DEVELOPMENT OF A BRAKING MODEL FOR SPEED SUPERVISION SYSTEMS


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1 DEVELOPMENT OF A BRAKING MODEL FOR SPEED SUPERVISION SYSTEMS Paolo Presciani*, Monica Malvezzi #, Giuseppe Luigi Bonacci +, Monica Balli + * FS Trenitalia Unità Tecnologie Materiale Rotabile Direzione Tecnica Sperimentazione Viale S. Lavagnini Firenze Italy # University of Florence  Department of Energetics "Sergio Stecco" + FS Trenitalia Unità Tecnologie Materiale Rotabile Summary The monitoring curves of a spee control system are generally implemente through a moel escribing the braking performances of trains in terms of initial elay an braking eceleration. The paper presents a braking moel evelope to be employe in a new ATP system for the Italian railways. Such moel, base on the brake weight percentage, was erive from the UIC regulation so as to be applicable to all kin of trains an braking systems. Taking also into account the variability of the major parameters affecting the braking performances of trains, a probabilistic analysis is presente with the purpose of etermining the proper safety margins to be applie to the basic braking moel. Abstract To improve safety an efficiency in the management of railway traffic, a new spee control system name SCMT is currently being carrie out at the Italian Railways. Other innovative spee supervision systems are being evelope in Europe, such as the ETCS/ERTMS, which will be also installe on the new high spee line RomaNapoli. All traffic management systems are generally base on a set of supervision curves relating the permitte velocity of the train to the running istance, in orer to ensure the respect of spee restrictions on the line by intervention of an emergency braking in case of train velocity exceeing the permitte one. These supervision systems must apply to all trains: passenger an freight trains of ifferent lengths, with various braking equipment, for all ranges of velocity. To elaborate this set of supervision curves, the onboar unit nees train eceleration epening on time an spee, as basic information about the braking behaviour of the train. For high spee trains the knowlege of a eceleration profile is currently available, because moern UIC regulation requires the experimental assessment of this parameter. But the braking performances of conventional trains are usually expresse by the brake weight percentage, calculate through the weights an the brake weights of the vehicles which make up the train. The brake weights are usually etermine on the basis of stopping istances obtaine in tests, accoring to UIC leaflet The brake weight percentage is therefore an effective way of
2 expressing the capacity of the train to stop over a certain istance, when running at a efinite initial spee. However, this parameter oes not provie any information about the actual eceleration characteristics, which can wiely vary epening on the brake equipment. Therefore the implementation of a spee supervision system requires the preliminary efinition of braking moels which allow to convert the general parameters affecting the braking performances of trains (such as brake weight percentage, goos/passenger brake position, brake equipment, train length etc.) into a basic eceleration profile as function of time, uring the eceleration rise phase, an of spee, uring fully evelope braking. But this basic eceleration profile is not yet sufficient to buil the emergency brake intervention curve, because this one nees a guarantee eceleration profile, which can be obtaine from the basic profile using appropriate safety coefficients. The paper presents the braking moel evelope for the SCMT system, base on the UIC evaluation metho an applying to all trains, which enables to transform the available information concerning train an brake features into the essential input ata for a spee supervision system. This moel takes also into account the scattering of the eceleration ue to the main factors involve in the braking, in orer to obtain the require safety level of the emergency brake intervention curve. To evaluate the safety margins, a complete analysis of parameters affecting the braking performance was carrie out. For the major parameters, the probability istribution was etermine on the basis of technical knowlege an experimental results, in orer to establish the combine probability istribution for ifferent types of trains, thus enabling the assessment of the safety egree as function of a reuction factor applie to the basic eceleration. Keywors Braking moel; spee supervision system; brake weight percentage; safety margins; probabilistic analysis.. DEVELOPMENT OF THE BRAKING MODEL. Introuction For all the conventional rolling stock an for spee until km/h the basic parameter currently use for national an international traffic in orer to etermine the braking performance of a train is the brake weight percentage as efine by the UIC regulation. For this reason the braking moel evelope for SCMT is base on this regulation. Nevertheless, the information about the eceleration uring a fully evelope braking cannot be irectly erive from the UIC brake weight percentage. The brake weight of vehicles is usually preetermine by calculations an verifie by tests in line carrie out either on a single vehicle or on a trainset, using the evaluation iagrams of the UIC leaflet Such iagrams establish a relationship, for ifferent initial spees, between the brake weight percentage an the average stopping istance obtaine in emergency braking on level track, with a train having a nominal length an a braking equipment in normal operating conitions. Incientally it is also important to consier the tolerance of the friction coefficient of the braking components. Therefore, the brake weight percentage only gives information about the average performance of the train in emergency braking an it oes not inclue any safety margin. This funamental consieration has to be taken into account for the efinition of a braking moel applicable to spee control systems. In aition, a braking moel that is evelope for this purpose has to be applicable both to stop braking an to spee reuction braking.
3 . Definition of the braking moel for SCMT In general a braking moel can be represente as follows (fig. ): an initial elay, a linear or step transient, a series of constant eceleration steps within establishe spee ranges. v v v a a 3 a s Figure General braking moel As a particular case of that general representation, a braking moel with a step transient an one level of eceleration was efine for SCMT (fig. ), for the spee range until km/h. a t e t Figure Basic braking moel for SCMT The basic parameters of this moel are the following: t e braking equivalent time, eceleration of fully evelope braking. The step transient variant, even if less accurate than the linear transient variant, is wiely an effectively use for braking moels because it allows simpler calculations. For reasons of simplicity this representation refers to the situation on level track. The complete moel must obviously take into account the effect of the graient on the eceleration. As a consequence of the ifferent brake systems use on the vehicles (isc brake; cast iron block brake with one pressure level; cast iron block brake with two pressure levels etc.), also consiering 3
4 the possibility of mixe trainsets, the typical curves of the instantaneous eceleration may be ifferent for the same brake weight percentage, as shown in figure 3. isc brake block brake pressure levels block brake pressure level a v nom v Figure 3 Instantaneous eceleration of various types of brakes Possible evolutions in braking systems must be consiere, such as the replacement of cast iron brake blocks with composite brake blocks on freight wagons for noise reuction. A braking moel that will be use for a spee control system must also be applicable in intermeiate ranges of spee. For that reason the constant eceleration profile chosen for SCMT has to prevent that real eceleration profiles are overestimate in significant spee areas. The equivalent time epens on the goos/passenger brake position an on the length of the train. The eceleration epens on the type of brake equipment (isc brake, block brake etc.) an on the brake force. For this parameter it is necessary to etermine a relation with the brake weight percentage..3 Braking equivalent time The braking equivalent time t e is obtaine by aing the absolute elay an a half of the eceleration buil up time. The absolute elay represents the elay with which the eceleration ue to pneumatic braking appears, from the moment in which the braking is activate. The time necessary in emergency braking to reach the maximum eceleration ue to the pneumatic brake has been obtaine from iagrams of the ERRI report B6 RP5, proviing the braking time at the en of the train epening on the train length. The following formulas aopte for the present braking moel give the braking equivalent time as function of the train length L (m), in emergency braking: Passenger position: t e = (L / ) (.) Goos position: t e = (L / ) (.) Over a certain train length the equivalent time calculate by means of the passenger position formula excees the value calculate by means of the goos position formula: in this case even for the goos position the first formula is to be use. The iagrams relating to these formulas are shown in fig. 4. 4
5 Braking equivalent time 6 Equivalent time (s) 8 4 P position G position Train length (m) Figure 4 Equivalent time use for the braking moel.4 Relation between the brake weight percentage an the eceleration uring fully evelope braking.4. General consierations The purpose of the proceure escribe in this section is to establish a relation between the brake weight percentage an the eceleration uring fully evelope braking, base on the UIC evaluation metho (UIC leaflet 544) an applicable to all kin of trains an braking systems. Therefore such relation oes not supply an average eceleration value linke to a certain brake weight percentage, but the minimum common value among those potentially linke to the same brake weight percentage, for ifferent braking systems an ifferent initial spees. The main criteria on which the evaluation curves of the UIC leaflet 544 are base were taken into account: the evaluation curve for km/h is especially applicable to freight trains with braking in passenger position; the evaluation curves for spee between an 6 km/h are base on the braking performance of trains with cast iron block brake, but they have general application for all passenger trains; the evaluation curves for 8 an km/h are specifically applicable to passenger trains with isc brake. The knowlege of such characteristics prevents from following the UIC increasing eceleration at spee lower than km/h for trains with isc brake or composite block brake, to avoi an overestimation of their braking performances..4. Deceleration relate to the UIC brake weight percentage The general formula of the UIC evaluation curves is the following: 5
6 S C = λ + D () where S is the stopping istance, λ the brake weight percentage, C an D the coefficients epening on the initial spee. For practical reasons in this proceure such formula was expresse in the following way: S C' V = λ + D (3) where V is the initial spee of braking an C' a new coefficient having a tren that is easier to interpolate for initial spees ifferent from the nominal ones. The corresponing eceleration values uring steay braking were calculate by the formula: = V 3.6 V S te 3.6 (4) where the total stopping istance S, given by the basic UIC formula, is reuce by the istance covere uring the braking equivalent time t e. An example of the results obtaine from the evaluation curves of the UIC leaflet 544 an of the ERRI report B 6 RP 7 is shown in fig. 5. UIC eceleration on steay braking with eq. time 4.35 s eceleration [m/s ] brake weight perc. 6% 4% % % 8%.6 6% initial spee [km/h] Figure 5 Deceleration uring fully evelope braking, calculate on the basis of UIC/ERRI iagrams Consiering the basic criteria establishe for this proceure, the minimum value of equivalent time relate to a train of nominal length is to be use for the calculation. In this way, a shorter istance being covere uring the equivalent time an a longer istance being covere uring constant braking, the calculation prouces a eceleration value that is minimum within the possible range. 6
7 .4.3 Basic relation between eceleration an brake weight percentage Since ifferent eceleration values epening on the initial spee of braking correspon to the same UIC brake weight percentage, it is important to choose the most suitable nominal conition for which the two parameters must be relate to each other. Among all the possible solutions, the following two were taken into particular consieration: the evaluation spee of km/h, accoring to the latest UIC irectives that are vali in particular for isc brakes; the stopping istance of m, accoring to the current UIC regulation. In case of high braking performances the first solution has the isavantage of proucing higher eceleration values on shorter braking istance; passenger trains equippe with cast iron block brake are not able to prouce such ecelerations in the spee range between an 6 km/h. The secon solution has the following avantages: it correspons to the UIC irective officially applicable until now, accoring to which most of the vehicles that are in service have been evaluate; it is generally applicable to ifferent braking systems such as isc brakes an block brakes; it allows to etermine a relationship between eceleration an brake weight percentage vali for initial spees increasing in a coherent way with the braking performances. Due to these significant avantages the secon solution has been aopte for the following proceure. This proceure consists of four steps: evaluation of the brake weight percentage values proucing a stopping istance of m for initial spees between an 6 km/h, through the UIC formula an iagrams; calculation of the constant values of maximum eceleration from the above mentione initial spees, using the minimum braking equivalent time, as explaine in the previous paragraph; these eceleration values are associate to the previously etermine brake weight percentages, etermination of the linear regression representing the eceleration as function of the brake weight percentage; evelopment of a further stuy to calculate the reuction of such basic eceleration above a spee threshol for the constant value. As concerns the braking equivalent time two ifferent hypotheses have been mae: t e = 3.5 s, as minimum absolute value; t e = 4.35 s, as minimum value for a train having a nominal length of 3 m. The linear relations which link the eceleration to the brake weight percentage in these two hypotheses are: I hyp. b =,647 λ +,3 (5.) II hyp. b =,685 λ +,94 (5.) that are characterize by a goo correlation coefficient. The couples of values "initial spee  brake weight percentage" for which these relations have been etermine are shown in table (approximate values). 7
8 λ % V in km/h Table Relation between the nominal spee an the brake weight percentage The iagrams of the linear regressions (5.) an (5.) are shown in fig. 6 an 7..3 Relation between brake weight percentage an steay eceleration for a stopping istance of m with an equivalent time of 3.5 s 6 km/h eceleration (m/s ) y =.647x +.35 R = km/h km/h km/h 4 km/h 5 km/h km/h brake weight percentage Figure 6 Linear regression for t e = 3.5 s.3 Relation between brake weight percentage an steay eceleration for a stopping istance of m with an equivalent time of 4.35 s eceleration (m/s ) y =.685x R = km/h km/h km/h 4 km/h 5 km/h 6 km/h km/h brake weight percentage Figure 7 Linear regression for t e = 4.35 s 8
9 .4.4 Deceleration as function of the initial spee of braking To take into account the behaviour of isc brake an composite block brake, in this braking moel the basic eceleration etermine by the proceure inicate above is applie as a constant value from initial spees lower than the spee threshol. For initial spees higher than the spee threshol it is necessary to apply a graual reuction of the eceleration uring fully evelope braking; this reuction can be assume as linear with a goo approximation, in coherence with the UIC evaluation curves. Even in case of a braking carrie out from a spee higher than the threshol, the new value of eceleration is consiere as constant over instantaneous spee. This characteristic behaviour is shown in fig. 8. b a V nom V in V max V Figure 8 Deceleration for initial spee higher than the spee threshol The relations between the eceleration an the brake weight percentage for initial braking spees exceeing the spee threshol, in the two previous hypotheses, become respectively: I hyp. = (,647 λ +,3) * (,(V V thr )) (6.) II hyp. = (,685 λ +,94) * (,(V V thr )) (6.) if V > V thr where such spee threshol is obtaine from the formula: V thr = 6,7 λ,443 (7) Compare to the ecelerations calculate from the UIC / ERRI evaluation curves, the formulas of this braking moel present the tenency shown in fig. 9. 9
10 eceleration [m/s ] Comparison between UIC eceleration for a t e = 4.35 s an braking moel eceleration initial spee [km/h] brake weight perc. 4% UIC 4% br.mo. % UIC % br.mo. 8% UIC 8% br.mo. 5% UIC 5% br.mo. Figure 9 Comparison between UIC/ERRI eceleration an braking moel eceleration As an alternative to the constant eceleration moel, the possibility has been consiere of replacing the constant eceleration uring the fully evelope braking with an instantaneous eceleration profile ecreasing in a linear way. The total ecrease shoul be equal to a fixe percentage (e.g. %) of the eceleration value calculate by the basic formula, being equivalent to this one from the point of view of the total braking istance (example in fig. ). This behaviour can be more appropriate in case of a spee reuction, for braking systems having a lower performance in a higher spee range. % a V o V in V inst Figure Alternative moel with a ecreasing eceleration profile.5 Remarks With the basic eceleration an the braking equivalent time etermine by the proceure escribe above the basic braking moel is achieve. Moreover, in orer to reuce the possibility of interference in the normal behaviour of the river, a ynamic reuction of the braking transient was inclue in the complete moel for SCMT, taking into account the eceleration alreay evelope by the safe brakes of the train.
11 However the basic eceleration values o not inclue any estimation about the necessary safety margins, so they cannot be irectly use to calculate the braking curves of a spee control system. The assessment of such safety margins, to be carrie out on the basis of an appropriate probabilistic analysis, is the subject of the next section.. PROBABILISTIC ANALYSIS APPLIED TO THE BRAKING MODEL. Braking moel The braking moel use in this analysis is the basic moel escribe in section. an represente in fig. : ' t t t Figure Braking moel use in the probabilistic analysis t represents the initial elay time; t is the time necessary to obtain the maximum eceleration; is the nominal eceleration value; is the eceleration value obtaine applying a safety coefficient. In this analysis, t an t variability is neglecte an the ispersion of the braking performance is exclusively relate to the variability of eceleration, that is consiere the funamental parameter an is consequently reuce using a proper safety coefficient. In this paper the analysis was carrie out using information relative to a passenger train equippe with isc brake. Similar consierations can be mae for ifferent types of trains (e.g. freight trains equippe with cast iron or composite block brakes).. Moel use to calculate the eceleration During a braking, the train eceleration can be approximately calculate applying the first ynamic equation to the train (projecte on the horizontal irection of motion) an the secon ynamic equation to each axle. From these equations the following expression can be obtaine:
12 where: n F i= = M + bi r R + R J R n i i= is the train eceleration; Fbi is the braking force applie on the ith axle; n is the number of axles; r is the istance between the rotation axes an the point where the braking force is applie; R is the wheel raius; Rm is the motion resistance (aeroynamics an internal resistance); M is the mass of the train; J is the moment of inertia of the ith axles. i The braking force applie to each axle is given by: where: pi S i Fmi is the pressure in the brake cyliner; is the cyliner surface; is the brake cyliner spring force; τ is the brake rig ratio; η is the brake efficiency; µ is the brake friction coefficient; ρ is wheelrail ahesion coefficient. Neglecting the following terms: R m ; F mi ; n J i = ; i R F bi m (8) = ( p S F )τηµρ (9) the eceleration can be expresse using the following simplifie expression: i i r psτηµρ = R. M () r In this stuy, besies the surface S an the ratio τ, also the ratio R was consiere constant, while the variability of pressure p, efficiency η, friction coefficient µ, mass M an ahesion coefficient ρ were taken into account. When each parameter assumes its nominal value p, η, µ, ρ, M, the eceleration is given by: mi
13 r psτηµ ρ = R. () M To guarantee the require safety level of the system, the eceleration value to be use in the braking moel for the calculation of the monitoring curves is given by: ' = k () where k is a safety coefficient ( k < ). The objective of this analysis is the evaluation of the probability that the real eceleration is smaller than ', i.e. the ratio is smaller than a given k. The ratio between real an nominal eceleration can be expresse as: p p η η = (3) M M µ µ ρ ρ The probability istribution of the ratio can be calculate as a combination of the probability of each parameter..3 Probability istribution of the parameters A probability ensity function was suppose for each of the ratios in the preceing formula. For the major parameters, the probability istribution was etermine on the basis of technical knowlege. p η The probability ensity of the ratios an is not known exactly, only the extremes of the p η ispersion fiels are known with goo approximation. In these cases, normal probability istributions were suppose, with mean m = an extremes given by the mean value ± 3σ, in particular ± 5% both for pressure an efficiency. µ The ratio can be expresse as: µ µ µ µ me = (4) µ µ me µ where µ, µ me an µ represent respectively the real, mean (for a given friction material) an nominal value of the friction coefficient. µ The probability istribution of the ratio was efine using a wie set of experimental ata: µ me from the experimental results, a probability istribution was evaluate, then the experimental istribution was approximate by a normal istribution. The mean an the variance of the normal istribution were tune to minimize the ifference between the experimental an the approximate cumulative curve (fig. ). 3
14 ashe curve = experimental istribution.5. continue curve = approximate istribution µ/µ µ/µ me me a) b) Fig. Distribution of the ratio µ me.4 µ, a) istribution obtaine from experimental ata, b) comparison between the experimental an the approximate normal istribution. For the ratios µ me M an µ M only the extremes of the variability fiels were known with goo approximation. Also in this case normal probability istributions were suppose, with mean m = µ an extremes given by the mean value ± 3σ, in particular ± 5% for me M an ± 8% for. µ M ρ To take into account the ratio, two cases were separately analyze. The first one is relative to ρ normal ahesion conition; in this case the ratio ρ ρ is equal to an its variability was not taken ρ ρ. The two cases into account. The secon case is relative to egrae ahesion conition; in this case the ratio was suppose to belong to a normal istribution with mean m =. 95 an 3 σ =. 5 were combine together, supposing that in the 95% of the cases the ahesion conitions are normal, while in the remaining 5% are egrae..4 Distribution of the ratio between the real an the nominal eceleration The istributions of the parameters were then combine to fin the istribution of the ratio. Since the terms σ of each istribution were small, the calculation was quite simplifie. The fraction between the real an the nominal eceleration can be expresse as the ratio between two terms: the numerator of the fraction represents the ratio between the real an the nominal braking force: F F p η µ ρ = (5) b b p η µ ρ while the enominator is the ratio between the real an the nominal mass: M. M 4
15 The numerator is the prouct of four ranom variables; each of them is characterize by a normal probability istribution, with given mean an variance values. Given a series of ranom variables ( x, x,..., x n ), belonging to normal istributions with mean m, m,..., m n an variance σ, σ,..., σ n, the istribution of their prouct can be approximate with a normal istribution whose mean is the prouct of the means of each parameter an whose variance is given by: = = σ = σ i i. (6) j, j i The istribution of the ratio between the braking force an the mass can be evaluate using the Taylor series expansion arreste at the first orer term. In particular, given two ranom variables x num an x en, which belong to two normal istributions N( m num, σ num ) an N( m en, σ en ), the xnum istribution of the ratio can be approximate with a normal istribution characterize by mean xen mnum m = (7) men an variance: 4 4 ( m j ) ( ) m num σ = σ + num σ en. (8) men men The istribution of the numerator epens also on the number of vehicles in the train: in particular, if n is the number of vehicles an m an σ are the parameters relative to the istribution of the F b ratio, evaluate consiering only one vehicle, the resulting istribution has mean m n = m Fb an variance σ σ n =. (9) n M It was suppose moreover that the istribution of the enominator is not influence by the M number of vehicles..5 Results In table cumulative probability istribution values corresponing to various safety coefficients an various numbers of vehicles are shown. The results are a combination between the istribution obtaine neglecting the variability of the wheelrail ahesion coefficient an the one obtaine taking into account the variability of ahesion. In fig. 3 the istribution of the ratio obtaine neglecting the variance of the ahesion coefficient is shown, while in fig. 4 the results are obtaine consiering the istribution of the ρ parameter. ρ 5
16 In fig. 5 the effect of the number of vehicles on the istribution of the parameter is shown. k vehicle vehicles 4 vehicles 8 vehicles 6 vehicles.6.5e9 6.6E5.83E 3.3E E E7 7.4E.53E7.8E3 6.3E E6 8.8E9 8.7E E7 9.4E E4.36E6.8E9.38E 6.5E5.8.8E3 9.3E5.5E6.8E8 4.5E E.69E3.5E4.5E5.4E6.9.E 4.73E.84E 6.89E3 3.E E.36E.85E.45E.9E Table  Cumulative istribution values for various safety coefficients an various numbers of vehicles vehicles 8 vehicles 4 vehicles vehicles vehicle vehicle vehicles 4 vehicles 8 vehicles 6 vehicles k k a) b) obtaine without the parameter ρ, Figure 3 Distribution of the ratio a) probability ensity function, b) cumulative istribution function. 6 vehicles 8 vehicles vehicles vehicles vehicle vehicle vehicles 4 vehicles 8 vehicles.. 6 vehicles k Figure 4 Distribution of the ratio k a) b) obtaine consiering the istribution of the parameter ρ, a) probability ensity function, b) cumulative istribution function. 6
17 4 6 vehicles 8 vehicles vehicles vehicles vehicle vehicle vehicles vehicles 8 vehicles 6 vehicles k k obtaine after combination of the two istributions with an without parameter ρ ; effect of the number of vehicles on the istribution, a) probability ensity function, b) cumulative istribution function. Figure 5 Distribution of the ratio Besies the probability analysis relative to the passenger train equippe with isc brakes, other probability analysis were unertaken for freight trains equippe with cast iron block brakes or with composite block brakes, in empty an in loae conitions. 3 CONCLUSIONS The evelopment of the new spee control system SCMT for the Italian railways require the implementation of a braking moel that was able to escribe with sufficient precision an proper safety margins the braking performances of passenger an freight trains. With this purpose a braking moel having a general valiity has been erive from the UIC evaluation metho; such moel allows to convert the general input parameters (brake weight percentage, goos/passenger brake position, train length etc.) into a basic eceleration profile as function of time an spee. In aition a complete analysis of parameters affecting the braking performance was carrie out. For the major parameters, the probability istribution was etermine on the basis of technical knowlege an experimental results, in orer to establish the combine probability istribution, thus enabling the assessment of the safety egree as function of a reuction factor applie to the basic eceleration. Following the same proceure, further analysis coul be unertaken in orer to choose the proper safety factors on the basis of a complete evaluation of the risks, also taking into account external factors. BIBLIOGRAPHY [ ] A. Singer: "Neue LZB/FZBBremskurven", (New LZB/FZB braking curves), Universität Hannover, Institut für Schienenfahrzeuge un maschinelle Bahnanlagen, Report No. 4/997. [ ] D. Jaenichen, R. Jaensch: "Neue LZB/FZB für Neubaustrecke KölnFrankfurt/M.", (New LZB/FZB for the new line Cologne  Frankfurt am Main), Technische Universität Dresen, Institut für Schienenfahrzeugtechnik, Final report,
18 [ 3 ] P. Presciani: "Stuio i un moello i frenatura per sistemi i controllo ella velocità" (Research on a braking moel for spee control systems), Ingegneria Ferroviaria No. 8/. [ 4 ] ERRI B 6 RP 3 "Existing an future train control an comman systems on the European railways Moels for eceleration curves", April. [ 5 ] "Description of the brake curve calculation", Ref. EEIGERTMS: 97E88 version 5D, /7/. [ 6 ] P. J. Bickel, K. A. Doksum: "Mathematical statistics. Basic ieas an selecte topics", vol. I, Prentice Hall,. [ 7 ] M. Biloeau, D. Brenner: "Theory of multivariate statistics", Springer,
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