road network infrastructures have components with a very long technical lifetime (of the order of 40 to 120 years (see e.g. Ref. [2]), therefore each decision must consider the usage of the assets in question for at least 40 years into the future. Long term plans, nowadays, are of the order of 10 years, even with the awareness that the remaining live time can be longer.

The complexity of these systems is that they are a mixture of components of different age and status that have to work together. Replacement of components is also a continuous and on-going process, and changes must be carefully executed.

On the other hand, the pressure for increases in traffic volume leads to a higher utilization of the existing infrastructure and hence leads to more severe degradation, which therefore requires more maintenance actions on the network.

Additional objectives set by the European Commission such as the increase of passenger and freight traffic by 2020, the reduction of travel time by 25-50% and life-cycle cost (LCC) by 30% and at the same time increasing safety (decreasing fatalities) by a 75% have put strong demands on operational and maintenance optimization (see Refs. [3], [4] and [5]). This has been in fact, subject of investigation in numerous European projects [3].

In the case of the railway transport, operational quality can be measured by punctuality or lack of train delays. In the road case, this concept could be also applicable by considering the planned speed of corridors. Maintenance activities are programmed or scheduled in time slots in which traffic load is low or inexistent, so the maximum priority is to keep the traffic flow in its highest possible values. Maintenance activities must therefore be performed near the capacity limit. Time between asset renewals should be long enough to balance maintenance costs and acquisition costs, and components replaced by deferred or planned maintenance.

On top of that, Infrastructure Managers (IMs) must keep infrastructure highly available so that the railway undertakings, i.e. train operating companies, can deliver a highly quality service at affordable price to the end users.

Studies have shown (see for instance Refs. [6], [7] and [8]) that roughly a 14% of the time delays in railway infrastructures are related to S&Cs failures. However there are other factors also accounting for such traffic delays (overhead wire 15%, track circuit 12%, signals 11% and track 10%).

It is known that the use of RAMS and LCC analysis is the way to achieve the goals that assure the performance of the network. These analyses allow, in particular to IMs, to assess the safety and availability of the system so that train operators, for instance, can deliver a transport products at an affordable price. In this sense they seek for high operability at an optimum maintenance cost.



2 Objectives

The purpose of the RAMS&LCC expert-based toolkit is to improve the reliability of the infrastructure under study and, in the long run, to decrease the maintenance costs. In order to achieve this aim, a research and exploratory process is necessary, in which a thorough study of the data currently collected in the maintenance systems is carried out. Thereafter a scientific model based on failure statistical theory is constructed. Obviously not all the failures can be described by the theoretical model, and ways have to be defined to explain those deviations.

RAMS&LCC combined analysis rests on two basic pillars: statistical analysis of maintenance interventions and information from the accounting system to estimate costs. These two building blocks are the foundation for a combined RAMS&LCC model, being the basic hypothesis the fact that using LCC modelling will help in making more appropriated business decisions in the future.

In order to focus our research, a preliminary analysis of S&Cs in the railway system has been carried out. This choice has the end of guidance in our analysis at the same time that emphasising the importance of this component of the railway infrastructure. Nevertheless, the S&Cs discussion can be perfectly extrapolated to other railway or road element.

FIGURE 1: WORK FLOW IN THE RAMS/LCC ANALYSIS

Data Mining

data cleaning

exploratory and

Maintenance Costs

Parametric Cost Model

System RAMS Characteristics

RAMS/LCC Simulation

Accounting System

Field Data Gathering

Data Farm

Life-Cycle Cost Prediction

Maintenance Policy

Data Mining

cleaning exploratory

and statistical analysis



Figure 1 shows a work flow diagram for the different phases of the analysis. The idea is to create an optimized maintenance policy to be implemented in the real system in accordance with the needs of the system itself.

In order to generate such a policy two kinds of information are needed, access of the maintenance costs and also maintenance activities carried out in the system. The cost data, together with the result of the RAMS statistical analysis are the inputs of the parametric cost models that will implement the LCC calculation. By means of a simulation, where input parameters take a stochastic character, a prediction is obtained for the Life-Cycle Cost of the system and therefore compared with the current maintenance policy. If an improvement in terms of costs can be achieved with the analysis, this policy has to be revised and incorporated in the maintenance activities. This process is repeated iteratively until optimum performance is reached.

Therefore, the key facets can be defined in base of the above described process:

Combined RAMS and LCC: optimize an existing system on the basis of costs. Costs are related to investments, operation, maintenance and non-availability. Therefore RAMS and LCC combined analyses give goal oriented indicators for optimization.

Statistical Models: optimization requires not only mean values of failures or rates of the different components but the distribution of the long-term behaviour of the whole system.

The knowledge of RAMS parameters filter out bad designed components. The LCC analysis allows identifying the most important cost drivers and necessary improvements. This process is of vital importance to avoid trials and errors and to economical optimization.



3 RAMS at System Level

In a previous document (see D5.1 Ref. [1]) the concept of RAMS at component level was introduced. Here the RAMS analysis at system level is discussed, including suitable models that depend on the system under consideration.

3.1 RELIABILITY MODEL

The reliability model tries to assess how often the system cannot achieve its operability and how long this situation takes. Two key points in the construction of the model are the definition of failure criteria (what is defined as non-functionality of the system), and the inherent nature of the system itself (whether the system is categorized as one-time-non-repairable or on the contrary it is reusable or repairable).

Concerning failures, these can be consequences of technology, materials, weather, third parties, or a combination of these factors. Therefore the following is needed:

To establish a definition of system failures (as generated by failures of components) and a classification of them.

To determine the frequency of the different types of failures and non-available time windows. These parameters influence traffic disruptions, timetables, number of cancelled or delayed trains, traffic flow, etc.

Once the definition of failure has been established, a decision has to be taken concerning the statistical treatment of the system. A natural choice is to consider repairable systems, for which different processes can be defined:

Renewal processes (RP): for repairable items, when maintenance actions restore the items to an operating condition that is as-good-as-new (perfect repair), and for non-repairable items when their failure data is identical and independently distributed (i.i.d.).

Nonhomogeneous Poisson processes (NHPP): for repairable items when a maintenance action restores the item to an operating state that is as-bad-as-old (minimal repair).

General renewal processes (GRP): for repairable items when a maintenance action restores the item’s operating state to be somewhere in between as-good-as-new and as-bad-as-old (imperfect repair), and for non-repairable items with trends in their historical data.

Most complex systems, such as aircrafts, automobiles, and by extension road and railway systems, are repaired and not replaced when they fail. When these systems are subjected to a use environment, it is often of considerable interest to determine the reliability and other performance characteristics under these use conditions. The basic data in reliability of repairable systems are the time between failures. These data can be classified in complete or censored, where censored data is by definition a group of data in which the value of a measurement or observation is only partially known. For example, it may be known when a unit has failed but there may be lack of information about when the unit was put in operation or its installation time.

Using the above statistical models, the failure rate or the number of failures in a specific time can be calculated. However an essential prerequisite in all these models is the estimation or assumption of the distribution function for the time to failure or time between failures. In any case, the methodology for analysis of the system reliability is based on a set of important tasks:



Field data collection, identification and formulation of covariates.

Identification of an appropriate statistical approach to estimate the number of failures in a specific time.

Identification of the distribution of failure data considering covariate effects.

Calculation of the reliability characteristics for the system.

The first stage is field data collection. Field data should include technical information concerning failures (unit id, serial numbers and operation time), descriptions of failures and their signatures, environmental conditions, suspected causes of failures, repaired items, repair times and root causes. Moreover, additional data may be available, such as specific working conditions of the assets. These data constitutes what is known as covariates.

After the data has been collected, the next step is to check whether there is an appropriate amount of failure data (enough times between failures for each unit). Lack of data is a common major problem in reliability data analysis because, as the reliability of the system is increased, the expected number of failures is reduced, leading to a small set of failure data. Methodologies, such as Bayesian approaches or model mixture, have been developed to deal with small collections of data. These methodologies are relevant when less than five failures are recorded per unit. Nevertheless, in our analysis it is consider that enough failure data has been recorded.

The next stage is the identification of the appropriate statistical approach to estimate the number of failures. At this point it is crucial to determine whether the data set is homogeneous or non-homogeneous. A set of units is homogeneous when the units are identical and their operational and environmental stress is comparable. When the set is homogeneous, it is appropriate to use Non-Homogeneous Poisson Process (NHPP), on the contrary for non-homogeneous data sets, a covariate-based model, such as the Proportional Hazard (PH) or the Accelerated Failure Time (AFT) models are the one to be used. Usually test statistics (such as Laplace, Military Handbook or TTT-statistics) are used to discern among models [20, 21]. A classification of the different models depending on the asset working conditions is shown in Figure 2.

FIGURE 2: STATISTICAL MODELS

Asset’s

working

conditions

Identical/comparable environmental conditions

(NHPP)

Power law

NHPP

Linear

NHPP

Log-linear

(Cox-Lewis)

Non-homogeneous conditions

Proportional Hazard

(PH)

Cox PH

(Repairable)

Andersen-Gill

(Non-repairable)

Accelerated Failure Time

(AFT)

Distributions

(Weibull, Log-normal, Gumbel)



The identification of the appropriate distribution of failure data is an important step. In general a lifetime distribution, such as the Weibull distribution, should not be used to address the study of repairable systems. Instead, to assess the reliability characteristics of a complex repairable system a point process should be used instead. The reason lies in the fact that these kind of distributions treat time between failures as identical and independently distributed (i.i.d.), and are unable to assess trends or inter-occurrence times, therefore being unable to recognize the differences between happy or sad systems (i.e., unable to recognise reliability grow).

The calculation of reliability characteristics for recurrent events (see Appendix D for a description of counting processes and the models commonly used) is modelled in base of the Rate of Occurrence of Failures (ROCOF) or intensity of the process that is denoted as ω(t) here, and which is defined as the derivative of the expected number of events:

In the NHPP with intensity ω(t), the number of events in (0,t] is Poisson-distributed with expectation value W(t). As stated above, the advantage of using this type of process is the ability to model trends in the rate of failures, because

( ) .

Therefore,

: deteriorating system (“sad system”),

: improving system (“happy system”),

(Constant): Homogeneous Poison Process (HPP).

The NHPP is also popular because it has practical foundations in terms of minimal repair, which consists on the situation in which the repair of the failed components is enough to get the whole system back in operation. Moreover, the repair of a single failure mode does not greatly improve the system reliability from what it was just before the failure. Under minimal repair assumption for a complex system with many failure modes, the system reliability after the repair is the same as it was just before the failure. In this case, the system is brought to an “as-bad-as-old” state and the sequence of failures at system level follows an NHPP.

Suppose that a system consists of many components and that a failure in any of the components constitutes a failure of the system. Each component repair (or replacement) is a renewal process governed by its respective distribution function. When the system fails due to a failure in a component, this is replaced and the system is again as good as new. The system has been repaired. However, because there are many other components still operating with various ages, the system is not typically put back into a like new condition after the replacement of a single component. Therefore, distribution theory does not apply to the failures of a complex system because, in general, the intervals between failures for a complex repairable system do not follow the same distribution. Distributions apply to the components that are replaced, but not at the system level. At the system level, a distribution applies to the very first failure. There is one failure associated with a distribution. For example, the very first system failure may follow a Weibull distribution.

In fact, it can be shown mathematically that the minimal repair assumption as defined above, corresponds to an NHPP with ROCOF that equals the hazard rate for the time to first failure, ω(t)=z(t), where z(t) is hazard rate to first failure. Therefore, in what follows ROCOF and hazard rate can be understood as the same thing.



For many systems in a real world environment, and in particular in the road and rail industry, a repair may only be enough to get the system operational again. This is the concept of minimal repair. For a system with many failure modes, the repair of a single failure mode does not greatly improve the system reliability from what it was just before the failure, so the system reliability after a repair is the same as it was just before the failure. In this case, the sequence of failures at system level follows the NHPP.

In the next subsections a description of state-of-the-art mathematical models implementing the concept of minimal repair is provided. A distinction has been done, namely, parametric vs. non-parametric and models with or without the inclusion of covariates. The application of a particular model depends on the situation and the available data.

3.1.1 PARAMETRIC NON-HOMOGENEOUS POISSON PROCESS (NHPP)

Point processes can be modelled using parametric or non-parametric models, each having their own advantages and disadvantages. Parametric models allow obtaining more reliable results in case of inference, but they pay the price of introducing a set of unknown parameters that need to be fitted. Among the parametric models, the most commonly used are the ones shown in Table 1 (see also Ref. [9]).

TABLE 1: MODELS FOR RECURRENT EVENTS DATA

Model Intensity Function Cumulative Function

Crow-AMSAA

Homogeneous

Log-linear

Power law

(

)

(

)

Proportional intensity

The parameters β and λ are the shape and scale parameters respectively. From these ROCOF, the reliability of the repairable unit is given by,

{ ∫

}

And the MTBF by,

∫ ∫

The last expression can be evaluated numerically once the ROCOF is known. Notice that the advantage of having a parametric form lies in the fact that more reliable estimates for the MTBF can be obtained. Therefore, as a rule of thumb, a parametric model will be used to fit the data and subsequently compared with a non-parametric estimate.



The estimation of the parameters in the ROCOF is done by Maximum Likelihood Estimation (MLE). From the fact that the expected number of failures is Poisson-distributed, the likelihood function for exactly observed failure times from a single repairable unit can be calculated to be:

{∏

} { }

If there are data from a group of m multiple (independent) repairable units, whose failure data follow the NHPP with the same intensity function , then the likelihood is the product of likelihoods of each unit:

∏({∏ ( )

} { })

In the previous formula, the assumption that all the units were put into operation at time t=0 is implicit1. Here the jth unit has experienced Ni failures in its time horizon and τj is the total operation time of the jth unit, and sij is the failure occurrence of the jth unit at the ith failure time; i=1,…,Nj, and j=1,…,m. The log-likelihood function for the m systems is then

{ } ∑∑

∑

The desired model for the ROCOF can then be plugged-in into the above function and find the set of parameters that minimize l(θ), that is, solve the set of equations (see Refs. [10], [11], [12] and [13] for more details):

𝜕

𝜕

Although close analytics expressions can be obtained depending on the ROCOF functional form used from Table 1, a numerical solution can also be found using an appropriate optimization algorithm. These models will be implemented in the algorithms of the RAMS-LCC expert toolkit.

3.1.2 NON-PARAMETRIC ESTIMATORS

Non-parametric estimators of the reliability function have the advantage of being model-independent. These types of models do not require any specific distribution assumption for the underlying processes, and therefore errors coming from selecting an incorrect distribution function are avoided.

There are numerous studies addressing the problem of non-parametric estimators for recurrent data events. In particular, a Nelson-Aalen (NA) estimator for the cumulative ROCOF W(t) can be straightforwardly constructed (see Appendix C for a description of the Nelson-Aalen non-parametric estimator). In the particular case when data are regularly spaced, that is, failures occur regularly in time, an estimator of the ROCOF may be built using kernel functions, as described in Ref. [14]. Estimators for the reliability function have been investigated by Wang and Chang in Ref. [15], and by Peña and collaborators in Refs. [16] and [17]. These are extensions of the broadly used Kaplan-Meier

1 In the case where the units are put into operation at different initial times,

, the cumulative number of

failures W also depends on . An integral of the ROCOF in the interval [

] needs to be evaluated.



(KM) estimator for non-recurrent events. These models will be also implemented in the expert toolkit.

3.1.3 PROPORTIONAL HAZARD (PH) MODEL

The reliability characteristics of the system are influenced by different factors such as the operational conditions, including climate conditions (temperature, wind, snow, dust, ice, etc.), the skills of the operators and maintenance crew, the history of the repair activities carried out, maintenance policy, etc. Reliability Regression Models (RRMs) include these factors in the form of covariates. The RRM can be categorised in two groups, parametric or non-parametric models.

A model for a lifetime T is called parametric if the distribution functions, probability or density, are written in functional forms , etc, where is a vector of unknown parameters. That is, these are models where the reliability function is assumed to have a specific distribution such as Weibull or log-normal. On the other hand, a model is called non-parametric if it allows any shape for the function f or F.

A major contribution to the concept of non-parametric models including covariates came from the notion of PH, where the effect of the covariates is multiplicative in the hazard rate. Of particular importance is the Cox’s PH in which the hazard rate is written in the form:

where is the baseline function, X is a vector of covariates, , and the

vector defines the set of regression parameters to be determined.

In principle, the PH model is a non-parametric regression method where there is no assumption about the baseline hazard rate. However, the baseline can be modelled as well, with normal, log-normal or Weibull-type distributions for instance, giving rise to a full set of parametric PH models.

In general, if failures are i.i.d, the Mean Time To First Failure (MTFF) is a time-independent function and can be calculated as follows:

∫

∫

∫

,

where f(t)=-R’(t) is the density, and R(t) the reliability function calculated with the desired method. For the HPP, the MTBF can also be estimated as the previous integral,

∫

However, for NHPP, the MTBF is a function of time. Often, the repair duration is relatively short compared to the time between failures and can be ignored. For a repairable system, when the repair durations are ignored, the MTBF can be estimated by using the following equation:

,

where t is the cumulative operating time, and N(t) is the observed number of failures by time t.

3.2 AVAILABILITY AND MAINTAINABILITY ANALYSIS

System Availability (A) and Maintainability (M) are correlated concepts that have a big influence on LCC. In fact, the frequency of maintenance or inspection considerably influences both the availability performance and the operating cost, i.e., man-hour cost, spare parts consumption cost, etc.



Availability can be extracted from system Reliability and Maintainability characteristics. Maintenance may be broadly categorized in two main types: Corrective maintenance (CM) and Preventive maintenance (PM). The definition of these concepts can be found at the Glossary of Terms and also in Deliverable 5.1 (Ref. [1]). Sometimes, a PM based on condition monitoring by modern measurement and signal-processing techniques is called predictive maintenance. Generally speaking, system’s reliability can be extracted from failure analysis of CM interventions while both CM and PM interventions give information about availability and maintainability characteristics. This process is schematised in Figure 3.

FIGURE 3: RAMS FROM CORRECTIVE AND PREVENTIVE MAINTENANCE

In simple terms, Availability and Maintainability can be measured in base of a combination of factors collected from historical intervention data (see Table 2):

Mean Up Time (MUT)

Mean Down Time (MDT)

Mean Time Between Failures (MTBF)

Mean Time To Repair/Restore (MTTR)

TABLE 2: AVAILABILITY AND MAINTAINABILITY PARAMETERS

According to Ref. [18], maintainability can be defined as “the ability of an item under given conditions of use, to be retained in, or restore to, a state in which it can perform a required function, when maintenance is performed under given conditions and using stated procedures and resources.” It is remarkable, however that most of traditional models do not consider the important influence of the “given conditions” in the calculations of maintainability. External conditions include factors such

Availability (A) Maintainability (M)

Availability:

Maintainability: ∫

Unavailability:

MTTM =

MTTR =

Operational Availability:

Repair rate

(when constant)



as cold weather, rain or snow, suppliers distance, staff skills, etc. As a consequence, if these factors are not included, the performance prediction may be imprecise and inaccurate, resulting in a poor availability prediction as well.

From a statistical point of view, the factors that influence the maintainability characteristics are covariates, and therefore one has to implement regression models to extract these A and M characteristics of the system, in the very same fashion than RRM described in the previous section. Moreover, these covariates might be time-dependent or time-independent.

In maintainability analysis one attempts to model the probability that a successful repair or preventive maintenance action be performed within a stated time interval. The random variable to access the probability is the time to repair or restore (TTR) and in general can be: time to failure recognition, time to failure isolation and time to failure correction or removal. The goal is to model the so called maintainability or restoration rate as compare to the hazard rate in reliability. As in the case of reliability, the maintainability analysis methods can be classified into parametric and non-parametric.

3.2.1 PARAMETRIC MAINTAINAB ILITY MODELS

These kind of models assume that the TTR follows a given parametric probability density function m(t) (e.g. Weibull, Gamma, Exponential, Lognormal, etc.). The probability that the repair will be accomplished within a time t is given by the cumulative distribution function M(t):

∫

The MTTR, the variance of the repair distribution and the repair rate are given by

∫

∫

,

∫

,

and

.

In parametric maintainability analysis, it is assumed that some identical repair actions are tested under identical conditions. However, the repair rate can have a dynamical nature that may differ according to the environment conditions. Therefore the inclusion of covariates can increase the predictive power and remove statistical biases.

3.2.2 PROPORTIONAL REPAIR M ODEL (PRM)

The PRM is basically a proportional hazard model where the repair rate is modelled as a product of two components; a time dependent baseline repair rate , and a time-independent function of the desired covariates , that is, the model takes the following form:

.

The function may take different functional forms (linear, exponential, and logistic). If the exponential-type function is taken, then the repair rate takes a Cox-type form given by,

,

and the maintainability function is,



{ ∫

}

,

with

{ ∫

}.

As in all the PH models, if is modelled, then the PRM is parametric, if not, then semi-parametric.

3.2.3 EXTENDED PRM WITH TIME-DEPENDENT COVARIATES

The previous PRM can be extended to include time-dependent covariates. In this case, all the previous formulas are equally valid but with the difference of considering the following hazard function,

.

This generalization has the advantage of incorporating seasonal effects in the covariates as for instance the fact that during winter, some systems may take more time to repair than in summer due to extreme weather conditions.

3.3 SAFETY ANALYSIS

In the safety analysis, the aim is to carry out an analysis of safety risks for persons and the possible hazards for the various risks. These system characteristics can be extracted from failure data leading to dangerous or hazardous conditions. The following can be considered:

Qualitative analysis of the hazards for the various risk-bearers in terms of possible accidents (derailments, collisions, unsafe situations due to third parties), and the causes of these accidents.

Quantitative analysis in terms of safety levels (risk of fatality per time unit or per kilometre for the various risk-bearers).

Unfortunately safety data is rarely available because of private reasons. Nevertheless, with the appropriate data base, failure rates can be obtained using similar methodologies as the ones described in the preceding sections. In general, the following set of failure rates can be defined:

- λdd : Failure rate of dangerous failures that are revealed (i.e. detected).

- λdu : Failure rate of dangerous failures that are unrevealed (i.e. not detected).

- λsd : Failure rate of safe failures that are revealed (i.e. detected).

- λsu : Failure rate of safe failures that are unrevealed (i.e. not detected).



4 LCC Assessment at System Level

Life-Cycle Cost (LCC) analysis has been used since the late 60’s and it has its roots in the American defence industry [19]. It is an economic technique for decision making and by assessing the total cost of acquisition, ownership and disposal of a product [20]. In this section, a description of how the method can be applied to railway and road infrastructure systems is provided.

4.1 LCC METHODOLOGY OVERVIEW

The life cycle of an asset can be subdivided into six phases according to the IEC 60300-3-3 standard (see Ref. [20]). From the ownership point of view, these phases are connected with the LCC of the asset as shown in Table 3. While acquisition and termination costs are usually fixed or not subject to ownership time variations, ownership costs depend on operation conditions and maintenance policies. The combined RAMS & LCC analysis described here focuses exclusively on the ownership LCC.

TABLE 3: RELATION BETWEEN ASSET PHASES AND OWNER C OSTS

The first step in the LCC analysis consists in identifying the so-called cost elements that considerably influence the total LCC of the system. According to the LCC international standard (Ref. [20]), it is recommended to develop a Cost Breakdown Structure (CBS) as a basis to the definition of the cost elements in the LCC analysis. This enables the analyst to find the cost drivers and also simplifies the work involved in setting up correct equations evaluating these costs.

The CBS may be developed by defining items along three independent axes, which are Life cycle phase, Product/work breakdown structure, and Cost categories. Figure 4 shows a visual representation taken from Ref. [20].

Life-Cycle Phases (IEC 60300-3-3) Life-Cycle Costs Nature

- Concept and definition - Design and development - Manufacturing - Installation

Acquisition Fixed

- Operation - Maintenance

Ownership Variable

- Disposal Termination Fixed



FIGURE 4: COST ELEMENT CONCEPT (IEC-60300-3-3)

The CBS depends on the system under study, being difficult to define a generic structure for the cost elements in the LCC analysis. Therefore, the CBS has to be tailored to the specific system or subsystem under study.

For the railway/road system, a possible CBS is shown in Table 4 and Figure 5, although many structures are possible depending on the system and the analysis to be carried out.

TABLE 4: EXAMPLE OF COST IN LCC ANALYSIS

AC (Investment Or Acquisition Costs)

AMC (Annual Maintenance And Operating Costs)

ADC (Annual Delay-Time Costs)

AHC (Annual Hazard Costs)

• Equipment and material purchase

• Engineering • Installation • Initial spares • Initial training • Disposal and

reinvestment

• Corrective maintenance (CM)

• Calendar based PM • Condition based PM • Operating cost • Energy consumption

• Short term delay • Long term delay

• Human safety • Environmental threat • Cleaning • Rebuilding

Once the CBS and the cost drivers have been identified, the next step deals with building a model to quantify the cost elements encompassed in a LCC analysis. That means to find appropriate relations among input parameters and the cost elements. Ideally, a system should be modeled from many points of view such as availability, maintainability, logistics, risk, human error in the system, etc. However, it is commonly assumed that availability and maintainability are the most significant cost drivers in LCC analysis, because they may have a wide range of impact on the cost elements categorized as ownership cost. It is therefore very important to properly model the system availability and maintainability.

In order to calculate costs, relevant data is needed, for example, cost per action taken. However, it is difficult to find data sources for operation data and cost data available to the public, because most of these data are stored in operating companies’ in-house databases. Nevertheless, when actual data related to an analysed system is not available, this may be estimated using some methods such as stochastic models, parametric techniques, and analogous techniques.



FIGURE 5: EXAMPLE OF COST BREAKDOWN STRUCTURE IN LCC ANALYSIS

In this work, the CBS of Figure 5 will be adopted as starting point in the LCC analysis.

4.2 COST MODEL FOR COST DRIVERS

In this section a model for the cost elements in the CBS of Figure 5 is proposed. It is important to notice that, the cost model is highly dependent on the system under study, therefore this model is subject to improvements and adaptations.

According to Figure 5, the LCC can be modelled as follows:

LCC = LCCA + LCCO + LCCT

A description of each of these cost elements is now given. See Appendix E for transcriptions and symbol legend used in this section.

Life-Cycle Cost

Acquisition Cost

Development

Material

Installation

Reinvestment

Ownership Cost

Support Cost

Investment

Operational

Energy cost

Clearance

Operational Staff

Yearly Maintenance Yearly CM

Yearly PM

Yearly Installation

Periodical PM

Man hours

Spare Parts

Equipment

Consequential Cost Sort Term Delay

Long Term Delay

Termination Cost



4.2.1 ACQUISITION COST (LCCA)

The acquisition cost consists on the costs that the owner has to invest to own the infrastructure. In theory, this is made of the cost of design/development, materials that compose the system, the cost of assembly and installation, and reinvestment. The most difficult cost element to estimate here is reinvestment, because it depends on multiple unpredictable factors.

LCCA = Design + Material + Assembly/Installation + Reinvestment

For the purpose of this project, this cost will be assumed fixed.

4.2.2 TERMINATION COST (LCCT)

This is the cost due to disposal of the asset, which includes the machinery or equipment and the transportation.

LCCT = Machinery + Transportation

For the purpose of this project, this cost will be assumed fixed.

4.2.3 OWNERSHIP COST (LCCO)

This cost collects the support cost and consequential cost.

LCCO = LSC + LUC

Data collection in this life-cycle phase allows optimizing these costs.

4.2.3.1 Support Cost (LSC)

These include operational costs, which are usually well-defined and fixed, and maintenance, which depends on the maintenance policies for the assets and varies among companies. Moreover, it also contains an initial cost for investment on maintenance equipment/machinery.

Initial investment on maintenance equipment (CI):

This cost depends on the system and contains all the costs due to purchasing new equipment at the starting stage of the asset life in order to carry out the maintenance activities.

Operational cost (CO)

As shown in Figure 5, the operational cost has been divided in

Energy cost (EC): basically heating, oil and electricity.

Clearance (CC): snow, trees, stones, etc.

Operational Staff (OSC): the one that make the system run.

The modelling of this cost is made as follows:

CO = EC + CC + OSC

where

EC (Energy Cost) = Power consumption in kW * Cost of Energy (currency/kWh).

CC (Clearance Cost) = Number of operators * Salary/hour * Clearance Hours.



OS (Operational Staff) = Number of operators * Salary/hour * Operation Hours.

Ownership Maintenance Cost: Yearly + Periodical

Finally, the costs of maintenance actions are included, and here is where the previously calculated RAMS characteristics of the system enter into play. Here a model is proposed, but it may be revised or modified to accommodate to particular systems.

Maintenance actions can be preventive or corrective. Corrective Maintenance (CM) is assumed to be carried out annually while Preventive Maintenance (PM) will be divided into annual and periodical. This is done because there may be PM actions carried out off the annual planning with well-defined and fixed costs. The annual costs of maintenance are functions of time while the periodical costs are assumed constant.

Yearly costs (identified with the letter “Y”) are subject to a Net Present Value (NPV) calculation. The NPV of a T-period project is calculated as:

∑

,

where CI is the initial cash outflow (investment), the Ck’s are the successive cash inflow in period k (specific yearly cost), and r is the expected interest rate.

Annual Corrective Maintenance (CM) Cost

The costs derived from CM assume failures in the system that lead to replacements or repairs of the components. The replacement/repair cost is therefore of the form:

∑∑ [ ]

where is the failure frequency of action i and unit j, MRT is the Mean Repair Time, and MLT is the

Mean Logistic Time.

Annual Preventive Maintenance Cost

The cost from PM may include the cost of inspections, condition based and periodical maintenance.

∑∑ [ ]

where is the maintenance frequency of action i and unit j, MAT is the Mean Action Time, and MLT

is the Mean Logistic Time.

Annual Installation Cost

Within maintenance, there is a cost of installation of each of the assets that need to be repaired. This cost will be denoted .



Periodical Preventive Maintenance Cost

∑∑ [ ]

which is very similar to the yearly PM cost but with a different frequency . This cost is assumed to

be fixed and not subject to the NPV multiplication factor.

4.2.3.2 Consequential or Unavailability Cost (LUC)

Delays can be sort or long term delays, and they are assumed to be associated with system failures, because the corridors have to be closed in order to carry out an urgent maintenance. Of course, not all the failures (or actions) will cause a delay in the system, so there is an associated probability

that a given failure or action i leads to a delay in the unit j. The LUC is then modelled as follows:

∑∑

where MDT is the Mean Down Time. This can be considered as a yearly cost and therefore multiply it by the NPV factor.

Clearly, costs depend on RAMS, basically through the failure frequency, MTTF, MTBF, MTTR and MDT. In order to carry out the analysis, the total period and the nominal interest rate have to be fixed beforehand.

Therefore, the inputs of the LCC analysis are:

1. Investments.

2. Results of the Maintainability analysis: cost of preventive/corrective maintenance.

3. Time window and interest rate during the LCC time window.

4. Results of the RAMS analysis: number of disruptions and timetable affecting errors.

4.3 ASSET DEGRADATION AND MAINTENANCE POLICY

It is important to notice that the number of maintenance actions will be proportional to the degradation of the assets. Traditionally, degradation has been modelled using simple proportional power laws. For instance, for track degradation, it can be estimated that the amount of degradation since the last restoration E is proportional to the total tonnage since last restoration T in MGT, the dynamic axle load P, and the speed in this track section V:

,

being empirical constants in the order of 1-3.

More sophisticated models have been proposed however. Gamma processes have been widely used in reliability studies to model deterioration (see for instance Refs. [21], [22], and [23]) and also in modelling track geometry [24]. A gamma process is a stochastic model for degradation occurring randomly in time, and characterized by independent and nonnegative increments distributed by using gamma distributions with identical scale parameters. Because of its properties, the gamma



process is considered as one of the most appropriate processes for the stochastic modelling of monotonic degradation accumulated over time, in a succession of tiny increments such as wear, stress, corrosion, erosion or the degradation process growth. A broad survey about gamma processes and its applications to maintenance can be found in Ref. [25]. Most recently inverse Gaussian processes are being explored to model maintenance under degradation.

In any case, the costs of maintenance, corrective or preventive, will generally depend on the degradation process, and a realistic model should correct the costs by a factor taking into account degradation. However, the inclusion of advanced degradation models is a bit beyond the scope of the present work.

4.4 PROPAGATING UNCERTAINTIES: SENSITIVITY ANALYSIS

In the previous section it has been pointed out that not all the necessary input parameters for the LCC analysis are known with precision, they follow a probability distribution. For instance, in roads viability studies2, general recommendations exists for parameters such as estimates for fuel or tyre cost, mean number of accidents, etc. These parameters may vary from one road to another or in between seasons, and as a consequence one has to talk about confidence intervals or error bars associated with these parameters. Generally speaking, there are different kinds of uncertainties:

1. Parameters are not well known due to missing data for existing systems.

2. Parameters are not well known due to missing experience of new components or systems.

3. Parameters like lifetime of components, failure rates, and other RAMS parameters are described by probability distributions.

The idea of propagating uncertainties consists on taking into account the parameters PDFs in the calculation. For instance, the NPV outcomes change as the input parameters (interest rate, time period, cash in/out flows) vary. Moreover, if RAMS parameters are described by PDFs, the related costs can be described by PDFs as well. A Monte Carlo Simulation can be performed to account for the variation of all these input parameters in the LCC Analysis. As a result a probability distribution can be derived for the LCC.

2 See for instance the World Road Association Guidelines (http://www.piarc.org/en/)

http://www.piarc.org/en/



5 Combined RAMS and LCC Analysis

The aim of WP5 is to finally carry out a combined analysis using RAMS in the LCC calculation. This is to calculate the cost for the non-availability and maintainability of the system, and to determine what Reliability, Availability, Maintainability quality and Safety level can be achieved for what Life-Cycle-Cost. The analysis aims at finding options with optimum level of RAMS/LCC combined, that is, a trade-off between the highest levels of R, A and S, and the minimum levels of M at the lowest Life-Cycle-Costs. From the previous section, it should be clear how this calculation is carried out, namely, LCC formulas intrinsically depends on the RAMS characteristics of the system and therefore stochastic LCC are extracted. For clarification purposes, a cartoon of this process is shown in Figure 6.

FIGURE 6: SCHEMA OF THE COMBINED RAMS/LCC CALCULATION FRAMEWOR K

Therefore, the combined RAMS/LCC analysis needs the following inputs:

- Results of R and A analyses: frequency of failures (rates) and R and A system parameters, number of canceled lines (trains or road interruptions), delay times, etc. Estimates of costs derived from non-availability of the system.

- Results of M analysis: system M parameters and maintenance costs. - Results for S analysis: S system parameters and achieved safety levels. - Results for LCC analysis: estimation of the system life cycle cost. - Estimation of costs derived from un-availability or maintenance activities: costs due to

cancelled or delayed trains/roads containing number of trains/roads, duration, average number of passengers or vehicles per line affected.

The output of the combined analysis is the costs of the various maintenance options compatible with current RAMS and required LCCs taking into consideration the intrinsic uncertainties associated with the elements:

- LCC: investment cost and maintenance cost. - Costs derived from cancelled lines or delays as a cause of system failure.

StochasticRAMS

StochasticLCC

Cost = Cost element x Failure Probability

Combined RAMS/LCC calculation



6 Case Study: Railway S&Cs Analysis

In this case study, a preliminary analysis of the railway database is demonstrated. The case study does not pretend to be an exhaustive analysis, because most of the functionalities are still in development in task 5.5. The aim of this example is to show some features of data collection, pre-processing, and exploratory data analysis which are part of the algorithms to be finally implemented. At present, only corrective maintenance (CM) actions have been analysed, and reliability of the system obtained, leaving the extraction of maintainability and availability characteristics for a later stage. Although some exploratory analysis of CM actions carried out has been made, the actual LCC models (which also need preventive maintenance interventions) are still under construction.

6.1 THE CASE STUDY DATABASE

Relevant data for road and railway demo cases are available from IP and TRV for use as Case Study in INFRALERT. In this section a preliminary analysis is carry out in the railway case.

The railway demo case (see Ref. [26] for more details) is a rail corridor in Sweden under the management of TRV, called Iron Ore Line (Malmbanan), in northem Sweden. The asset register BIS is a tool used by TRV to store and retrieve information about assets such as location, type, year of installation, etc. BIS is connected to a number of Swedish Transport Administration systems, such as Ofelia, Bessy, Optram, etc.

Sample of basic data provided for this case study is shown in Appendix F for the railway case. Specifically, there are three types of databases as shown below:

Asset_register: mimics the asset management database BIS. Data for track, sleepers, switches, and level_crossings have been provided.

o There are two track sections (labelled as 1 and 2), and according to the S&Cs asset register, the first one is 112km long and goes from km 472 to km 584, the second one is 147km long and goes from km 202 to km 349. There are two types of tracks, being a 93% of the international standard type UIC60, and a 7% of them of the Swedish standard type BV50.

o The sleepers are labelled type B and T. The 97% of them are of type B, and the 3% of them are of type T.

o There are four types of fastenings, Pandrol UIC60 (93%), Pandrol BV50 (4%), Heyback (3%), and Spik (0.4%).

o The sleepers’ data contain information about: the track section, position (km, m, and mm), model, etc.

o The switches data contain relevant information about the track section, position (km, m), some geometric information (length, radius), type of switch, year of installation, etc.

o Level crossings contain also information about the track section, position, road barrier, lights presence, traffic flow, volume, etc.

In order to simplify the study here, only S&Cs will be considered.



Work_orders: mimics the corrective actions database Ofelia. A rich set of 6664 data from maintenance actions has been provided. The registers have information concerning the type of asset, the track section in which the asset is located, the component that registered the failure, actions taken, etc.

Rail_geometry: mimics the track recording car database Optram and contains specific data about the geometry of the track.

The infrastructure system that is being analysed, can be divided into different technical subsystems, i.e., substructure, track, electrical system, signalling system and telecom system. A schema of the railway system is provided by the hierarchy tree of Figure 7.

FIGURE 7: RAILWAY SYSTEM ASSET HIERARCHY TREE

From the list of assets provided by the railway use case, the following elements can be identified:

1. First layer:

a. Infrastructure: “Substructure”, “Superstructure” and “Stations”.

b. Signalling and communication: “Signal”, “Signalling control, rbc and line block system”, “Telecommunications Transmission System”, “Balise group”, “Traffic Management System - ARGUS”, “Traffic Management System”, “Railyard Lighting”, “Positioning system”, “Detector”.

c. Power supply: “Cable equipment”, “Alternative Power Supply”, “Electric power system”, “Overhead line”, “Converter station”.

d. Rollingstock: “Bank”

2. Second layer:

a. Substructure: “Tunnel”, “Level crossing”.

b. Superstructure: “Track”, “Switch”, “Derailer”.

Railway system

Rolling stock

Signalling & Communication

Infrastructure

Substructure

Subgrade

Protective layers

Embankment fill

Base soil Bridges

Tunnels

Superstructure

Plain track

Ballast

Sleepers

Fastenings

Rail pad

Base plate

Clip

Screw

Rails

Joints

Switches & Crossings

Stations

Power supply



c. Stations: “Real Estate”, “Watches”.

INFRALERT focuses on the analysis of the Infrastructure, and particularised to the superstructure. An analysis of the superstructure can include the following elements: Track, Switch, and Derailer. The present section outlines the analysis of the Switches (a.k.a. S&Cs).

6.2 COSTS OF MAINTENANCE

On June 14, 2013, the Swedish Transport Administration submitted a proposal for a 2014-2025 national transportation plan to the Swedish Government. The Government decided on 522/55 Billion SEK/Euro for measures in the transportation system between 2014 and 2025. An important area in this plan is operation and maintenance of the railways (see Ref. [27]) . The total operational volume of maintenance was approximately 813 Million Euro according to the TRV 2015 annual report, and the measures carried out in rail maintenance by TRV to reduce costs from 2013 to 2015 are dissected in Table 5.

TABLE 5: OPERATIONAL VOLUMES FOR RAIL MAINTENANCE IN M€ (TAKEN FROM REF. [27])

2013 2014 2015

Maintenance M EUR M EUR M EUR

Track 252,2 306,7 242,7

- of which reinvestment 142,3 138,9 86

Switches and crossings 44,6 47,9 44,3

- of which reinvestment 17,2 18,6 9

Bridges 11,5 22 40,2

- of which reinvestment 8,1 14,6 35,4

Tunnels 9,8 3,3 4,7

- of which reinvestment 4,6 -0,1 3,2

Overhead contact line 31,9 41,2 23,8


Other electrical infrastructure 38,5 48,2 52,2


Signalling and telecommunications infrastructure 27,3 40,9 49,2


Other maintenance measures 59,1 61,6 75,3

- of which other reinvestment measures 19,9 22,1 22,9

Fixed portion in functionally procured BAS maintenance 110,7 124,4 117,8

Winter services 38,8 34 33,8

Damage repair 28,7 27,3 24,9

Governance and support in implementation of maintenance measures

16,1 17,4 18,4

Total maintenance measures, incl. reinvestments 669,2 774,9 727,5

- of which reinvestments 254,5 271,8 225,1

0

Property and station management 22,2 19,5 21,5



Other costs within maintenance operations 70,4 55,3 63,6

Total operational volume, maintenance – railway 761,9 849,7 812,6

TRV’s cost of operation and maintenance together with reinvestments has been estimated to be around 728 Million Euros during 2015. Of this total cost, rail maintenance cost amount a 40% and S&Cs amount at least for the 10% of the maintenance costs according to Ref. [27] (in Refs. [28] and [29], S&C maintenance cost was estimated to be around 13% in 2006). S&Cs, as one of the railway subsystems, cause most train delays. This highly cost is due to several factors, its complexity and degradation of this asset, and the fact that S&Cs are very important components which need to be maintained regularly to keep high safety levels. Unfortunately, the costs for individual S&Cs are difficult to estimate due to lack of information.

Planned work takes place at nights and on weekends, but for unplanned work such as remedial work, the problem becomes extremely clear. Such work has major consequences for traffic, as there are no margins in the capacity allocation. Maintenance costs are therefore also affected by accessibility. The rate of replacement for track switches has decreased during the last period. The backlog of need is 660 track switches that must be replaced; over the next 10–20 years an additional 1,430 track switches that are not part of the standard and management range of products need to be replaced. The low rate of replacement means that an increased percentage of the infrastructure is passing the end of its technical lifetime, which can clearly affects robustness and punctuality.

As it has been stressed along this document, cost benefit analyses based on life-cycle cost are a great tool to access maintenance cost information on S&Cs. This information can reveal, for instance, which types of S&Cs are more prone to record a failure depending on external conditions such as weather, location, or other important factors.

6.3 BASIC CONCEPTS ABOUT S&CS

S&Cs are very important subsystems in the railway system because they allow trains to shift from one track to another. They also allow trains to meet, and slower trains to be overtaken, therefore contributing to increase the capacity of single and double track networks. Due to its important role, they have been and they are the subject of study of numerous research projects world-wide.

An S&C is a complex composed system which is made up of different subsystems. In Figure 8 a cartoon of a typical S&C is shows, a list of its main components is provided in Table 6, and a visual tree hierarchy of the S&C system is shown in Figure 9.



FIGURE 8: DIAGRAM OF A SIMPLE S&C (TAKEN FROM HTTP://WWW.RAILWAY-TECHNICAL.COM/)

TABLE 6: SUBSYSTEM COMPONENTS OF AN S&C

Ballast Heating (de-icing system) Rail joints (welded and insulated) Base plates Locking device Sleeper (bearer) Check rails Stock rails Snow protection

Crossing Switch device (stretcher/motor/control bars)

Switch blade

Fasteners Signalling system Slide plates and switch rollers

FIGURE 9: SYSTEM BREAKDOWN OF AN S&C

These components can be therefore identified in our Case Study database. Obviously, the information in the database is not so well organized (sometimes missing, other richer, duplicated or redundant) and the analysis is time consuming because it requires a cumbersome data cleaning and exploratory data analysis. This is a potential problem in terms of generality of the algorithms to be developed in the subsequent tasks 5.3 of this WP, because the aim of INFRALERT is for them to be applicable independently of the database under study is. For instance, a list of components obtained when filtering data for the S&Cs is summarized in Table 7. Sometimes each component is by itself a system, as for example the signalling system.

S&C

Switch drive

Actuator general

Motor

Gearbox

Bar

Contact spring

Actuator box

Locking device

Control device

Electronic

Relay

Magnet

Crossing Switch blade

Blade

Lifting support

Fastener

Rail

Stock rail

Check rail

Join Fastener Heating

Control

Element

No sub system

Unknown

Known



TABLE 7: S&C COMPONENTS IN THE CASE STUDY DATABASE.

Set of components from the railway use case

Heating (*) Control device 920 (Others) (*)

Conversion Device (Other) Switch (Other) (*)

Gear Drive Connector (reed switch)

Switch heating (Other) (*) Low voltage switchboards

Wing rail Crossing (Other) (*)

Control ruler (*) Cable

Fastener (*) Tension rod Lift the heel

Duty Order (Other) Heavy

Plenum Stock rail (*)

Joining 850 (Others) (*) TRIAC

Magnet part Leader 838 (Others)

Snow shield Control Rod (*)

Wires Frog

Electronic Connector Fortification 843 (Others)

Gear Set

Terminal block Ball and lever

Controller Fiber Spacers

Heat Resistance Screw, Nut

Link Contact Bleck

Guardrail Stock Rail

Protection cover Railway sleepers (*)

Ballast (*) Lock control rod (*) Heavy Lace Braces

6.4 RELIABILITY MODEL FOR S&CS

S&Cs are considered repairable systems, that is, they can be characterized as a system which is repaired rather than replaced after failures. The repair brings the system to operation state. It is therefore considered that the reliability of the system does not improve after the repair, and the probability of failure is the same as before the failure happened.

A clear picture of the repairable characteristic is provided by the event plot as shown in Figure 10¡Error! No se encuentra el origen de la referencia.. In this plot, each event, represented by a point in time, corresponds to a corrective maintenance action. As can be seen, there are assets (or units) that suffer a larger number of failures than others. This may correspond to failures in different components of the given asset. After the failure occurs, the asset is put back to operating state until another failure happens in the same or a different component.

FIGURE 10: EVENT PLOT FOR THE SYSTEM OF S&CS



As explained in the previous sections, the reliability model used for this kind of systems is the NHPP. The NHPP is a counting process that can be used for repairable systems where the condition after repair is as-bad-as-old, that is, the repair does not change the condition of the system and the probability of failure is the same as just before the repair. A description of counting processes and in particular the NHPP is given in Appendix D.

The reliability function can be calculated for the whole system or for a specific S&C. As it has been explained at length in the text, the reliability can be calculated using non-parametric or parametric models. Figure 11 shows the result of applying different non-parametric estimators to assess the reliability of the whole system of S&Cs. These estimators are the ones that have been described in this document.

FIGURE 11: NON-PARAMETRIC ESTIMATORS FOR S&CS RELIABILITY

However, parametric models allow better estimates and more reliable results in case of inference. Table 8 and Figure 12 show an example of results obtained for the different parametric models considered here when they are applied to a specific unit3. Table 8 shows values of the fitted parameters, the likelihood function at the minimum, and the Akaike Information Criterion (AIC) for each of the NHPP models considered. Figure 13 shows the cumulative number of failures of this particular unit (Appendix G shows a sample of the R source code that performs these fits).

TABLE 8: ESTIMATED PARAMETERS FOR UNIT S&C NO.23

Model β (shape) [95% CI] λ (scale) [95% CI] Min-Log-L AIC

Crow-AMSAA 1.66 [1.27 2.05] 23.06 [14.99 31.13] 185.17 -366.34

Log-linear 0.74 [0.30 1.19] 2.77 [2.19 3.35] 183.24 -362.49

Power law 1.66 [1.27 2.05] 0.15 [0.06 0.24] 185.17 -366.34

Proportional intensity 1.66 [1.27 2.05] 3.14 [2.79 3.49] 185.17 -366.34

3 In this particular case unit 23 has been selected because it has registered a large enough number of events.



FIGURE 12: RATE OF OCCURRENCE OF FAILURES MODELLED TO UNIT 23 DATA

FIGURE 13: CUMULATIVE DENSITY OF FAILURES OF UNIT 23

6.5 COST ANALYSIS OF MAINTENANCE ACTIONS ON S&CS

The assets in our DB have been given an ID (Asset_id). In the Work Order file, the registers are failures reported for the different components of each of the assets. Here, only S&Cs have been selected for the analysis, and there are recurrent failures in the S&C component tree.

Table 9 shows some information about the tracks being considered here, namely, the start and end stations, length of each track, the number of switches and crossings that can be found on each track, and the averaged Million Gross Tones (MGT) on each track per year.

An important feature that can give relevant information in terms of maintenance is the study of failures/maintenance actions per type and age of S&C. Let us concentrate here on the analysis by type.

In Sweden there are different types of S&Cs, some of them have great similarities with European standards. On the other side there are others with their particular characteristics. To identify the type of S&C it is needed to combine information coming from different databases (i.e., different data files in our Case Study). Filtering WO’s by Switch data, there are a total number of 1409 maintenance interventions for S&Cs. The Asset Register contains only 61 registers for S&C. This means that there is



either incomplete information or several maintenance interventions carried out on a particular switch or switch component (sometimes it is a combination of both).

TABLE 9: DATA FOR S&C ACCORDING TO ASSET R EGISTER

Track Section Station Id (Start-End) Length [km] Number of S&C Averaged MGT/Track/year

1 Ap – Ac 112 26 24

2 Bq – Bb 147 35 17

There are 9 different types of S&Cs in or Case Study (retrieved from the Asset Register). These are shown in Table 10 together with the number of items present for each type, proportion respect to the whole database and some geometry characteristics.

TABLE 10: TYPES OF S&CS IN THE RAILWAY USE CASE

S&C Type Number Of Items

Proportion (%)

Radius [m]

Length [m]

Length [m]

(including diverting

track)

Switch blade length [m]

Rail weight

[kg]

Angle [degree]

EV−UIC60−300−1:9 24 39 300 33.2 49.8 13.0 60 63

EVR−UIC60−760−1:15 17 28

EV−UIC60−760−1:15 13 21 760 54.2 83.1 21.5 60 3.8

EV-SJ50-12-1:12 2 3

EV−SJ50−11−1:9 1 2

EV−UIC60−500−1:12 1 2

EVR−UIC60−760−1:14 1 2

EV−UIC60−760−1:14 1 2 760 54.2 81.3 21.5 60 4.1

EVR−60E−760−1:15 1 2

Using the above information one can study what type of S&C is more prone to register a failure and therefore give relevance to monitor those assets in the future. In order to identify the type of each S&C that has been maintained, the following steps have been followed:

Select a given track sector (Track_section). Select the station (From_station). Match the Asset_id in the WOs file with the Switch_number in the asset register file.

Unfortunately, the information is incomplete and the exact type of every S&C that has been maintained cannot be extracted. As a result, there are up to 1041 S&Cs that has been labelled as “Not specified”, for which the type is not known. Although this proportion of is quite high (74%), it is possible to have an idea of the maintenance activities carried out per type.

Next, a survey of the different maintenance actions has been carried out. These are recorded in the column Action in the Work_Order file. A total of 19 different actions have been identified. Each action is associated with a cost that would need to be provided by the corresponding maintenance company (in our case TRV). Although this information is important to build a realistic LCC model, it is very difficult to obtain due to privacy reasons. Nevertheless, there is always the possibility of simulating these costs per action. An example is given in Table 11 (these can be always substituted by more realistic numbers in a later stage). Some important remarks about these costs are in order:

Costs/event (Euro) for different maintenance actions have been generated randomly.



It is assumed that each maintenance action cost is known. Costs have been generated from a uniform random distribution U(0, 1000Euro). Some of the costs have been set to zero but they can be changed in future simulations.

TABLE 11: RANDOMLY GENERATED MAINTENANCE COST/ACTION IN EURO

Maintenance Action Cost [Euro] section 1 Cost [Euro] section 2

Cleanup 946.718 607.271

Replacement 189.138 766.496

Restoration 508.856 757.675

Provisional_repair 273.279 115.594

Inspection 115.078 456.250

Cleaning 830.038 914.679

No_action 0 0

Lubrication 782.213 469.050

Full_Repair 98.547 277.439

Restart 639.195 332.445

Adjustment 599.029 555.432

Snow_removal 151.546 765.922

Not_specified 0 0

Consulting 0 0

Speed_reduction 599.626 34.583

Taken_out_of_service 245.163 200.450

Lack_of_maintenance_Snow_removal 106.398 977.808

Software_installation 0 0

Disposal 742.592 246.394

The next step in the calculation, which will be incorporated in the algorithms of task 5.5, consists on the calculation of the Mean Times to Repair and Mean Logistic Times associated with these actions. To calculate these mean times it is needed to solve an integral numerically using the reliability model previously determined. Once these are known, they can be plugged-in into the equation of CYCM given in Section 4.2 to extract the combined RAMS&LCC associated with these CM actions.

At this point an important remark should be noted. In many cases there is an uncertainty in the input values of the LCC-model. For instance, costs may vary per year, region, or either they may have an inherent uncertainty due to lack of information. This uncertainty can be handled by using probabilistic distributions instead of fixed values, and propagate them in the LCC-model. Different distributions can be used being the more common ones the triangular, normal or either truncated distribution functions. For explanation purpose, some shapes for these tentative distributions are shown in Figure 14.



FIGURE 14: D ISTRIBUTIONS USED IN MONTE CARLO SIMULATIONS OF LCC

In Figure 15 and Figure 16 the bar plots for the different specified maintenance actions taken on individual assets are shown. The different track sections are shown in different panels. As can be seen, each asset is given an asset ID and there are some assets for which more maintenance has been carried out than others. These assets need to be monitored more closely.

FIGURE 15: MAINTENANCE ACTIONS ON S&CS OF TRACK SECTION 1

FIGURE 16: MAINTENANCE ACTIONS ON S&CS OF TRACK SECTION 2

It is also interesting to see the maintenance activity by type of S&C. These are shown in the bar plots of Figure 17 and Figure 18.



FIGURE 17: MAINTENANCE ACTIONS BY TYPE ON SECTION 1

FIGURE 18: MAINTENANCE ACTIONS BY TYPE ON SECTION 2

In the implementation of the specific Cost Model for these maintenance actions all the relevant cost elements and each of the individual events must be considered. Then, when summing them up together the overall cost per section and/or S&C type can be obtained. This method implies the knowledge of individual action costs that at times are not so well defined. This type of calculations will be developed and implemented in the RAMS&LCC toolkit.



7 Conclusions

This deliverable has presented the general RAMS&LCC methodology that is intended to be applied to the railway and road infrastructure systems under study in the INFRALERT project. This methodology will be the core of the working algorithms that will feed the eIMS with RAMS and LCC characteristics of the system including uncertainties. For this purpose, the deliverable has deeply described all the mathematical technology behind the models to be implemented in the expert RAMS&LCC toolkit.

In order to place the on-going work, the deliverable has used a case study as an application example. This example does not represent the final analysis yet, but it provides a feeling of how the methodology is applied. The case study consists on a preliminary analysis of the S&Cs present in the railway database, which will be part of a more ambitious application, pretending to apply the same methodology for the whole infrastructure, including railways and roads.

The next step ahead is to construct efficient algorithms that materialize all the presented methods. This will be done in the next task 5.5 and presented in a future deliverable. Four sets of algorithms are envisaged:

Data collectors

Data pre-processing tools (cleaning, exploratory and failure data analysis)

RAMS calculators implementing suitable models

LCC calculators implementing targeted models depending on system/subsystems

RAMS&LCC integrators

Tracker of the KPIs derived from RAMS&LCC

The information presented herein is a framework based on the current knowledge and state of the art, but its final implementation depends on the actual data availability. Nevertheless, the content presented here may be subject to improvements, revisions or modifications during the rest of the project.



Appendices

A. USEFUL RELATIONS AND NOTATION USED IN RELIABILITY FORMULAS

In this appendix relations between reliability functions are given so the reader can find connections in subsequent appendices. These are also in connection with deliverable D5.1 [1].

The lifetime T of an individual or unit is a positive and continuously distributed random variable.

The probability density function (PDF) is called f(t).

The cumulative distribution function (CDF) is called F(t) and given by

∫

.

The reliability (or survival) function is defined as

∫

.

The hazard function (also called hazard rate or failure rate) of T at time t is defines as

Some useful relations are the following:

,

,

,

∫

,

where ∫

is called the cumulative hazard function.

The Mean Time To Failure (MTTF) or expected lifetime is defined as,

∫

∫



B. KAPLAN-MEIER (KM) ESTIMATOR OF THE SURVIVAL FUNCTION

The Kaplan-Meier estimator is a non-parametric statistic used to estimate the survival or reliability function from lifetime data. This estimator may be used to measure, in a model-independent way, the time-to-failure of a system.

The Kaplan-Meier estimator is one of the most frequently used methods of survival analysis. The estimate may be useful to examine recovery rates, the probability of death or failure, and the effectiveness of a maintenance action for instance. It is limited in its ability to estimate survival adjusted for covariates; parametric survival models and the Cox Proportional hazards model may be useful to estimate covariate-adjusted survival.

To define this estimator, consider n individuals, where the ith individual has potential lifetime Ti and potential censoring time Ci. The pair ( is observed, where

,

{

If are the times with at least on failure, are independent and identically distributed (i.i.d) with common reliability (survival) function R(t), the estimator is constructed by defining the following quantities:

nr = number of items at risk at time Ti; i.e. the number that can fail at time Ti; counted immediately before Ti.

dr = number of failures at time Ti.

It follows that can be estimated by:

∏

C. NELSON-AALEN (NA) ESTIMATOR OF THE CUMULATIVE HAZARD

The Nelson-Aalen estimator of the cumulative hazard function can give useful information on whether the distribution of failure time T is IFR (increasing failure rate) of DFR (decreasing failure rate). This estimator is constructed as follows.

Recall from Section ¡Error! No se encuentra el origen de la referencia. that Z . Thus

if is the KM estimator (see Section A), then the following estimator can be defined:

∏

∏ (

)

The Nelson-Aalen (NA) estimator is the first order approximation when , i.e.,

∑



D. COUNTING PROCESSES

In this appendix some key concepts about the reliability of a repairable system as a function of time are briefly explained. The appendix tries to be a complement or introduction and in any case exhaustive. For a more detailed presentation the reader is advised to consult Ref. [14].

D.1 DEFINITIONS

Assume a repairable system for which it is desirable to know the system’s availability, mean time of failures, mean time to the first failure, and the mean time between failures.

The system is assumed to be put into operation at time t=0 and when the system fails it is repaired to a functioning state, where the repair time is considered negligible. There is a sequence of failure times Si and a sequence of failure inter-occurrence times Ti. Consider the random variable N(t), the number of failures in the time interval (0, t], this particular stochastic process {N(t), t≥0} is called a counting process. The relation between N(t), the time between failures Ti, and the calendar times Si is shown in Figure 19.

A counting process {N(t), t≥0} is therefore defined as follows:

,

is integer valued,

If , then

For , the quantity represents the number of failures that have occurred in the interval .

FIGURE 19: RELATION BETWEEN NUMBER OF EVENTS, INTEROCCURRENCE TIMES AND CALENDAR TIMES

The rate (or intensity) of the counting process is by definition,

Where is the expected mean number of failures in the interval .



D.2 TYPES OF COUNTING PROCESSES

In general, four types of counting processes can be defined:

1. Homogeneous Poisson processes (HPP)

These are counting processes where the inter-occurrence times are independent and exponentially distributed with the same failure rate λ.

2. Renewal processes (RP)

They are a generalization of HPPs where the inter-occurrence times have an arbitrary life distribution. Upon failure, the component is replaced or restored to an “as-good-as-new” state. This is often known as a perfect repair.

3. Nonhomogeneous Poisson processes (NHPP)

These are counting processes where the rate of occurrence of failures varies with time. Inter-occurrence times are neither independent nor identically distributed. The NHPP is often used to model repairable systems that are subject to a minimal repair strategy, with negligible repair times. Minimal repair means that a failed system is restored just back to functioning state. After a minimal repair the system continues as if nothing had happened, and therefore the likelihood of system failure is the same immediately before and after a failure. A minimal repair thus restores the system to an “as-bad-as-old” state.

4. Imperfect repair processes (IRP)

These are counting processes in between RP (“as-good-as-new”) and NHPP (“as-bad-as-old”).

A schema of these types of counting processes is presented in Figure 20.

FIGURE 20: TYPES OF REPAIRS AND STOCHASTIC PROCESSES

The RP and the NHPP represent two extreme types of repair: replacement to an "as-good-as-new" condition and replacement to "as-bad-as-old" (minimal repair), respectively. Most repair actions are, however, somewhere between these extremes and are often called imperfect repair or normal repair.

Because of the importance of these types of processes, a short review of their main characteristics is given. For simplicity and given the kind of systems studied here, only homogeneous and non-homogeneous Poisson processes are reviewed.

Type of repair

Perfect Repair or replacement

"as-good-as-new"

HPP RP

Imperfect repair (normal repair)

IRP

Minimal repair

"as-bad-as-old"

NHPP



D.3 HOMOGENEOUS POISSON PROCESS (HPP)

A definition of a HPP could be the following: a counting process {N(t), t≥0} is said to be an HPP having a constant rate of occurrence of failure (λ>0) if:

i. .

ii. The process has stationary and independent increments, i.e., the event may occur at any time in the interval, and the probability of the event occurring in (t, t+Δt) is λΔt.

iii. The inter-occurrence times of events T1, T2, … are independent and exponentially distributed with parameter λ, or equivalently, the number of events in any interval of length t is Poisson distributed with mean λΔt.

These types of processes have some important properties:

1. Reliability function:

If the inter-occurrence times are distributed exponentially, , the probability distribution is , and the reliability function , with .

2. Lack of memory:

If , therefore,

.

So, for any age s, the remaining life has the same distribution as the lifetime distribution of a new item.

3. If and , then .

4. If are independents each with distribution , then,

So a series system of n components with lifetimes that are independent and exponentially distributed with hazard rate λ, has a lifetime which is exponential with hazard rate nλ, and hence,

There are two alternative ways to test exponentiality of lifetimes, one is graphical (TTT-plot), and other statistical (Barlow-Proschan’s test). For more details on these tests see Ref. [14].

D.4 NON-HOMOGENEOUS POISSON PROCESS (NHPP)

The NHPP is a generalization of the HPP and can be defined as the counting process {N(t), t≥0} with time-dependent rate for , satisfying:

i. .

ii. The system will not experience more than a failure at the same time.



iii. The process has non-stationary independent increments, i.e., the event may occur at any time in the interval, and the probability of the event occurring in (t, t+Δt) is .

The basic parameter of the NHPP is the Rate of Occurrence of Failure (ROCOF) function . The cumulative rate of the process is,

∫

.

This function covers the case in which the rate is also a function of some explanatory variable that is function of t. It is important to note that the NHPP model does not require stationary increments, which means that the failures might be more likely to occur at certain times than others, and then, the inter-occurrence times are generally neither independent nor identically distributed.

Several studies of failure data from practical systems have concluded that the NHPP was an adequate model and that the systems that were studied approximately satisfied the properties of the NHPP defined above. However, when replacing failed parts that may have been in operation for a long time, by new ones, an NHPP clearly is not a realistic model. For the NHPP to be realistic, the parts put into service should be identical to the old ones, and hence should be aged outside the system under identical conditions for the same period of time.

From the definition of NHPP it is straightforward to verify that the number of failures in the interval (0, t) is Poisson distributed:

The mean number of failures in (0, t) is , and the variance is ( ) .

The cumulative rate W(t) of the process is therefore the mean number of failures in the interval (0, t) and is sometimes called the mean value function of the process. Poisson Processes can be parametric or non-parametric. For parametric models is specified as a function of a finite-dimensional parameter. Some of the more commonly used parametric and non-parametric NHPPs are discussed in what follows.

D.5 PARAMETRIC NHPP MODELS

Several parametric models have been established to describe the ROCOF of an NHPP. These models may be written in the common form

,

where λ0 is a constant, and determines the shape of the ROCOF curve. Some commonly used parameterizations are the following:

1. The power law model is defined as

A repairable system modelled by the power law model is seen to be improving (happy) if 0 < β < I, and deteriorating (sad) if β > 1. If β = 1 the model reduces to an HPP. The case β = 2 is seen to give a linearly increasing ROCOF.

2. The linear model is defined as



A repairable system modelled by the linear model is deteriorating if α > 0, and improving when α < 0.

3. The log-linear(or Cox-Lewis) model is defined as

A repairable system modelled by the log-linear model is improving (happy) if, β < 0, and deteriorating (sad) if β > 0. When β = 0 the log-linear model reduces to an HPP.

D.6 NHPP MODELS WITH TIME DEPENDENT COVARIATES

Traditional parametric methods (as the above) most often only consider time-independent parameters (or covariates). However, there are a more general family of NHPP models that consider time-dependent covariates. These are of two types:

Model the intensity of ROOF as a function of time dependent covariates:

where is a row vector of covariates at time t and the vector of parameters. Notice that this is just a generalization of the above Cox-Lewis model.

Model based on the Cox’s Proportional Hazard Model (PHM):

External covariates X(t), which include fixed covariates, can be incorporated in a Poisson process by specifying the ROCOF as a function of t and the covariate history X(t) = {x(u) : 0 ≤ u ≤ t}. This is usually done by defining covariate vectors Z(t) that are based on X(t) and then considering ROCOF of the form4:

where is an arbitrary unspecified function called baseline, , and the

vector defines the set of regression parameters to be determined. If the

baseline function is specified parametrically, the model is fully parametric, otherwise it is semi-parametric.

D.7 NON-PARAMETRIC ESTIMATORS

Recurrence times can be treated as a type of correlated survival data in statistical analysis. However, because of the ordinal nature of recurrence times, statistical methods which are appropriate for clustered survival data may not be applicable to recurrence time data. For instance, although the Kaplan-Meier estimator of Section A has been frequently used for exploratory analysis, this estimator is generally inappropriate for the analysis of recurrent events.

4 This model is due to Anderson and Gill (AG) [22] and represents an extension of the Cox proportional hazard

model from single event data to recurrent event data. See Ref.[13] for comments on the Cox’s model.



There are several non-parametric models available to study survival (reliability) of recurrent events. Among others, the following can be identified:

Peña-Strawderman-Hollander (PSH) model: estimation of survival function for recurrence time data by means the generalized product limit estimator (PLE) method. It considers a model wherein the inter-event times for a unit are not-correlated, i.e., under the assumption of a renewal or identical and independently distributed model. This generalizes the product-limit estimator to the situation where the event is recurrent.

PSH estimator under a Gamma Frailty model (MLE): considers a model wherein the inter-event times for a unit are correlated. This dependence among the inter-event times is induced by an unobserved latent or frailty variable. To describe this correlated recurrent event model, it is postulated that there is an unobserved Zi, with Z1, Z2, ..., Zn identical and independently distributed random variables from a distribution Hz, which is taken in particular to be a gamma distribution with mean 1 and variance 1/a, where a > 0 is unknown. For more details see Refs. [23, 24].

Wang-Chang (WC) estimator: a competing estimator that was proposed by Wang and Chang in Ref.[25] applies even if the frailty components are not gamma distributed; hence, their estimator is more general than the one by PSH.



E. SYMBOLS AND ABBREVIATIONS IN LCC FORMULAS

Symbol Description

LCCA Life-Cycle Acquisition Cost

LCCT Life-Cycle Termination cost

LCCO Life-Cycle Ownership Cost

LSC Life-Cycle Support Cost

LUC Life-Cycle consequential Unavailability Cost

CO Operational Cost

EC Cost of Energy

CC Clearance Cost

OSC Operational Staff Cost

CYCM Yearly Corrective Maintenance Cost

CYPM Yearly Preventive Maintenance Cost

CI Initial Cash Inflow

Ck Cash Inflow in period k

r Interest rate

λij / fij / ηij Failure / Maintenance / Periodical PM Frequency of action i and unit j

CL Labour Cost

CP Spare Part Cost

CE Maintenance Equipment Cost

MRT Mean Repair Time

MLT Mean Logistic Time

MAT Mean (Preventive Maintenance) Action Time

CYIN Asset Installation Cost

CPPM Periodical Preventive Maintenance Cost

pij Probability of delay of action i and unit j

CDelay Cost of delay

MDT Mean Delay Time



F. SAMPLE OF RAW DATA (FOR S&CS)

In this appendix a brief description about the data base provided for the railway demo case is described. A more extended description can be found in Ref. [26]. Only sample data for S&Cs is shown.

F.1 DATA FROM ASSET REGISTER DATABASE (BIS)

The data in shows selected data provided by railway use case from the BIS database. The following columns have been provided:

- Track_section - Km+m start

(2 columns) - Km+m end

(2 columns) - Item_length

- UNE (type of track) - Track no. - Station - Switch no.

- Switch type - Point_length

(length of blade) - Radius - LRswitch (L/R S/C)

- Max_speed - Installation_type - Installation_year

F.2 DATA FROM MAINTENANCE DATABASE (OFELIA)

The data in Figure 22 shows corrective maintenance data provided by railway use case from Ofelia database. The following columns have been provided:

- Reported_date - Traffic_affecting - Cause_code - Start_date - Finished_date

- Track_section - From_station - To_station - Asset_id - Asset_type

- Asset_part - Component - Unit - Actual_fault

- Fault_description - Root_cause - Description_root_cause - Action - Action_description

FIGURE 21: SAMPLE OF BASIC DATA FROM ASSET REGISTER (S&CS)



In reports it is not necessary to fill in asset numbers. Therefore Asset_id is not a good ID identifier for assets. This makes less straightforward to find search the asset ID from this database. The identification of the asset is done by selecting From_station and To_station.

F.3 DATA FROM INSPECTION REPORT DATABASE (BESSY)

Preventive maintenance data from inspection report system Bessy has also been provided. The following columns have been provided

- Year - Inspection Date - Inspection Report ID - Intervention Date - Proposed

Intervention Date - Track Type

- Track Section - Position From/ To - Km From/To - M From/To - UNE - Inspection Type - Asset Indenture

- Asset Type - Comments - Asset Model - Proposed

Intervention - Intervention

Notes

- Status - Priority - Remark Qty - Position

Start_Position - End_Position

This information is used to obtain periodic maintenance information.

F.4 DATA FROM TRACK RECORDING CAR (OPTRAM)

The data in Figure 23 shows an extract of geometry data provided by railway use case from Optram database. The following columns have been provided:

- Track_Section - Start/End - Segment - DateNum - Track_Name - Meas_Direction

- UNE - Percent_Measured_200 - Segment_Length - Measurement_Car

- Curvature - Radius - Speed - StDev_LL - StDev_AL

- StDev_AL - CA - Gauge_Average - Cant_Average - StDev_LL_25_70m - StDev_LL_70_150m

FIGURE 22: SAMPLE OF CORRECTIVE MAINTENANCE DATA (S&CS)



FIGURE 23: SAMPLE DATA OF TRACK RECORDING CAR



# * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # ROCOF det er mi nat i on wi t h MLE # Case: LTU S&Cs # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ### Funct i ons sour ce( " ut i l i t i es. R" ) ### Sel ect S&C WOs swi t ch. df <- get Dat a( " . . / LTU_Rai l wayDat abase/ Wor kOr der s/ WO. t xt " , " Swi t ch" ) ### Feat ur e sel ect i on col sel ec <- c( " Asset _i d" , " Repor t ed_dat e" , " Tr ack_sect i on" ) df <- swi t ch. df [ , col sel ec ] #df $event <- sampl e( c( 0, 1) , nr ow( df ) , r epl ace = T) # r andoml y sel ect some as censor ed df $event <- r ep( 1, nr ow( df ) ) # sel ect al l as f ai l ur es names ( df ) <- c( " i d" , " t i me" , " sect i on" , " event " ) dat e <- as. POSI Xct ( df $t i me) ear l i est <- mi n( dat e) # " 2008- 01- 03 10: 14: 51 CET" # set t i me or i gi n ( f i r st day i n 2008) or i gi nT <- as. POSI Xct ( st r pt i me( " 2008- 01- 01 00: 00: 01" , " %Y- %m- %d %H: %M: %S" ) ) # t r ansf or m dat e t o numer i c as t i me di f f er ence t o or i gi n df $t i mest aps <- as. numer i c ( dat e) - as. numer i c ( or i gi nT) df $t i mest aps <- df $t i mest aps / mean( df $t i mest aps ) ### Spl i t dat af r ame based on i d: cr eat e a l i st l i st Of uni t s <- spl i t ( df , df $i d) ### cumul at i ve number of f ai l ur es f or a gi ven uni t uni t <- " 23" #sampl e( uni que( names( l i st Of uni t s) ) , 1, r epl ace = T) # sel ect one uni t r andoml y uni t . df <- l i st Of uni t s [ [ uni t ] ] p <- cumNumFai l ur es ( uni t . df ) [ [ 1] ] ct <- cumNumFai l ur es ( uni t . df ) [ [ 2] ] cf <- cumNumFai l ur es ( uni t . df ) [ [ 3] ] ### Fi t s # - Cr ow- AMSAA: " cr ow" # - homogeneous: " homogeneous" # - Power - l aw: " power " # - Log- l i near : " l ogl i near " # - Pr opor t i onal : " pr opor t i onal " #f i t power - NHPP ( ppPower <- f i t NHPP( t i mes=uni t . df $t i mest aps [ 1: nr ow( uni t . df ) - 1] , Tend=uni t . df $t i mest aps [ nr ow( uni t . df ) ] , i nt ensi t y=" power " ) ) [ 1: 5] #f i t l og- l i near - NHPP ( ppLogl i n <- f i t NHPP( t i mes=uni t . df $t i mest aps [ 1: nr ow( uni t . df ) - 1] , Tend=uni t . df $t i mest aps [ nr ow( uni t . df ) ] , i nt ensi t y=" l ogl i near " ) ) [ 1: 5] ### Pl ot f i t t ed NHPP model s ct <- uni t . df $t i mest aps [ 1: nr ow( uni t . df ) - 1] cf <- 1: l engt h( ct ) t i messeq <- seq( mi n( ct ) , max( ct ) , l engt h. out =100) model LogLi n <- exp( ppLogl i n$par amet er s [ 2] ) * ( exp( ppLogl i n$par amet er s [ 1] * t i messeq) -1) / ppLogl i n$par amet er s [ 1] model Power <- ( t i messeq/ ppPower $par amet er s [ 2] ) ^ppPower $par amet er s [ 1] pl ot ( ct , cf , t ype=" p" , x l ab=" Ti me" , y l ab=" Cumul at i ve r ecur r ences" ) l i nes ( t i messeq, model LogLi n, col =" gr een" , l t y=3, l wd=2) l i nes ( t i messeq, model Power , col =" bl ack" , l t y=4, l wd=2) l egend( " t opl ef t " , pch=" - " , col =c( " gr een" , " bl ack" ) , l egend=c( " Log- Li near " , " Power " ) , bt y=" n" , y. i nt er sp=1, l t y = 1, l wd=2)

G. SAMPLE OF R CODE FOR ROCOF DETERMINATION

In this appendix an extract of the code used to calculate the rate of occurrence of failures is given. The aim is to show the cleanliness of the R programming language to perform complicated tasks. The code implements the steps shown in the flux diagram of Figure 24.



FIGURE 24: BLOCK DIAGRAM FOR ROCOF DETERMINATION

Raw DatagetData

cleanDataRawData

S&CSelect

DifferentComponet

renameAssetIDs

ROCOF Model

optimize SelectUnit

results



Glossary of terms

Asset The physical transportation infrastructure (e.g. travel way, structures, etc.); more generally can include the full range of resources capable of producing value-added for an agency (e.g. human resources, equipment, materials, financial capacity, real state, corporate information, etc.).

Availability The percentage of time that a system is able to perform its required functions at a stated instant of time or over a stated period of time. The faster the system can be repaired after a failure, the greater the availability (EN50126, 1999).

Corrective Maintenance

The maintenance carried out after a failure has occurred and intended to restore an item to a state in which it can perform its required function [BS 4778]

Failure Departure of a component’s functionality targets from specification (Smith, 2005). Termination of the ability of an entity to perform a required function under specified conditions (Villemeur, 1992).

Linear asset An asset not specific to a single location representing a network. For example, oil and gas pipe lines, roads, highways, rail tracks and utility lines (water, sewage and power [30] transmission). They can cross over with other networks and hold many non-linear assets (traffic control systems, stations, power generating stations).

Maintainability The ability of an item, under stated conditions of use, to be retained in, or restored to, a state in which it can perform it required functions. The probability that a failed item will be restored to operational effectiveness (Smith, 2011; EN50126, 1999).

Predictive Maintenance

Preventive maintenance based on condition monitoring by modern measurement and signal-processing techniques.

Preventive Maintenance

Maintenance carried out at predetermined intervals or corresponding to prescribed criteria and intended to reduce the probability of failure or the performance degradation of an item [BS 4778].

Reliability The probability that an item will perform a required function, under stated conditions, for a stated period of time (Smith, 2011; EN50126, 1999).

Repairable item Items that can be repaired after a failure has occurred.

Safety Freedom from those conditions that can cause an unacceptable damage risk such as death, injury, occupational illness, or damage to or loss of equipment or property (EN50126, 1999).



References

[1] INFRALERT consortium, «Deliverable D5.1: RAMS data collection and failure rate analysis at component level,» INFRALERT project, H2020, 2015.

[2] Ministerio de Fomento. Dirección General de Carreteras., Instrucción de Acciones a considerar en Puentes de Ferrocarril IAPF-07, Madrid: Centro de Publicaciones Secretaría General Técnica. Ministerio de Fomento., 2010.

[3] Shift2Rail Joint Undertaking, «Shift2Rail,» 07 07 2014. [En línea]. Available: http://shift2rail.org/. [Último acceso: 30 01 2017].

[4] Shift2Rail Joint Undertaking, «Annual work plan,» S2R JU Governing Board, Brussels, 2016.

[5] E. Commission, «Research & Innovation,» 30 01 2017. [En línea]. Available: http://ec.europa.eu/research/transport/index.cfm?pg=transport&lib=rail. [Último acceso: 30 01 2017].

[6] C. Stenström, A. Parida y D. Galar, «Performance Indicators of Railway Infrastructure,» International Journal of Railway Technology, vol. 1, nº 3, pp. 1-18, 2012.

[7] R. Granström, Management of condition information from railway punctuality perspectives, Luleå: Luleå University of Technology (Doctoral Thesis), 2008.

[8] B. Nyström, Aspects of Improving Punctuality, From Data to Decision in Railway Maintenance, Luleå: Luleå University of Technology (Doctoral Thesis), 2008.

[9] W. Q. Meeker y L. A. Escobar, Statistical Methods for Reliability Data, New York: John Wiley & Sons, 1998.

[10] M. Rausand y A. Hoyland, System reliability theory: models, statistical methods, and applications, John Wiley & Sons, 2004.

[11] B. H. Lindqvist, «On the Statistical Modeling and Analysis of Repairable Systems,» Statistical Science, vol. 21, nº 4, pp. 532--551, 2006.

[12] G. R. Weckman, R. L. Shell y J. H. Marvel, «Modeling the reliability of repairable systems in the aviation industry,» Computers & Industrial Engineering, vol. 40, nº 1-2, pp. 51-63, 2001.

[13] L. H. Crow, «Methods for reducing the cost to maintain a fleet of repairable systems,» de Annual Reliability and Maintainability Symposium, Tampa, Florida, 2003.

[14] M. L. Gámiz, K. B. Kulasekera, N. Limnios y B. H. Lindqvist, Applied Nonparametric Statistics in Reliability, London: Springer Science & Business Media, 2011.

[15] M. C. Wang y S. H. Chang, «Nonparametric Estimation of a Recurrent Survival Function,» Journal of the American Statistical Association, vol. 94, nº 445, pp. 146-153, 1999.

[16] E. A. Peña, R. L. Strawderman y M. Hollander, «Nonparametric estimation with recurrent event data,» Journal of the American Statistical Association, vol. 96, nº 456, pp. 1299--1315, 2001.



[17] M. Hollander y A. Peña, «Nonparametric Methods in Reliability,» Statistical Science, vol. 19, nº 4, pp. 644-651., 2004.

[18] British Standards Institution, British Standards: Glossary of Terms Used in Quality Assurance Including Reliability and Maintainability Terms (BS 4778), London: BSI, 1991.

[19] Y. Kawauchi y M. Rausand, «Life-Cycle Cost (LCC) Analysis in Oil and Chemical Process Industries,» de Toyo Engineering Corp., Chiba, 1999.

[20] International Electrotechnical Commission, Dependability Management - Part 3-3: Application Guide - Life Cicle Costing, Geneva: IEC 60300-3-3 , 2004.

[21] M. Abdel-Hameed, «A gamma wear process,» IEEE Transactions on Reliability, vol. 24, nº 2, p. 152–153, 1975.

[22] A. Grall, L. Dieulle, C. Berenguer y M. Roussign, «Continuous-time predictive-maintenance scheduling for a deteriorating system,» IEEE Transactions on Reliability, vol. 51, nº 2, pp. 141-150, 2002.

[23] D. Zuckerman, «Optimal replacement policy for the case where the damage process is a one-sided Lévy process,» Stochastic Processes and their Applications, vol. 7, nº 2, pp. 141-151, 1978.

[24] S. Mercier, C. Meier-Hirmer y M. Rous, «Modelling track geometry by a bivariate Gamma wear process, with application to maintenance,» de Risk and Decision Analysis in Maintenance Optimization and Flood Management, Delft, IOS Press, 2009, pp. 123-136.

[25] J. M. van Noortwijk, «A survey of the application of gamma processes in maintenance,» Reliability Engineering & System Safety, vol. 94, nº 1, pp. 2-21, 2009.

[26] INFRALERT consortium, «Deliverable D1.2: Set up of case studies and evaluation framework,» INFRALERT project, H2020, 2015.

[27] Trafikverket, « The Swedish Transport Administration Annual Report. http://online4.ineko.se/trafikverket/Product/Detail/44395 (accessed September 20, 2016),» Swedish Transport Administration, Borlänge, 2013.

[28] A. Nissen, «Classification and cost analysis of switches and crossings for the Swedish railway,» Journal of Quality in Maintenance Engineering, vol. 15, nº 2, pp. 202-220, 2009.

[29] A. Nissen, «LCC analysis for switches and crossings: a case study from the Swedish Railway Network,» International Journal of COMADEM, vol. 12, nº 2, pp. 10-19, 2009.

[30] European Commettee for Electronical Standardization, «UNE-EN 50126-1:2010. Railway applications - The specification and demonstration of Reliability, Availability, Maintainability and Safety (RAMS). Part 1: Basic requirements and generic process. British Standards (BSI).,» Brussels, 2010.

Report on RAMS&LCC integrated modelsinfralert.eu/wp-content/multiverso-files/21_572b4a01a6b... ·...

Documents

Transcript of Report on RAMS&LCC integrated modelsinfralert.eu/wp-content/multiverso-files/21_572b4a01a6b... ·...