Tracks with concrete sleepers have a rubber pad is placed between the rail and the sleeper in order to reduce the ...... Rockhampton, Queensland. [23]. Budai G.
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RAMS management of railway track Narve Lyngby, NTNU; Per Hokstad SINTEF; Jørn Vatn, NTNU Department of Production and Quality Engineering, The Norwegian University of Science and Technology, N-7491 Trondheim, Norway Abstract This chapter will provide a review of ageing/degradation models relevant for railway track. Recent models for maintenance/renewal optimization used in railway will also be presented. Further, some of the methods/techniques are applied on a few case studies, using actual data. Finally some future trends are outlined.
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Introduction................................................................................................................ 2
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Railway track ............................................................................................................. 2
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Degradation modelling .............................................................................................. 7
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Methods for optimizing maintenance and renewal.................................................. 12
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Case studies on RAMS ............................................................................................ 16
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Conclusions and future challenges .......................................................................... 28
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References................................................................................................................ 28
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1
Introduction
In the past, railway maintenance procedures have been traditionally planned based on the knowledge and experience of each company, accumulated over many decades of operation, but without any kind of reliability- or risk-based approaches. With the major goal of providing a high level of safety to the infrastructures there were not much concern over the economical issues [Carretero 2003]. However, nowadays, limitations in budget force the railway infrastructure managers to reduce operational expenditures. Therefore, efforts are being made for the application of reliability-based and risk-informed approaches to maintenance optimisation of railway infrastructures. The underlying idea is to reduce the operation and maintenance expenditures while still assuring high safety standards [Carretero2003]. Optimising maintenance involve an estimation of the degradation of an object or a system and the consequence of this degradation, often in form of cost. Having knowledge about the degradation, we can estimate when measures are necessary, when life span reaches its technical and/or economic end, etc. Only by estimating these figures correctly, accurate life cycles including all maintenance work to be carried out throughout useful life can be drawn. Especially the possibility to predict residual lifetime of any asset is of high importance. The consequence of degradation relates to safety and operational expenditures as speed limitations and corrective maintenance actions.
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Railway track
After the Second World War, the national railway companies used design of their own. Later international standards were adopted that often implied heavier rail: UIC54 rails and in many countries today, UIC 60 rails [Zoeteman, 2004]. Instead of jointed rails, continuous welded rails (CWR) became the new standard, which resulted in greater passenger comfort and less maintenance. Improvements to the rails were complemented with innovations in the design of sleepers, fastening sections and ballast bed. In addition to these gradual improvements, completely modified track types have also been developed. Firstly, new ballast track sections have recently become available; these have a different sleeper design than conventional ballasted track. Germany developed a wide-sleeper track, where the weight of the sleepers themselves doubled, but the pressure on the ballast bed almost halved as a result of the new shape, resulting on considerable less (geometric) maintenance [Cronau, 1998]. Austria developed a frame sleeper track which adds a longitudinal beam to the traditional sleeper concept. Subsidence has been reduced by two-thirds, thanks to the more continuous support of the rails [Riessberger, 2002]. In addition, considerable technological progress has also been made in development of ballast less track sections that replace ballast by concrete or asphalt beds (slab-track). However, the most common type for a railway track is the ballasted railway track. Figure 1 shows a ballasted railway track consisting of rails and sleepers, laid in and fixed by ballast on an existing sub-grade. This economic design, which was chosen on the basis of experience, has remained virtually unchanged despite technological developments in the components. Tracks are long, large structures stretching hundreds or thousands of kilometres. In addition to economy, the design is a rational structure for supporting heavy fast trains on soft ground.
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Figure 1. Configuration of railway track
Rails are longitudinal steel members that guide and support the train wheels, and transfer concentrated wheel loads to the supporting sleepers spaced evenly along its length. The rails are held to the sleepers with steel fasteners ensuring that they do not move vertically, longitudinally, or laterally. Tracks with concrete sleepers have a rubber pad is placed between the rail and the sleeper in order to reduce the peak forces from the rails on to the sleepers [JBV 1998]. The sleepers provide a solid, even and flat platform for the rails, and form the basis of a rail fastening section. They hold the rails in position and maintain the designed rail gauge. Sleepers are laid on top of compacted ballast layer with a distance of typically 60 – 70 cm. Sleepers receive concentrated vertical, lateral and longitudinal loads from the wheels and rails and distribute them over a wider ballast area to decrease the stress to an acceptable level [JBV 1998]. The ballast transmits these vertical forces into the sub-grade [JBV, 1999]. The secondary function of the ballast is to give the track a good lateral stability. The lateral strength of a railway track is to a large extent defined by ballast lateral resistance. After maintenance and renewal actions of the ballast layer, the ballast particles are not well enough consolidated and lateral resistance is low. This means that the geometrical settlement of the track is quite fast in the first period before the track settles. Sub ballast is a layer of aggregates between the ballast layer and the sub-grade, and usually comprised of well-graded crushed rock or sand/gravel mixtures. It prevents penetration of ballast grains into the sub grade, and also prevents upward migration of fines into the ballast layer. Sub ballast therefore, acts as a filter and separating layer in the track substructure, transmits and distributes stress from the ballast layer down to the sub grade over a wider area, and acts as a limited drainage medium. Sub-grade is the ground where rail track structure is constructed. It may be naturally deposited sub-grade or specially placed fill material. The sub-grade must be stiff and have sufficient bearing capacity to resist traffic induced stresses at the sub-ballast/sub-grade interface. Instability or failure of sub grade will result in an unacceptable distortion of track geometry and alignment, even with excellent ballast and sub ballast layers. In addition the track consists of drainage section (sub drain, trench etc.) to avoid moisture problems in the track. Moisture in track is by many regarded as the main parameter affecting the degradation progression, and is therefore a very important asset to consider in the optimisation of maintenance.
Railway track degradation A railway track is degrading even if it is not in use or maintained [Corshammar, 2005]. It is visible on disused railway tracks where the vegetation has taken over. The vegetation is forcing its way from the side of the track for finally to cover the track completely. This natural
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… degradation is slower on tracks with traffic, but will still make a contribution to the overall degradation. Vegetation and remnants of vegetation bind up moisture which can lead to freezing and worsen stability of the track. Together with this natural degradation, the traffic is contributing to the degradation of the track. Forces from trains passing, put stress on the railway track components. The whole track is degrading at the same time, but the different maintenance points are degrading with different speed [Corshammar, 2005]. It would be too great of a task to describe the degradation of all railway track components and their interaction in this chapter. Instead a description of settlement of track and degradation of rails is made in a few words. These degradation processes will later be used as examples in some case studies on RAMS. Track settlement Trains subject the track structure to repeated loading and unloading as they pass. Each load-unload cycle causes deformation of the track, part of which is elastic and recovers; while part suffers permanent deformation. Track settlement is an integrated process in which settlement of one component affects that of the other. As soon as the track geometry starts to deteriorate, the variations of the train/track interaction forces increase, and this speeds up the track deterioration process. According to Dahlberg [Dahlberg, 2001] the track settlement occurs in two major phases. Directly after a major maintenance action of the ballast the settlement is relatively fast until the gaps between the ballast particles have been reduced and the ballast is consolidated. The second phase of settlement is slower and is caused by several basic mechanisms of ballast and sub grade behaviour, for example continued volume reduction, i.e. densification caused by particle rearrangement, sub-ballast and/or sub grade penetration into ballast voids, volume reduction caused by particle breakdown, volume reduction caused by abrasive wear, inelastic recovery on unloading due to micro-slip between ballast particles, movement of ballast and sub grade particles, and so on. Measurement of track geometry irregularities is the most used automated condition monitoring technique in railway infrastructure maintenance. Most problems with the track (at least the ones concerning the ballast and substructure) are revealed as track geometry irregularities [Berggren, 2005]. Rail degradation Rail degradation is basically due to wear and fatigue [Larsson, 2004]. These mechanisms vary in strength depending on the track load. In this respect the track curvature has a major influence with the following relationships to degradation: -
Narrow curves implies wear
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Tangent track implies fatigue
Expressed in a plot, it will look like figure 2.
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Figure 2. Wear and fatigue mechanisms as a function of curve radii. The degradation index corresponds to a relative degradation rate [Larsson 2004].
Rail wear Wear occurs due to interaction of rail and wheel. It includes abrasive wear and adhesive wear. Adhesive wear is predominant in curves and dry conditions [Olofsson and Nilsson 2002]. Abrasive wear is observed at the wheel tread and rail crown, and adhesive wear is observed at the wheel flange and gauge face. In order to reduce the rate of wear, wheels and rail are lubricated. Lubrication helps to reduce rail gauge face wear and reduces energy or fuel consumption along with noise reduction [Kumar et al, 2006]. With other words; lubrication is good economy. Rail fatigue In the late 1990s fatigues accounted for about 60% of defects found by East Japan Railways, while in France (SNCF) and UK (Railtrack) the figures were about 25 and 15%, respectively. Fatigue is a major future concern as business demands for higher speed; higher axle loads, higher traffic density and higher tractive forces increase [Kumar 2006]. Effective ways to reduce the initiation and propagation of fatigue failures is an important field of research, [see Ringsberg and Bergkvist (2003), Ishida, et al (2003), Fletcher and Beynon (2000), Sawley and Kristian (2003) and Jeong (2003)]. Lubrication reduces wear rate and damage to the rails but on the other hand it also causes fluid entrapment in cracks that leads to crack pressurization and reduces the crack face friction that allows relative shear of the crack faces. This accelerates crack propagation. Presence of manufacturing defects in rail subsurface and the direction of the crack mouth on the rail surface are both responsible for guiding crack development direction [see Bower and Johnson (1991) and Bogdanski et al (1997)). Presence of water or snow on the rails may also increase the crack propagation rate. When these minute head checks are filled with water or lubricants they don’t dry up easily. During wheel rail contact, these liquids get trapped in the crack cavities and build up very high localized pressure which may even be greater than the compressive stress. If head checks are in the direction of train traffic, crack growth takes place due to liquid entrapment, but when head checks are in opposite direction of train traffic, the liquid is forced out before its entrapment.
Inspections and interventions Maintenance of railway track includes inspections and interventions. With the inspections, the infrastructure managers employ various methods to obtain information about the
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… track. These methods are complementary and provide a wealth of information for maintenance planning as well as ensuring track safety. Inspections The inspection methods used on railway tracks include both manually and automated methods. Visual inspection is a much used method. To accomplish this, a trained inspector walks the track or rides on a slow moving track vehicle and looks for track problems. However many rail defects cannot be seen with the naked eye, hence, other methods must be utilized to supplement the visual process. Special equipment travelling over the rails that incorporates ultrasonic or inductive methods for detecting those internal, small, and otherwise not visible defects is then used. Track geometry problems such as variations in track gage, cross level, twist, profile, and alignment can be measured manually with the use of hand devices, but it is a time consuming process. More efficient is a vehicle that continuously measures the track geometry as it travels along the track. Most of these vehicles are using laser measurements or accelerometers. The laser measurement can also generate transverse rail profiles. These profiles can be used to measure wear or identify areas of plastic flow. Interventions Intervention is here used as actions carried out to improve the quality of an asset, including both maintenance (preventive and corrective) and renewal actions. There are different types of interventions ordered by data from inspections or at fixed intervals, such as; tamping, grinding, track lifting, ballast cleaning, rail renewal, sleeper renewal etc. Some of the more important for permanent railway track are described bellow. Tamping Tamping is the common term for the operation of lining, levelling and tamping, since it is performed by the same machine. The maintenance action is performed by lifting the track and laterally squeezing the ballast beneath the sleeper to fill the void spaces generated by the lifting operation. Ballast tamping is an effective process for re-adjusting the track geometry [Salim, 2004]. However, some detrimental effects, such as ballast damage, loosening of ballast bed and reduced track resistance accompany it. Loosening of ballast by the tamping process causes high settlement in track. Tamping is eventually needed again over a shorter period of time, and in the long run, ballast gradually becomes contaminated by fines, which impairs drainage and its ability to hold the track geometry. Eventually fouled ballast will need to be replaced, or cleaned and re-used in track [Salim, 2004]. Ballast cleaning The ballast layer can be fouled due to: ballast breakdown, infiltration from ballast surface, sleeper wear, infiltration from underlying granular layers, and sub-grade infiltration. This will affect the bearing capacity of the ballast bed and the drainage function, which in turn will give an even worse function of the ballast. A ballast cleaner removes the fouled ballast and put the cleaned ballast back in the track. Rail grinding Grinding has been undertaken for many years to maintain rail to increase rail life. Rail grinding objectives have included the removal of corrugation (undulations on the rail surface that increase dynamic forces), the removal of rail surface damage (which also improves ultrasonic inspection), and rail re-profiling to improve vehicle steering. In the last two decades increasing emphasis has been given on grinding to remove cracks produced by rolling contact fatigue (RCF). Such cracks form on almost all railways, from transit sections to high-speed passenger railways and heavy-haul freight railways. While much theoretical and experimental work is in progress to understand RCF, most railways see grinding as the only tool that is currently available to control the development of small cracks into significant defects.
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Degradation modelling
Up till now the railway track and the degradation has been briefly described. When crossing to the area of modelling it, there may be beneficial to start looking at the railway track with reliability glasses. This stochastic point of view will also form the basis in the maintenance optimisation described later.
Stochastic modelling The track is considered to be reliable when it performs its intended function under operating conditions for a specified period of time. When this is not the case, the track “fails”. The probability that the track will fail in a small time interval from time t to t + Δt , given that the item has survived up to time t is called the hazard rate. The concept of hazard rate is involved in most methods and approaches to maintenance analysis [Vatn2002]. The hazard rate function can have several behaviours. As far as the railway is concerned, the most likely character is the socalled bath tub curve, as shown in figure 3.
Figure 3. Bathtub curve with a local time-dependent hazard rate
As illustrated in the figure, the bathtub curve may be divided into three phases: I.
Infant mortality (infancy) period
II. Useful life period III. Wear-out period. During the early life of an item (I), there are early failures caused by initial weakness or defects in material, poor quality control, inadequate manufacturing methods, human error, initial settlement etc. Early failures show up early in the life of an item and are characterized by a high failure rate in the beginning, which keeps decreasing as time elapses. Other terms for this decreasing failure rate period are burn-in period, break-in period, early failure period, wear-in period and debugging period. During the second part of the bathtub curve (II), the hazard rate is approximately constant. This period of life is known as the useful life during which only random failures occur. There are various reasons for the occurrence of failures in this period: power surges, temperature fluctuations, human errors, overloading, earthquakes etc. Screening techniques or maintenance practices cannot eliminate these failures. But by making the design of the item more robust with respect to the environments, the effects could be reduced.
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… After the useful life the wear-out period starts (III), when the failure rate increases. The causes for these ‘wear-out’ failures include wear due to aging, fatigue cracking, corrosion and creep, short designed-in life of the maintenance point under consideration, poor maintenance, wear due to friction and incorrect overhaul practices. In figure 3, the word “local time” is used to emphasize the fact that time is relative to the last failure. The bath tube curve indicates that the number of failures will be reduced if the item is maintained before we run into the right part of the curve. There exists also another bath tube curve related to the global section time as shown in figure 4, where we also have illustrated the local bath tube curves.
Figure 4. Bathtub curve with a global time-dependent failure intensity
Note that on the y-axis the dimension is failure intensity, or performance loss. This reflects that the important issue now is the number of failures (overall degradation) per time unit, or generally loss of performance, independent of what has happened up to time t. In figure 5 the numbers 1, 2, 3 and 4 are identified, where the following maintenance situations apply: 1. Point maintenance, related to the explicit failure modes of a maintenance point. 2. Life extension maintenance. The idea here is to carry out maintenance that prolongs the life length of the section. A typical example is rail grinding to extend the life length of rails. 3. Maintenance carried out in order to improve performance, but not renewal. A typical example is adding ballast to pumping sections to improve track quality and reduce the need for track adjustment 4. Complete renewal of major railway maintenance points or sections. Transferring this theory to railway track maintenance, the following distinguish between local curves as failures within maintenance points and global curves as failures in sections. Here maintenance points are defined as points along the track susceptible to inspection and/or interventions, whereas sections are longer parts of track, often with defined parameter values.
Degradation in local time As mentioned; the bath tub curve indicates a wear out zone. However, the shape of the degradation in this wear out zone differ due to which maintenance point that is observed. Vatn [Vatn, 2002] has classified different failure classes related to the characteristics of maintenance point degradation: o
Gradual failures progression 8
… o “Sudden” failure progression In addition to these failure models, Vatn described two models where the failure progression for some reason is not observable. These models are not included here, as we expect that it will be possible to observe the failure progression for all maintenance points of the railway track. The two different classes of failures are illustrated in figure 5.
Figure 5. Two different failure models
Observable gradual failure progression When we have a gradual failure progression, it is assumed that the progression of failure can be observed. The geometrical degradation of a railway track is a typical example. In order to ensure full capacity of the track, the geometrical deviations must be below a certain limit. If we exceed this limit, action must be taken in form of speed restrictions or full stop of the line. A failure occurs when the geometrical deviation exceeds a specified limit. Observable “Sudden” failure progression (PF interval) Now assume that the track section could operate for a very long time without any sign of a potential failure, but at some point of time a potential failure would be evident. In figure 5 a “P” is indicated for potential failure, i.e. the time where a coming failure is observable. The time interval from the failure is first observable, and till a failure occurs is very often denoted the PF interval. In practical terms, then, the PF interval is the grace time available to detect the presence of a defect before it causes the actual failure. An example could be a rail which is exposed to a combination of fatigue and a flat wheel which initiates a crack (potential failure, P). However, such cracks could be detected by ultrasonic inspection, and hopefully before the crack propagates to a failure.
Degradation of sections As we change our view from looking at maintenance points to look at sections, the life spans changes along with the notation on degraded state. From having a life span of 1 to 5 years regarding maintenance points, we now consider the whole section implying a life span of 30 to 60 years. When considering a section, the overall degradation must be considered. This implies a change from following the degradation of one maintenance point until failure to number of failures, e.g. loss of performance or quality. A great challenge looking at sections lies in the presence of many influential factors that culminate in the degradation. Underlying degradation processes of maintenance points together with changing effect of maintenance carried out within the life span. Figure 6 visualises the problem and shows a theoretical degradation pattern for the geometry of a railway track; the quality of track installation is reflected in the initial quality level. Track roughness increases with the amount of tonnage carried. At some point, intervention is needed. The broken lines show that replacing the can be postponed through tamping. After some time the effectiveness of this maintenance may become inadequate, which is reflected in a decreased interval. Replacement becomes necessary before the track condition passes the “operational limit” when safety of the traffic may be at stake. The figure also shows an entirely different option (at 120 megaton, MGT) 9
… which is to upgrade the section, e.g. with an improved ballast bed; the reduced amount of maintenance and extended life should be traded of against this investment.
Figure 6. Degradation of track geometry, adapted from [Ebersohn and Ruppert, 1998]
The degradation process is a complicated process. The rate at which the degradation occurs is a function of time and/or usage intensity. If all railway track sections were identical, operated under exactly the same conditions and in exactly the same environment, then, all sections would degrade in the exact same manner. However usage intensity or operating conditions together with environmental conditions, and material varies between sections. Several attempts have been made to make empirical models based on records or measurements made on track explaining the complicated degradation process. Four models explaining the track settlement are presented as examples; -
An empirical track settlement model based on Japanese experience [Sato, 1995].
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A statistical deterioration model made by the Office for Research and Experiments of the International Union of Railways (ORE).
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A series of equations predicting settlement rate from ballast pressure based on experiments at the Technical University of Munich [Demharter, 1982].
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An Austrian model looking at development of track quality from a passenger’s point of view.
The degradation process is in these models explained through a set of variables, which are more or less the same for all. The most important variables are according to the Japanese model [Sato, 1995] traffic, time, track condition and humidity. This choice of variables is supported by German and Austrian models with some deviations; the German and Austrian models do not consider humidity, however both models regard vehicle characteristics as an important variable. The UIC model [Dahlberg, 2001] contains no track parameters but only loading parameters such as traffic volume, dynamic axle load and speed. As can be seen from these models, the settlement is modelled quite differently, even though the variables used are more or less the same. In the models from Japan and ORE, the settlement grows linearly with respect to the loading, whereas in the German model the settlement grows only logarithmically with respect to loading [term containing log (N)]. The discrepancy between the two models might be large, especially if the parameters in the equations are determined for a relatively small number of loading cycles and the equations thereafter are used to determine the settlement after a large number of loading cycles. The difference is growing even larger comparing the German model to the Austrian which is proclaiming an exponential development of the settlement. Instead of saying that the sub-grade and ballast is being more compact over time (that it settles) which in turn slows the settlement, the Austrian model says that the rougher 10
… the track becomes the more dynamic forces are created when trains passes, which in turn increases the settlement. Japanese study In early 1960, Japanese railway companies published a relationship that enabled the estimation of the settlement of railway ballast when subjected to cyclic loading. Originally developed from laboratory results, the following equation is currently used to estimate the deformation, y, of both heavy haul narrow gauge and high standard gauge:
(
)
y = γ 1 − e − αx + β x Where x is the repeated number of loading or tonnage carried by the track, α is the vertical acceleration required to initiate slip and can be measured using spring loaded plates of the ballast material on a vibrating table, β is a coefficient proportional to the sleeper pressure and peak acceleration experienced by the ballast particles and is affected by the type and condition of the ballast material and the presence of water, and γ is a constant dependent on the initial packing of the ballast material. Heavy axle load study (ORE) ORE has suggested a model to estimate track deterioration, e [ORE, 1998]. The deterioration is divided into two parts: the first part describes the deterioration directly after tamping, e0 , and the second part describes the deterioration depending on traffic volume T, dynamic axle load 2Q and speed v. The relationship reads
e = e0 + hT α (2Q ) ν γ β
where h is a constant and the parameters α , β and γ have to be estimated from experimental data. Due to lack of data, UIC suggests α = 1 and β = 3 . Further, it is assumed that the influence of the speed can be neglected, implying that ν γ need not be treated separately, but may be included in the proportionality factor h. German study Experiments under well controlled laboratory conditions at the Technical University of Munich representative of vehicles passing a dipped joint have been used to establish equations to calculate rate of settlement (S) [Demharter, 1982]. Ballast pressure is multiplied by the log of the number of axle passes as follows:
S = a × p × ln ΔN + b × p 1.21 × ln N As can be seen from the equation a logarithmic settlement law has been used. The first term represents the fast settlement just after a maintenance action. ΔN express a pre loading period comprising the first passing axles. ΔN should be ≤ 10000 and N in the second part should express the total number of passing axles. The ballast pressure p should be calculated with the Zimmermann method [Demharter, 1982]. The parameters a and b are constants suggested to be in the value range; 1.57-2.33 (a) and 3.04-15.2 (b). Austrian study TU Graz has examined settlement developments in Austria by the quality figure MDZ, which represents accelerations in the vehicle caused by track imperfections [Promain, 2002]. The MDZ figure comprises of both horizontal and vertical deviations in track together with lack of super elevation and speed [Hummitszch, 2005]. An exponential development of the MDZ figure over time was found giving the following expression for track quality:
Q = Q0 × e − b×t 11
… where Q represent the track quality represented by the MDZ number and Q0 the initial track quality.
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Methods for optimizing maintenance and renewal
Using the models of degradation as basis, the maintenance in form of inspection and intervention intervals can be optimised with respect to the total cost of maintenance and risk.
Optimising point maintenance In the degradation models, there are defined a limit called “critical failure progression”. This is a limit saying that degradation passing this limit is assumed to be critical. However in real life there often exists more than just one level. In the Norwegian rail administration there are three limits. A maintenance limit (ML) indicates further investigation, an action limit (AL) says that maintenance is required and finally operational restriction and maintenance is mandatory when passing a safety limit (SL). The failure progression is a random variable. When the failure progression exceeds the maintenance limit a corrective maintenance action is performed which resets the section. If the failure progression exceeds the failure limit a failure occurs. The failure progress could be specified in various ways. Two common used models for the failure progression are the Wiener process, and the Gamma process. Both processes are continuous-time stochastic processes. A limitation in these processes is that the degradation is assumed to be linear with time. This is problematic when e.g. cracks are modelled, since the failure progression is believed to go faster and faster as the crack size increases. Yet another limitation is the assumption of independence which is often used in reliability analysis. It means that if one maintenance point in a section fails and is repaired, all the other maintenance points in the section will function as normal without regard to the repair going on. For many sections this assumption will be unrealistic, and other approaches have to be used to determine the section availability. An alternative approach is to base the analysis on Markov models. By using Markov models a wide range of dependencies can be taken into account. [Rausand 2003]. As a last example of methods used within the railway industry is the Monte Carlo simulation. The simulation is carried out by generating certain random and discrete events in a computer model in order to create a realistic or “typical lifetime scenario of the section. In the Monte Carlo approach a realization of the life process is simulated on the computer and, after having observed the simulated process for some time, estimates are made of the desired measures of performance, thus the simulation is treated as a series of real experiments. Simulation is an attractive alternative because it allows the modelling of virtually any time to failure distribution, as well as allowing behaviours that preclude analytic solution. Further, simulation may be a more efficient approach in complex sections where it would be to time consuming to develop analytical models. The major disadvantage of simulation is the long simulation times needed to achieve high accuracy. In addition you will, in many situations, create a sort of “black box” with little insight in the processes going on. Barrier modelling A risk model on a format that allows the prediction the risk level as a function of the maintenance level is a useful tool when calculating costs. An example of such a model is the barrier model, looking at the outcome in term of cost if barriers in connection with the maintenance point should fail. A barrier is defined as that part of a section that prevents the occurrence, or at least lowers the occurrence probability to a minimum, of the so called ‘topevent’, if a maintenance point fails. A ‘top-event’ describes the worst direct consequence of any failure. Normally there is more than one barrier preventing the occurrence of the ‘top-event’. The
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… idea of barrier thinking is illustrated in Figure 8, where the case of crack growth in rails is illustrated:
Figure 7. Barrier modelling [Vatn,2002]
Only because of the existence of these barriers, railways can be run in the way they are. Up to now for most maintenance points the existing barriers have been defined, but their influence is just estimated. It will cost a lot of further work to calculate their true factors explicitly. Still it is regarded as most important to integrate those barriers in the calculations, because it is always better to take facts that obviously exist into account even if the exact figures are not known, than not to include any value. Cost analysis In order to optimise the maintenance effort, the maintenance point performance measures and the cost model must be combined, and then balance with the maintenance cost. The maintenance cost is specified by:
PM Cost
Cost per preventive maintenance (PM) activity (intervention).
I Cost
Cost per inspection.
CM Cost
Cost per corrective maintenance (CM) action.
The cost per unit time is now given by:
C (τ ) = C R (τ ) + [1 / τ Int + rr (τ )]PM Cost + I Cost / τ Insp + λ (τ )CM Cost where CR (τ ) are costs calculated from risk model. λ (τ ) and rr (τ ) are effective failure rate and
renewal rate respectively. In the equation τ Int is used to denote the maintenance interval in case
of interventions, whereas τ Insp denote the maintenance interval in case of periodic inspections.
Further τ is used without any index when interval must be found from type of maintenance activity. To find the optimum maintenance interval C (τ ) from the equation for various values of the maintenance interval, τ , can be calculated and then chose the value for τ that minimizes C (τ ) . The output of such an optimisation task can also be plotted in a diagram as illustrated by figure 8.
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Figure 8. Optimisation of maintenance intervals
Here the cost of corrective maintenance as a consequence of failures is measured upon the cost of preventive maintenance.
Optimising section maintenance and renewal Here the objective is to establish a sound basis for the optimization of maintenance and renewal. The idea is to choose maintenance activities in time and space such that costs are minimized in the long run. Different “headings” are used for such analysis, e.g. LCC analysis, Cost/benefit analysis and NPV (Net Present Value) analysis.. As in the latter approaches the degradation as a function of time is the background of this approach as well. This degradation could be transformed into cost functions, and when the cost become very large it might be beneficial to perform maintenance or renewal activities on the infrastructure. In the following the notation c(t) is introduced for the costs as a function of time. These costs included in this function mainly include -
punctuality loss
-
accident cost
-
extra operational and maintenance cost due to reduced track quality
By a maintenance or renewal action the function c(t) is typically reset, either to zero, or at least a level significantly below the current value. Thus, the operating costs will be reduced in the future if investments in a maintenance or renewal project are made. Figure 9 shows the savings in operational costs, c(t) - c*(t), if maintenance or renewal at time T is performed. In addition to the savings in operational costs, savings due to an increased “residual life time” will also often be achieved.
Figure 9. Cost savings [Vatn, 2002]
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… Special attention will be paid to projects that aim at extending the life length of a railway section. A typical example is rail grinding for extending the life length of the rail, but also for the fastenings, sleepers and the ballast. Fig. 10 shows how a smart activity ( ) may suppress the increase in c(t) and thereby extend the point of time before the cost explodes and a renewal is necessary.
Figure 10.
Life length extension [Vatn, 2002)
Following this line of arguments, looking from a cost-benefit point of view, all projects with a cost-benefit ratio higher or equal one should be carried out. The challenge lies in finding the optimal time to carry out the project. However due to lack of financial resources and insufficient work capacity, the infrastructure managers might have to drop some of the projects. A tool design to help prioritizing projects so that is the Norwegian PRIFO tool. The results of the PRIFO tool are supposed to be listed and rated according to their cost-benefit ratio. The higher this figure, the more economically beneficial is the assessed project. Figure 11 illustrates the output of the PRIFO tool:
Figure 11.
Ranking of Projects
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Case studies on RAMS
Examples of optimisation of inspection intervals for the ultrasonic inspection car and the geometrical inspection car are presented in two cases.
Optimizing ultrasonic inspection intervals Introduction Failure and inspection data from Dovrebanen (a part of the Norwegian railway line from Trondheim to Oslo) are analysed. The degradation/repair process within the fixed inspection interval is modelled as a time continuous Markov chain. Also the change of state implemented at the end of an inspection interval is modelled as a (time discrete) Markov chain. The model will demonstrate how reliability depends on the inspection interval, and will thus support identification of the most cost effective preventive maintenance strategy for the railway line in question. The critical failures (i.e. broken rail) can either be seen as shocks (i.e. with no “warning”), or as a gradual degradation, where the line goes through various degraded states (with cracks) until it gets a critical failure. When a degraded failure occurs, the railway line is still functioning, and the crack can only be revealed by inspections of the line. Those inspections are performed at regular intervals by Ultrasonic Inspection Cars (UIC). However, at each inspection there is only a probability q of detecting a degraded failure. A piece of rail which is degraded is more prone to suffer a critical failure than a piece of rail not degraded (i.e. in the OK state). When a critical failure occurs, the failure has to be repaired in order to maintain regular traffic. More details are given in ref [Hokstad et. al. 2005], and we note that the same data has also been analysed in ref [Podofillini et. al.,2004 ]. A General Markov Failure Model A phase type distribution is used for time to failure. The failure model includes two different states for degraded failures and two different states for critical failures. In addition we have the OK state (see Figure 12);
Figure 12.
A general Markov failure model
The critical failures can be divided into two categories; failures due to degradation, denoted F1 and “shock” failures, denoted F2. The latter failures happen when the rail is exposed to large external forces like rolling stock. Those failures cannot be avoided by inspection. The critical failures due to degradation, however, can be avoided by performing preventive repair if they are discovered at inspections. The first degraded state, denoted D1, is for minor degraded failures (cracks). If a rail is detected in this state, the observations are made more frequent so that a critical failure due to degradation not should be possible. When the degraded state called D2 is detected (larger cracks) the failure is repaired immediately. The development of degraded and critical failures is modelled by a time continuous Markov chain, see Figure 1. If the railway line does not have a critical failure, there is a constant
16
… rate λ for getting a shock failure F2. In order to reach the critical failure state F1 the rail has to go through the degraded states D1 and D2. We partition the rail into small pieces of rail so that one piece is in only one of the states OK, D1, D2, F1 or F2, (and the above rates refer to these small pieces of length 1 m). A Specific Failure model for the railway case In the following illustrations we will for simplicity ignore the state F2. The effect of this failure category can afterwards be incorporated by just adding an additional failure rate, λ (cf. Figure 1). However, in order to later also to model the maintenance of the degradation failures we split the degradation states, according to whether these are detected or not. A subscript u on the degraded states indicates that a degraded failure is undetected (by UIC). Likewise, a subscript d indicates a degraded failure that is detected. Thus, D1u = Minor degraded failure (crack) being undetected. D1d = Minor degraded failure detected by UIC; then the observations are made more intensive (frequent) so that a critical failure due to degradation is not possible. D2u = Major degraded failure (crack) being undetected; (state believed to be OK). D2d =Undetected major degraded failure when the piece of line earlier has been detected to be in state D1; and is therefore it is closely monitored, but it is not known that the state D2 is reached. As soon as the state D2 is detected the failure is repaired immediately and the piece of rail goes to OK. At the beginning of an inspection interval, we can start in one of the states OK, D1u or D2u D1d. The Markov diagram is presented in Figure 13, which is valid for the complete inspection interval of length, T. Note that if we during the inspection detect that the line is in the state D1, then the next inspection interval will start in state D1d. Here ρ refers to the transition rate caused by additional inspection in a detected state D1 as explained above. The direct rate to OK follows by the assumption of immediate repair. For simplicity we introduce an absorbing state OK*, and transfer to OK is carried out at the start of the next inspection interval. So both the states F1 and OK* are made absorbing states in the time continuous chain, meaning that a fresh start in OK always takes place at the beginning of an interval. This means that the same piece of rail can never have two failures or visits to D2u within the same interval, and we do not start in OK in the middle of an interval. This is a computationally simplifying assumption that will not to a great extent affect our results.
Figure 13.
Markov model adapted to the railway case, (now ignoring state F2)
Further, note that the modelling allows transitions from D1 to D2 to have different rates, depending on whether the degradation to D1 is detected or not. The rate of failures (to F1) is however assumed to be the same for both D2u and D2d. 17
… Using transition rates as in Figure 3, we can now easily write down the intensity matrix, Q, of this time continuous Markov chain. Now numbering the states as OK = 1
D1u = 2
D1d = 3
D2u = 4
D2d =5
F1 = 6
OK* = 7
This 7x7 matrix is of the form ⎡A K⎤ Q=⎢ ⎥ ⎣0 0 ⎦
where A is a 5x5 matrix, and the “0”-s here are matrices consisting of zeros only. Then using a suitable method for solving Markov chains, for example the computer package Maple, we can easily find the transition probabilities of the process. Let Xn(t), n=1, 2, ….., be the state of the time continuous Markov chain at time t in n’th inspection interval, and let pjk(t) = P(Xn(t) = k | Xn(0)= j) We denote this matrix of transition probabilities by P(t) and get ⎡e tA P(t ) = ⎢ ⎣0
A −1 (e tA − I ) K ⎤ ⎥ I ⎦
where I is the identity matrix of appropriate dimension. The Maintenance Model Next we introduce the Markov Chain for transitions at the inspection. The states Xn(0) and Xn(T) are of particular interest, where T is the length of the interval. Now consider the transitions of state that may occur as the result of the inspection (occurring at times T, 2T, ….). In order to be able to fit the model to the failure/inspection data, we now introduce the variables Un and Vn. Un tells the true state for a small piece of rail immediately before inspection, and Vn tells the true state immediately after inspection, i.e. at the start of the next inspection interval. Thus actually Un = Xn(T),
n = 1, 2, …….
Vn = Xn+1(0), n = 1, 2, ……. These have the asymptotic distributions πk = P(Un =k);
k = 1, ….., 7
ψk = P(Vn = k);
k = 1, ….., 7,
and the corresponding row vectors (vectors being bold) are: π = (π1, π2, ……., π7) ψ = (ψ1, ψ 2, ……, ψ7) Further, we introduce probabilities that degraded failures are detected by inspection: q1 = Probability that state D1 of a line segment is detected. q2 = Probability that a degraded failure, D2 is detected by the inspection; not knowing in advance that the state D1 was reached. q3 = Probability that a degraded failure, D2 is detected by the inspection; knowing in advance that the state D1 was reached. We can then introduce the matrix, R, for the transitions at the inspections, i.e. transitions from Un to Vn.
18
… OK D1u D1d
0 ⎡1 ⎢ 0 1− q 1 ⎢ ⎢0 0 ⎢ R = ⎢q 2 0 ⎢q3 0 ⎢ 1 0 ⎢ ⎢⎣ 1 0
D2u
D2d
0 q1
0
0
0
0
1
0 1− q2
0
0 0
0
0 1 − q3
0
0
0
0
0
0
F1 OK*
0 0⎤ 0 0⎥ ⎥ 0 0⎥ ⎥ 0 0⎥ 0 0⎥ ⎥ 0 0⎥ 0 0⎥⎦
Now the transition matrix for Un equals R · P(T), and similarly the transition matrix for Vn equals P(T) · R. Now the asymptotic distributions of Un and Vn , are determined from the relations ψ = π ·R π = ψ ·P(T) Thus, the vector π is found from π = π ·R ·P(T) Overall Model and Assumptions As indicated above the two Markov chains can be combined into one total model, see Figure 14. Here we give the state numbers 1, …., 7 in addition to the notation OK etc., and again for simplicity we ignore the state F2. The solid lines represent the possible transitions within an inspection interval T, (cf. matrix P(T)). Recall that we make the simplifying assumption that one small piece of rail can only have one visit to OK* and F1 (and F2) within one inspection interval. Therefore we actually treat these as absorbing states in the time continuous Markov chain, and the process always start in state OK at the beginning of the next interval. The dotted lines of Figure 14 indicate transitions at the end of the test interval (cf. matrix R). With probability q1 we leave D1u and with probability q2 we leave D2u (thus observing state D2 but then going directly to OK which is the starting state at the beginning of next interval). Further, with probability q3 we leave D2d (thus actually observing D2), and there will also be transitions from the “absorbing states” OK* and F1 to OK.
Figure 14.
Overall failure/maintenance model, (state F2 not included)
19
… So in the total the model we have a time- continuous Markov chain in the time spans between inspections, while at the inspections we have transitions following a time discrete Markov chain. The model and the estimation are based on some assumptions: • • • •
•
The processes Un and Vn, are assumed to be stationary processes; (the actual railway line is quite old so this is a rather realistic assumption). The probability distributions of the time continuous processes are identical for all inspection intervals (i.e. stationarity is assumed also in this respect). Failures are equally distributed over the railway line (homogeneity). However, the estimated rates can be seen as averages for the line in question. We treat the critically failed states as absorbing states that are also repaired at the inspections. This implies that one small piece of rail can not fail critically twice inside one inspection interval. Mean Time To Repair (MTTR) = 0.
The Input Data The input parameters for the estimation of model parameters are listed in Table 1. Degraded failures have been recorded since 1st January 1991 and have been exposed to 8 tests by October 2002. Critical failures have been recorded since 1st January 1989, and assuming the same length of the test interval (T=4299/8=537 days) also for these, this implies that these have been exposed to 9.4 tests. In total 800 degraded failures were observed. For 22 of these the severity was not recorded (i.e. not categorized as D1 or D2), and these 22 failures were just distributed proportionally amongst the two categories, giving in total 331 failures of type D1 and 469 of type D2. Further, 20 of the D2 failures have already been observed in state D1 (i.e. coming from state D1d and are actually monitored closely). It is then assumed that the test detects the transition to D2, and so these failures are corrected, and there is a transition back to state OK. The 449 detected failures of type D2u represent transitions from D1u, i.e. the degradations are observed for the first time. These are a fraction q2 of the actual number of rail pieces in state D2u, and will by the test be brought back to the state OK. The other will remain in state D2u. Finally, the number of critical failures of type F1 and F2 are given as 249 and 81, respectively. In order to carry out the estimation of the unknown rates it is also required to estimate some parameters by expert judgments (operational experience); these are: The probabilities q1, q2 and q3 and the rate (ρ) of reaching state D2 under additional inspections when state D1 is detected.
20
…
Parameter definition
Parameter
Value
Length of rail
L
Number of tests/inspections 1989-2002
nTF
9.4
Number of tests/inspections 1991-2002
nTD
8
Number of days, 1989-2002
N1
5027
Number of days, 1991-2002
N2
4299
Length of test/inspection interval
T
537 days
Number of observations in state D1, (i.e. ND1 transitions from D1u to D1d ) Number of observations in D2 when it was not known that state was degraded, (i.e. transitions ND2a from D2u) Number of observations in D2 when it was known that state was degraded, (i.e. transitions from D1d) Number of observations in F1
365 km = 365 000 m
331
449
ND2b
20
NF1
249
Number of observations in F2
NF2
81
Probability of detecting D1 failure at test
q1
Probability of detecting D2 failure at test q2 (state D1 not detected previously) Probability of detecting D2 failure at test q3 (state D1 already detected) Rate of detecting D2 in additional inspections; (assuming on the average two additional ρ inspections within each interval T)
0.4 (expert judgement) 0.7 (expert judgement) 0.9 (expert judgement) (2/T)q3 (exp. judgem.)
Table 1. Input data to analysis
Estimation It is now quite easy to estimate the distribution of Un under stationarity. For instance, the estimate of π6 is given by the number of observations in F1. The equations for π4 and π2 are obtained similarly. Finally, the estimated π5 and π7 are found from the number of detections of state D2 given that degradation D1 is already known. One third of these observations are assumed to be carried out at the ordinary inspections (see bottom of Table 1), and two thirds at the additional inspections, thus resulting in transitions to state OK*. Further, the estimate of π3 can be obtained from the following relation: π3 = (π2 · q1 + π3) · (1- e-σT) The argument is that if Un = 3, then either Un-1 = 3 or Un-1 = 2, and D1 being detected, giving a transition from state 2 to 3, and in addition no transition has occurred in the last inspection interval. Finally we have the normalization equation. Now, a joint estimation of the stationary distribution, π, and the unknown rates (μ, ω, σ, ν), are obtained by a recursive approach (see ref [Hokstad et. al. 2005]), and we get the following estimates for the stationary probabilities πj (for 1 m rail): (πˆ1 , πˆ 2 ,..., πˆ 7 ) = (0.99935, 2.83·10-4, 7.25·10-5, 2.20·10-4, 2.54·10-6, 7.30·10-5, 4.57·10-6)
Further we get the following estimated rates (per day):
21
…
μˆ = 5.9 · 10-7 /m ωˆ = 1.6 · 10-3
νˆ
= 9.3 · 10-4
λˆ = 4.4 · 10-8 /m σˆ = 9.2 · 10-4 ρˆ = 3.4 · 10-3
Observe that the estimate for ρ was obtained directly from expert judgment (Table 1), and λ was estimated directly from the number of F2 failures, (NF2) the total number of days N1, and the
rail length, i.e. λˆ = NF2/(N1·L). Further, observe that now the distribution of Vn is also found using the R-matrix (ψ = π ·R). There are five possible states at the start of a test interval (ψ 6 = ψ 7 = 0) and
(ψˆ1 ,ψˆ 2 ,......,ψˆ 5 ) = (0.99958, 1.7·10-4, 1.9·10-4, 6.6·10-5, 2.5·10-7).
Maintenance Optimization When the estimates of the rates λ, μ, ω, σ and ν are established it is of interest to consider the effect of various levels of maintenance on basic reliability parameters like: Frequency (rate) of entries into failure states Mean Time To Failure (MTTF) Now introduce MTTFi = Mean Time To Failure for failure mode i (i.e. F1 and F2). The rate 1/MTTF = (1/MTTF1 + 1/MTTF2) could also be referred to as the (asymptotic) Rate of Occurrence of Failure, ROCOF 1 , (see ref [Ascher et. al., 1984]) when there is no maintenance. First observe that • •
MTTF1 = (1/μ+1/ω+1/ν) = 9.2 years MTTF2 = 1/λ = 62 years MTTF = (1/MTTF1 + 1/MTTF2)-1 = 8.0 years. There are two parameters related to the maintenance that we can control: • •
Length of inspection interval, T Frequency of additional inspections, (cf. ρ), when degradation in state D1 is detected
As the number of entries to the critically failed state F2 does not change with the maintenance level, we keep focusing on the failures due to degradation. In particular the asymptotic rate of entering F1, and its inverse (MTTF1) for various values of T is most interesting. To find such entry rates we need the over-all average probability of the time continuous process being in various states. The probability for the process X(t) to be in state k at time t equals; (remember that ψ5 = ψ6 = 0): 5
p k (t ) = ∑ j =1ψ j p jk (t ) Here pjk(t) are the elements of the transition matrix P(t). Now the “average probability” to be in state k equals 5
p k* = ∑ j =1ψ j p *jk where
p*jk =
1 T T 0
∫
p jk (t ) dt
1
Here we generally use ROCOX as the rate of occurrences of state X; (the asymptotic rate of entering state X)
22
… For example, the “overall” (average) probability of the time continuous process to be in state OK is found as P(OK) = ψ1 · p1* = ψ1 · [1- e-μT] / (μ ·T) ≈ [1- e-μT] / (μ ·T) The rate of entries into the degradation state D1 (actually D1u) is then found as ROCOD1u = P(OK) · μ ≈ [1- e-μT] /T The most interesting results is obviously the rate of entries into the critical failure state F1. We then need the probabilities to be in states D2u or D2d. Now P(D2u) = ψ1 · p14* + ψ2 · p24* + ψ4 · p44* P(D2d) = ψ3 · p35* + ψ5 · p55* can be derived. The asymptotic rate into F1 equals ROCOF1 = [P(D2u) + P(D2d)] · ν So for the maintained system, the asymptotic MTTF1 = 1/ ROCOF1 is given by these formulas. Now ROCOD1u, ROCOD2u, ROCOF1 and MTTF1 are computed for a few values of T, and also for some alternative values of ρ. Table 3 gives the results for T=365 days and 730 days together with the actual value of the observations (T=537 days). The mean number of observations in state F1 for a time span corresponding to the actual data (i.e. 5027 days) is also given. In Table 4 we use T = 537 days, but vary the rate ρ. The ρ is given as a percentage of the value used in the analyses above. Parameter
Inspection interval, T 365 days 537 days 5.88 · 10-4 5.88 · 10-4 -4 2.98 · 10 3.50 · 10-4 0.98 · 10-4 1.42 · 10-4 28.0 19.3
ROCOD1u (per day and km) ROCOD2u (per day and km) ROCOF1 (per day and km) MTTF1 for 1 km rail (years) Mean no. of failures (in 180 5027 days)
261
730 days 5.88 · 10-4 3.88 · 10-4 1.85 · 10-4 14.8 339
Table 2. Reliability parameters for various values of T
ρ (rate of increased ROCOF1 inspection by (per day and km) detecting state D1) 25% 1.50 · 10-4 50% 1.47 · 10-4 100% 1.42 · 10-4 200% 1.36 · 10-4 400% 1.31 · 10-4
MTTF1 for 1 km rail (years) 18.3 18.6 19.3 20.1 20.9
Mean no. of failures (in 5027 days) 276 270 261 250 240
Table 3. Reliability parameters for various values of ρ
So the estimated model shows for instance that the MTTF for critical degradation failures of 1 km length of rail decreases from 28.0 years when inspection interval is 1 year, to 14.8 years when the inspection interval is 2 years (Table 3). Similarly, mean number of failures is almost doubled when T increases from 1 year to 2 years. The frequency of increased inspections when a degraded failure is detected has less effect on the results (Table 4). We note that the computed estimates are valid for the given line only; for another line with a different state of the line and the environment one would obviously get different estimates
23
… for the transition rates. The approach is, however, generally applicable, and the usefulness of analytic models to optimize maintenance is demonstrated.
Optimizing track maintenance In this second case we will address the optimisation of track geometry inspections on the Norwegian railway network. The critical failure is twist, and only one failure mechanisms are considered; critical failure occurs as the result of a degradation process. In addition, three limits are set in advance; ML=maintenance limit, AL=action limit and SL=safety limit. If the maintenance limit is reached, an action plan to maintain the degraded area should be planned so that the degraded area is maintained before the action limit can be expected to be reached. If the action limit is reached, the degraded area should be maintained as soon as possible, and latest before the next inspection. An action limit can also require restrictions such as reducing quality class of the area which involve speed restrictions. Maintenance should be carried out immediately if the safety limit is reached. Various types of inspection and maintenance are performed on the line. Inspection by a self-propelled engine that uses three point laser sections to measure the track geometry (the ROGER 1000) is carried out at regular intervals. Additional inspection will be initiated on a segment of the line if degradation above a certain level is observed. The degradation/repair process within the fixed inspection interval is modelled as a time continuous Markov chain. Also the change of state implemented at the end of an inspection interval is modelled as a (time discrete) Markov chain. The model is based on actual inspection data for a specific railway lines in Norway. These data are used to estimate the parameters of the model. The given failure/maintenance model and estimation technique should generally be useful for systems that experience deterioration and are subject to imperfect inspection. Basic model description This case resembles the case with broken rails with some difference. In this case we follow the following assumptions: -
The system is subjected to a degradation process. The degradation is modeled as a Markov chain with r states. The MTTF in each state is based on empirical degradation
-
The system is periodically inspected. However, the inspection interval is not constant. The time of next inspection depends on the system state revealed by the previous inspection.
-
Initial twist depend on the designed twist due to changing cant in transition curves
-
PM and CM are modeled as perfect in the sense that the system is put back with to a design twist level.
Estimation of parameters The lengths of the main states are retrieved from a simplified model of track settlement made on the result from regression analysis of geometrical inspection data. The data were collected with the ROGER 1000 in the period from October 2003 to June 2005 on four Norwegian tracks; the Dovrebanen, Nordlandsbanen, Meråkerbanen and Sørlandsbanen. The settlement model reads:
Twist1 = Twist 0 + 2 ×10−5 × e
(
0 , 0246× Twist 0 − Twist Design
)
×T
Where Twist1 is the resulting twist, Twist 0 is the initial twist retrieved from inspection, Twist Design is the designed twist due to change in cant and T is tonnage run on track. The MTTF
for state 1,2 and 3 can then easy be calculated;
24
…
MTTF1 (Tons ) = MTTF2 (Tons ) = MTTF3 (Tons ) =
ML − Twist 0 −5
2 × 10 × e
(
0 , 0246× Twist 0 − Twist Design
AL − Max(Twist 0 ; ML ) 2 × 10 −5 × e
(
0 , 0246× Twist 0 − Twist Design
SL − Max(Twist 0 ; AL ) 2 × 10 −5 × e
(
0 , 0246× Twist 0 − Twist Design
) ) )
×T ×T ×T
Note that MTTF is in term of # of tons. In order to get the time in year, the MTTF must be rewritten:
MTTFi (Years ) =
MTTFML (Tons ) Ton/year
As can be seen from the formulas, we have a situation with increasing degradation. By obtaining
v=e
(
0, 0246× Twist 0 − Twist Design
)
Then we use the basic summation rule for a geometric series, and obtain:
λ0 =
1− 1
vr (1 − 1 ) MTTF v
in order to fulfil MTTF = ∑i1/λi. We could also easily verify that Var(T) is given by:
Var (T ) =
1− 1
v 2r (1 − 1 2 )λ20 v
If the calculated variance does not match the variance we are aiming at, we could change the number of states. An increase in the number of states (r) will give smaller variance. Specification of cost elements To optimise inspection intervals and intervention level we specify the following basic cost elements given in table 4: Cost element
Description
CI
The cost per inspection
CF
The (unavailability) cost per system failure
CCM
The cost of repairing a system failure
CRC
The cost of renewing the system at state l
Table 4. Specification of cost elements
The total cost per unit time is given by:
C(τ,l) = CI/τ + (CF+CCM)×λFP(τ,l) + CRC×r(τ,l)
25
… The costs are expressed as their current value, i.e. the present value (PV) is calculated for all future costs. For the calculation of the present value of a future amount C in year y, the following equation is used:
CPV = C × (1 + d )
−y
where d is the discount rate. Input Data Now, after having defined the final Markov process, the equations presented in the other case can be used to calculate performance measures. The chosen lengths of the inspection intervals τ , τ and τ are assumed to be 36 months, 6 months and 1 month, respectively. 1
2
3
In the presented example the analysis period is 30 years. The expenses were estimated for the inspection, renewing, failure, and repairing costs are given in table 5
CI CRC CF + CCM
1 30 100 2,0 % 7,0 %
Infl. rate Disk. rate
Table 5. Costs related to track maintenance
For twist (9 m basis of measuring) the following limits applies (in Norway) • ML=20 • AL=31 • SL=34 In this case we assume to be in a transition curve with designed twist of 5. We also assume to start at this state, giving a MTTF of -
15,6 years in state 1
-
8,3 years in state 2
-
1,9 years in state 3
-
0,5 years in state 4
In order to find a good inspection strategy, the length of the inspection interval τ , τ and 1
2
τ must be found. To do this we introduce a reduction parameter. This reduction parameter tell 3
how much to reduce the inspection interval when reaching the maintenance limit (ML). To find the best reduction parameter, it is changed from 1 (no changing in inspection interval) to 0,2 and compared to find which gives the lowest present value (PV) of the total cost. The total costs (present value calculated over a time horizon of 30 years) are plotted in Figure 15.
26
…
100 90
PV of total costs (1000 EUR)
80 70 1
60
0,3 0,25
50
0,2
40
0,15
30 20 10 0 1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
Inspection interval (years)
Figure 15.
Total PV costs for different reduction parameters
The plot shows clearly that the optimal inspection strategy is given by a reduction parameter of 0,2 and inspection interval of 7 years, i.e. inspection intervals around 7 years, 18 months and 3 months, respectively. The different cost drivers and the total costs are shown versus the inspection interval for the optimal value of the reduction parameter in Fig. 16.
PV of toatl costs (1000 EUR)
80
CI
70 CF
60
CCM
50
Tot kost
40 30 20 10 0 0
5
10
15
20
Inspection interval (years)
Figure 16.
Present value of total costs and costs for CM, PM and inspections
27
…
6
Conclusions and future challenges
In this chapter we have introduced both ageing/degradation models and recent models for maintenance/renewal optimization that are relevant for railway track. Some of the models for maintenance optimisation are illustrated by a couple of case studies using data retrieved from inspection on Norwegian railway tracks. However there are some future challenges which are not addressed, but should require attention. The possibility for a degradation process to reverse is not included in the Markov modelling presented above. However, this could very well be the case in geometrical degradation, and in particular with twist, as the forces in the track may to some degree reverse the development. There may also be a challenge that maintenance can reduce the quality of the system. Maintenance action like tamping does not improve the quality of the ballast, but rather worsen it, making the degradation process change after the maintenance action. Finally, the grouping of maintenance activities should be addressed. Such a grouping will not change the maintenance itself but will reduce the total costs as the logistics are improved.
7
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