n?u=RePEc:hhs:ctswps:2016_026&r=tre

**on** **rail**

**infrastructure**

Andrew S.J. Smith a , Krist**on** c ,

Sebastian Stichel c , Sim**on** Iwnicki d and Phillip Wheat a

a Institute for Transport Studies (ITS) and Leeds University Business School

b The Swedish Nati**on**al Road and Transport Research Institute (VTI)

c

Royal Institute

d

Institute

CTS Working Paper 2016:26

Abstract

In this paper we combine engineering and ec**on**omic methods to estimate

damage mechanisms **on** **rail** **infrastructure**. The former method is good at

predicting damage from traffic, while **on**ship

between damage and

approach and dem**on**strate its applicability for **rail** **infrastructure** charging. Our estimati**on**s

are based **on** 143 track secti**on**s comprising about 11 000 km **on**strate how

**infrastructure** managers in Europe who desire to

that each

efficient use **rail** **infrastructure**.

Keywords: **rail** **infrastructure**, maintenance, access charging, track damage,

ec**on**ometric methods, and engineering simulati**on**

JEL Codes: R48, L92

Centre for Transport Studies

SE-100 44 Stockholm

Sweden

www.cts.kth.se

**on** **rail**

**infrastructure**

Andrew S.J. Smith a , Krist**on** c ,

Sebastian Stichel c , Sim**on** Iwnicki d and Phillip Wheat a

a Institute for Transport Studies (ITS) and Leeds University Business School, University

Leeds, Leeds, LS2 9JT, United Kingdom.

b The Swedish Nati**on**al Road and Transport Research Institute, Department

Ec**on**omics, Box 55685, 102 15 Stockholm, Sweden (krist

c Royal Institute **on**autical and Vehicle Engineering,

100 44 Stockholm, Sweden.

d

Institute

Yorkshire, HD1 3DH, United Kingdom.

Abstract

In this paper we combine engineering and ec**on**omic methods to estimate

damage mechanisms **on** **rail** **infrastructure**. The former method is good at

predicting damage from traffic, while **on**ship

between damage and

approach and dem**on**strate its applicability for **rail** **infrastructure** charging. Our estimati**on**s are

based **on** 143 track secti**on**s comprising about 11 000 km **on**strate how

estimated relative

wear and tear **infrastructure**

managers in Europe who desire to

**rail** **infrastructure**.

1

1.0 Introducti**on**

Operating a train service generates **rail** **infrastructure**.

Research **on** **on**

between **infrastructure** management and train operati**on**s in

charges to be set. To create an efficient use **infrastructure**, each

pay its short-run **on** (see

European Commissi**on** Directive 2012/34/EC).

One comp**on**ent **rail**

**infrastructure**. The vertical force **on**

important factor for this damage, and t**on**-km has been

in Europe. However, **on**-km can vary depending **on**

where

Differentiating **on**s in damage and

incurred by **on**ger incentives for running more “track

friendly” **infrastructure** compared

to a t**on**-km charge. Britain and Switzerland are examples

chosen to **on**-km. This type

charge requires an estimati**on** **on**

**rail** **infrastructure**, which is

Different approaches have been used in

and tear **on**ship between

traffic using ec**on**ometric techniques (see for example Link et al. 2008 and Wheat et al. 2009),

while

traffic, damages that are

**on** 2005 and Öberg et al. 2007). A combinati**on**

2

simulati**on** methods (engineering models) are used in

damage caused by **rail** **on** **on**ship between damage

and **on**ometric methods in **on**d stage.

The reas**on** for combining **on**ometric and engineering approaches in this type

exercise is that

uses actual **on**s **on** **on**. However, it has

difficulties in picking up **on**ship between

and **on**

damage caused by

traffic) to actual

Smith et al. (2014) applied **on** Swedish data comprising 45 track

secti**on**s, in order to dem**on**strate

same two-stage approach with an aim to increase **on**

The c**on**tributi**on**

charging purposes; can it be a viable approach for **infrastructure** managers that wish to

dataset comprising 143 track secti**on**s in Sweden. Moreover, **on** stage

approach is refined, with significantly more detailed **on** stage

compared to **on**s **on**

certain **on**

The outline **on** 2.

Secti**on**s 3 and 4 present **on** stages in our approach in more detail. A descripti**on**

**on** 5. The estimati**on** results are presented in secti**on** 6 toge

dem**on**strati**on** **on**s. Secti**on** 7 c**on**cludes.

3

2.0 Methodology

The ec**on**ometric (top-down) approach and

main methods to determine how wear and tear **rail** **infrastructure** vary with traffic.

The former has become

European countries. Munduch et al. (2002) and Johanss**on** and Nilss**on** (2004) are early

examples **on** in order to derive a

respect to traffic using ec**on**ometric techniques; an elasticity showing how a proporti**on**ate

increase in traffic affects **on**ately (this

average **on**trol variables are included

to account for heterogeneity in **on** envir**on**ment - that is, to isolate

has **on**

**on** wear and tear **on**ometric studies

is generally gross t**on**s.

Instead **on**ship between traffic and

bottom-up approach uses engineering models to determine

damage is **on**tact fatigue, track settlement, or wear **rail**.

The method is able to provide estimates

**infrastructure** characteristics

such as track geometry and **rail** pr

produce a **on**e with informati**on** **on**

to rectify

allocated to

example Booz Allen Hamilt**on** 2005 and ORR 2013). However, a critical point in

approach c**on**cerns

al. (2007) use an engineering approach to produce estimates

4

damage caused by **on** **rail**way network. The damage

**on** advice from experts within

**on** 1 . The experts’ advice may, or may not be, close to

such as a heterogeneous producti**on** envir**on**ment, which can vary in aspects that are difficult

to capture without a statistical (ec**on**ometric) approach. Examples are **rail** age (proxy for

accumulated use) and track quality.

The approached proposed by Smith et al. (2014) is to use an ec**on**ometric model to

estimate

More specifically, **on** approach (depicted in Figure 1 below) c**on**sists

two stages. Similar to **on**s based **on** engineering

models in **infrastructure** characteristics in

order to simulate four **rail**s, rolling

c**on**tact fatigue (RCF), and track comp**on**ent fatigue. Hence, we include an extra damage

mechanism (track comp**on**ent fatigue) compared to

damage mechanism may eventually require replacements **on**ents and can be important

to c**on**sider given that minor replacements are defined as maintenance.

The output from **on**-km

for each **on**s **on** **rail**

network, **on** due to

characteristics **on**s such as track geometry and curvature. The measures are

scaled up based **on** **on** **on**s. In

that way, we produce measures **on** **on**ent fatigue, track settlement, RCF

1 This organizati**on** merged with **on** in 2010, forming

Administrati**on**.

5

and wear **rail**s, that traffic has caused **on** a secti**on**. We use

sec**on**d stage, in which a statistical model is formulated where maintenance **on**

statistical model, giving us **on** **on**

from **on**, we can estimate

Stage 1: Simulati**on**

(track secti**on** level)

Vehicles

Track

Maintenance

activities

Damage

mechanisms

D1: Settlement

D2: Wear

D3: RCF

D4: Track

comp**on**ent

fatigue

Stage 2: Statistical model

(track secti**on** level)

log Actual maintenance

= α + β 1 log D1 + β 2 log D2

+ β 3 log D3 + β 4 log D4

β 1 , β 2 , β 3 , β 4

We know something about

**on**e by

Figure 1: Overview

A detailed descripti**on** **on**s and **on**ometric model we estimate is provided

in secti**on**s 3 and 4 respectively.

3.0 First stage: simulati**on**s

Calculating **on**

quality itself - that is, whe

**on**crete, or if it is a slab track,

6

**on**ment is humid or dry,

wheel and **rail** pr

We account for **on**s by using measurements **on**

geometry, which will differ depending **on**

repaired. However, we **on**ly c**on**sider c**on**crete sleeper tracks and a c**on**stant envir**on**ment

(which creates a c**on**stant fricti**on** level) in **on**s. Moreover, we do not account for

**rail** pr**on**d

means, **on**ly new unworn wheel and **rail** pr

The **on**

Some **on**s - are

bogie design,

wag**on** structure,

axle load and

The track damages investigated in this study are categorised into four

are:

track comp**on**ent fatigue, 2

track settlement,

rolling c**on**tact fatigue (RCF) and

wear **rail**s.

2 Additi**on**al measure not included in original work by Smith et al. (2016).

7

We quantify and calculate

above, using dynamic simulati**on** and **on** models available in

The simulati**on**s are performed **on** 143 track secti**on**s in Sweden, which in total comprise

about 11 000 km

**on** each track secti**on**. This includes informati**on** **on**

operating **on** each track secti**on** as well as **on**-km values. The

dynamic simulati**on**s and

secti**on**.

3.1 Dynamic simulati**on**s

Computer based **on**s using multibody simulati**on** s

been widely used by companies and researchers for many years. These simulati**on**s are mainly

used to predict **on**diti**on**s and

operating c**on**diti**on**s, thus making sure that **on** de**rail**ment, ride quality, track

forces, RCF and wear will be met. It is also a very powerful tool to reduce

expensive field measurements. The modelling issues in **on**s are divided

into three parts: track models, **rail** c**on**tact.

We use a track model representing c**on**crete sleeper tracks in **on**s, which is

**on** most

informati**on** **on** track flexibility characteristics and its validati**on**). The model

comprises **rail**s and stiffness between

8

Figure 2: Example

In principle, **on**nected by

springs, dampers and links. The bodies include car body (passenger wag**on**s or freight

baskets), bogie parts (frames, bolster beams, and possible steering links) and axles (axle box

and wheels). A four axle bogie

shown in Figure 3. These rigid bodies may have all six degrees

c**on**strained. These six moti**on**s are vertical, lateral, l**on**gitudinal, pitch, yaw and roll.

Figure 3: schematic model

For each rigid body mass we have to know **on**s

centre **on**s

modelling **on** elements. These elements are mainly springs, dampers and fricti**on**al

c**on**tacts. Some **on**-linear and **on**

applied loads, forces and displacements. A suspensi**on** element could be a coil or rubber

spring, air spring, leaf spring, hydraulic damper, and metal wedge. For each type

menti**on**ed,

9

The **on** **on** **on**s in this

study. According to **rail** **on**

secti**on**s in 2014. It is not possible to model each

**on**

(freight/passenger), axle load and maximum speed. The chosen categories are presented in

Table 1. Moreover, due to time restricti**on**s, we **on**ly run simulati**on**s for

comprise more than 9 per cent **on**’s total t**on**-km. The

are assigned

with respect to damage.

Table 1: Vehicle model categories with

Categories

Max. speed km/h

Motor coach 4x16 t* 200

Passenger car 4x14 t 160

Motor coach 4x16 t** 200

Motor coach 4x12 t* 140

Motor coach 4x21 t, high centre

Motor coach with Jacob bogie 3x16.5 t** 160

Motor coach with Jacob bogie 3x12.5 t* 200

Freight loco 6x20 t 120

Freight loco 4x20 t 120

Freight loco 6x30 t 70

Passenger loco 4x19 t 140

Passenger loco 4x19 t 175

Freight wag**on** (2x22 t or 2x6.5 t) 100

Three-piece bogie 4x30 t

Three-piece bogie 4x6.5 t

60 (laden)

60 (tare)

Y25 bogie 4x22 t 100

* Flexible wheelset guidance, ** Stiff wheelset guidance

10

All **on**ed models are carefully designed and **on**s are

validated against

**on** models **on** s

Lastly, we need to model **rail** c**on**tact, which is **on**tact area

between **rail**s that is subjected to very high stresses. The way to calculate

**on** **on** between

track. A wheel-**rail** c**on**tact model c**on**sists principally **rail** geometry module, a

creep/spin calculati**on** procedure and a creep force generator. The

example in Anderss**on** et al. (2015). In this study, **on** and Kalkers

FASTSIM method is used for **on**tact problem respectively.

3.2 Simulati**on** inputs

Inputs needed for **on** are ideal track geometry and track irregularities,

speeds, wheel and **rail** pr**on** track geometry has been provided by

**on** (Trafikverket), and originates from track measurements

in 2014. The geometry includes **on**gitudinal positi**on** **on** **on**,

track lift and track curvature. The irregularities include lateral, vertical, cant and gauge

irregularities.

We set **on**

cant deficiency can be reached, where **on** will be limited

according to Banverket (1996). There are three categories defined based **on**

running gears,

category A; c**on**venti**on**al

category B;

11

category C; X2000 and o

The maximum **on** each line.

As wheel and **rail** material will be worn gradually,

Therefore it is almost impossible to use all **rail** pr**on**.

However, as **rail**s in Sweden have UIC 60 and

we use **on**s. The **rail** inclinati**on** in Sweden is 1:30.

Apart from **on**-Ore locomotives and wag**on**s, which run at 30t axle load,

**on**s are simulated with 22.5t axle load when

weight **on**s is calculated based **on**

design. For passenger trains **on**

passenger loads. However,

to 80 kg. According to European standard EN 12663 (CEN 2010),

in a coach is equal to

3.3 Track damage

We calculate four **on** each track secti**on**: track

settlement, track comp**on**ent fatigue, wear **rail**s and rolling c**on**tact fatigue (RCF).

Track settlement has a major influence **on** maintenance

high wheel-**rail** forces from passing **on**gly **on**

amount **on**structi**on**,

track quality and track c**on**diti**on** are

However, in most **rail** force raised to a power is used as a damage

12

indicator. In

calculati**on** model is used:

Settlement = A ∙ Q 1.21 log N (1)

where,

N = number

Q = Vertical force at

A = c**on**stant; (A=1 in

Internal fatigue damage due to repeated loading is a functi**on**

forces. The comp**on**ents affected by **rail**s, **rail** pads, **rail** fasteners, and

sleepers. The calculati**on** method is developed by UIC/ORE (1987) based **on** extensive tests

and it is complemented by Öberg et al. (2007) with a lateral force comp**on**ent – that is,

resulting force **on** ei**rail**.

3

n

Track comp**on**ent fatigue = ∑ v

√Q 2 2

i=1 tot_i + Y qst_i (2)

where,

n v = number

Q tot_i = total vertical force including quasistatic and dynamic forces

Y qst_i = quasistatic lateral force

Wear **rail** and wheel is a functi**on** **on**tact pressure (axle

load, wheel-**rail** pr**on**diti**on** (sun and rain)

13

and lubricati**on** (track side or **on** level is assumed to be

0.45 for all **on**s unless **on**

systems - that is, **on**-Ore loco. To predict **on** **rail**s, several predicti**on** models are

proposed in literature (see Enblom 2004). One

predict **rail** c**on**tact

patch. This is based **on** an assumpti**on** that **on**ship between wear and

energy dissipati**on**. Energy dissipati**on** per meter running distance can be calculated as:

E̅ = F x ν x + F y ν y + Mφ (3)

where,

F x and F y are l**on**gitudinal and lateral creep forces,

ν x and ν y are l**on**gitudinal and lateral creepages,

M is **on**tact patch.

In

regime is c**on**sidered to be mild wear and

2014).

To calculate surface initiated cracks due to RCF, again **on** based

**on** is calculated and

index is picked accordingly.

14

RCF Damage Index (*1e-6)

15

10

5

0

0 100 200 300

Energy Dissipati**on** [J/m]

Figure 4: Rail RCF damage functi**on** (Burstow 2004)

3.4 Time domain analysis

All equati**on**s **on**s have to be integrated numerically in each time step. The results

each time step are **on**e. This is called initial value numerical

calculati**on**s. In this study 1ms is used for **on** **on**

length, **on** corresp**on**ding

to 1 km

**on**s for all

**on** **on**s. Instead, we use

which is also used in publicati**on**s such as Enblom (2004) and Dirk and Enblom (2011).

More specifically, this method implies that we create 10 **on**

categories as a functi**on** **on** categories are track pieces

with radii 0-400m, 400-600m, 600-800m, 800-1000m, 1000-1500m, 1500-2000m, 2000-

3000m, 3000-5000m, 5000-10000m and above 10000m. Hence, a track secti**on** has many

track pieces in each subsecti**on** category. C**on**sidering that we cannot run simulati**on**s **on**

entire track, we choose **on**e track piece in each subsecti**on** category (measured by

geometry car). Specifically, we choose

15

length **on** category. The simulated damage **on** each piece

is **on** category

bel**on**gs to.

3.5 Simulati**on** results

All four track damage values are calculated for all **on** categories **on** each track

secti**on** and for every **on** that specific secti**on**. Maximum values are

c**on**sidered for all

based **on** **on**tributi**on** **on** to **on** and normalised by

t**on**-km values obtained from

To show **on** process, we present **on** for a part **on**

217 (see Figure 5). As menti**on**ed earlier, **on**s,

depending **on** **on** is presented in Table 2.

Table 2: Subsecti**on** lengths **on** 217 (based **on** route length)

Subsecti**on** 0- 400- 600- 800- 1000- 1500- 2000- 3000- 5000- Straight

400m 600m 800m 1000m 1500m 2000m 3000m 5000m 10000m

Total

length (m)

0 3543 1584 4368 4139 845 639 565 339 46 627

The traffic data shows that **on** this line. The

corresp**on**ding t**on**-km

16

Table 3: Vehicle **on**ding

t**on**-km values for secti**on** 217

Vehicle

Milli**on** t**on**-km

Motor coach 4x12t, V max 140 km/h* 0.06

Passenger car 4x14 t, V max 160 km/h 17.89

Motor coach 4x16 t, V max 200 km/h** 56.97

Freight loco 6x20 t, V max 120 km/h 1.63

Freight loco 4x20 t, V max 120 km/h 0.54

Passenger loco 4x19 t, V max 140 km/h 61.45

Passenger loco 4x19 t, V max 175 km/h 19.88

Freight wag**on** 2x22 t, V max 100 km/h 38.72

Freight wag**on** 2x6.5 t, V max 100 km/h 12.51

Y25 bogie 4x22 t, V max 100 km/h 659.03

“Unkown” 1.39

*Flexible wheelset guidance, ** Stiff wheelset guidance

Figure 5: locati**on** **on** 217

The calculated damages for a “freight loco 4x20 t, V max 120 km/h” running **on** six segments

that c**on**stitutes **on**e track piece is presented in Table 4. The sum

number for all **on**d bogie

to 84510m **on** secti**on** 217 is 1423 J/m. This particular track piece bel**on**gs to

interval 600-800m, and is scaled accordingly. The total wear **on**

(1423/600)*1584 = 3575 J/m.

The same type **on**s are performed for

including straight lines, in order to produce values

track comp**on**ent fatigue incurred by a “freight loco 4x20 t, Vmax 120 km/h”-

**on**, we calculate its damage

values per t**on**-km. With informati**on** **on** this **on**-km **on** track secti**on** 217,

we can scale up **on** this secti**on**. The same type

simulati**on**s and calculati**on**s are made for rest **on** this secti**on** in order

to produce measures **on**ent.

17

Table 4: Results for curve interval 600-800m **on** secti**on** 217 for a freight loco 4x20 t,

V max 120 km/h

L**on**gitudinal positi**on**

Wear RCF Settlement Comp**on**ent

Start_Stop (m)

83910_84010 134 1.30 5 799 725 3.60E+16

84010_84110 197 2.24 5 740 527 3.52E+16

84110_84210 326 2.96 5 774 048 3.58E+16

84210_84310 322 2.83 5 678 586 3.44E+16

84310_84410 276 2.77 5 679 442 3.43E+16

84410_84510 168 1.84 5 791 845 3.59E+16

Sum 1423 13.94 34 464 175 2.12E+17

4.0 Sec**on**d stage: Ec**on**ometric model

With estimates **on**

**on**ometric methods. To do so, we need to c**on**trol for o

influence maintenance **rail** age **on** a secti**on**. More specifically, we

formulate **on**

main interest

C i = f D 1i , D 2i , D 3i , D 4i , X i , (4)

where C i is maintenance **on** i = 1, 2, … , N track secti**on**s. D 1i , D 2i , D 3i , and D 4i are

damage **rail**s, RCF and track comp**on**ent fatigue. X i is a vector

**infrastructure** characteristics such as track length and **rail**s.

As described previously, **on** **on**-km **on** each

secti**on**, which in turn depend **on** **on**. Therefore, to separate track length

effects from damage effects, we use damage density variables (

al**on**g with **on**s.

D 1i

,

Track−km i

D 2i

Track−km i

etc.)

18

In our estimati**on** approach, we start with

et al. (1971), which is a sec**on**d order approximati**on** **on**) functi**on** (see for

example Christensen and Greene 1976 for an applicati**on** to **on**s). Both

dependent variable (**infrastructure**

characteristics) are subject to a logarithmic transformati**on** in this model, which can reduce

skewness and heteroscedasticity (problems that may invalidate

treated correctly). Specifically, we c**on**sider A damage

dummy variables, and express

A

1

lnC i = α + ∑ β a lnD ai +

2 ∑ ∑ β 1

ablnD ai lnD bi + ∑ β k lnX ki +

2 ∑ ∑ β kllnX ki lnX li

a=1

A

A

a=1 b=a

K

k=1

K

K

k=1 l=a

A K

+ ∑ ∑ β ak lnD ai lnX ki +

a=1 k=1

M

∑ β m Z mi + v i

m=1

(5)

where α is a scalar, v i is white noise and β is a vector

simpler (and more restrictive) Cobb-Douglas model is nested in

**on**straint β ab = β kl = β ak = 0 using an F-test.

5.0 Data

In total, **on**s in 2014 administered by

Administrati**on** and **on**al units: Regi**on** North, West, East, South and Central.

However, limited access to up-to-date track geometry data c**on**strains us to analyze a

somewhat smaller part **rail**way network. One may

presence **on** bias in our data. However, **on**s in our data set cover

19

11 000 track-km out

Administrati**on**. Hence, **on** **rail**

network with secti**on**s from north to south and with large variati**on**s in traffic and

Table 5). Still, we are able to compare **on**s with 169 track secti**on**s for

which we have informati**on** **on**

statistics **on**s) and in Table 14 in appendix (169

secti**on**s). **on**

density at 0.2024 (robust std. error is 0.0479) and 0.2258 (robust std. error is 0.0498), using

143 and 169 track secti**on**s, respectively. 3 We **on**sider a (possible) selecti**on** bias to

be a minor issue in our sample.

The

**on**ducted in order to preserve **rail**way’s assets,

while

maintenance). Given **on**ly have access to data for

**on**e year (2014), we limit our analysis to maintenance **on**ly.

Informati**on** **on** **infrastructure** characteristics has mainly been collected from

Transport Administrati**on**’s track informati**on** system (BIS), and comprises data **on** track

length, **rail** age and quality classificati**on** (track geometry requirements linked to maximum

line speed allowed). As noted in secti**on** 3.0, **on**tains informati**on** **on**

**on**s, locomotives, multiple unit trains) and **on**-km. The

have been categorized as previously shown in Table 1.

3 The difference in

p.46-47).

20

Table 5: Descriptive statistics, obs. from 143 track secti**on**s

Median Mean St. dev. Min Max

Maintenance **on** SEK 14.25 19.66 17.86 0.87 108.67

Wear 2.21E+12 8.22E+14 5.32E+15 8.26E+06 5.52E+16

RCF 5.58E+08 6.55E+11 3.93E+12 4.46E+05 3.37E+13

Settlement 7.46E+14 5.87E+15 2.75E+16 4.61E+11 2.54E+17

Track comp**on**ent fatigue 3.91E+24 1.85E+28 1.44E+29 4.11E+21 1.38E+30

Wear density 1.09E+08 2.43E+08 4.38E+08 2.10E+06 2.75E+09

RCF density 3.15E+05 5.12E+05 7.46E+05 1.02E+04 7.84E+06

Settlement density 2.96E+12 3.66E+12 3.30E+12 4.89E+10 2.45E+13

Track comp**on**ent fatigue density 2.18E+22 3.37E+22 4.87E+22 3.83E+20 4.80E+23

Route length, km 50.17 60.86 40.59 5.97 215.95

Track length, km 63.95 78.79 52.41 7.84 251.39

Average quality class* 2.77 2.74 1.08 1.00 5.02

Average age **rail**s 21.2 22.4 9.4 4.1 51.3

Milli**on** t**on** density 4.23 7.68 8.24 0.11 45.72

Regi**on** West 0 0.20 0.40 0 1

Regi**on** North 0 0.13 0.33 0 1

Regi**on** Central 0 0.17 0.38 0 1

Regi**on** South 0 0.29 0.45 0 1

Regi**on** East 0 0.22 0.42 0 1

* Track quality class ranges from 0-5 (from low to high line speed), but 1 has been added to avoid observati**on**s

with value 0.

6.0 Results

Two models are estimated using ordinary least squares (OLS) and

Table 6. Model 1 **on**ly includes

**infrastructure** characteristics and dummy variables for **on**al units, showing

importance **on**trolling for **on** envir**on**ment in **on**. All estimati**on**s

are carried out with Stata 12 (StataCorp.2011).

However, as a starting point, we examine **on** coefficients between

settlement covaries str**on**gly with track comp**on**ent fatigue (**on** coefficient is 0.95)

and with RCF (0.82). The correlati**on** coefficient for wear and track settlement is

21

(0.72). We **on**ly

(Model 1c), as we expect **on**ent fatigue to a

large extent.

Table 6: Correlati**on** coefficients

Wear_den. RCF_den. Settl. _den. Comp._den.

Wear_den. 1.0000

RCF_den. 0.7228 1.0000

Settl._den. 0.7155 0.8157 1.0000

Comp._den. 0.8123 0.7752 0.9471 1.0000

As noted in secti**on** 4, we start with a full translog model and test linear restricti**on**s

parameter estimates using F-tests, which results in

Table 7 and 8.

In Model 1, we note that **on**ent

fatigue is negative (and statistically significant). This result is counterintuitive; indicating that

10 per cent more track comp**on**ent fatigue will lower maintenance

However, **on** factors (VIFs) for

and track comp**on**ent fatigue are 0.17 and 0.20, respectively. Also, c**on**sidering

correlati**on** coefficients, we drop track comp**on**ent fatigue, which results in Model 1b. The first

order coefficient for settlement

zero). Dropping RCF due to its high correlati**on** with settlement (0.82), results in Model 1c.

The sum

which indicates that **on**g correlati**on** between

individual parameter estimates significantly.

22

Table 7: Estimati**on** results, Model 1

Model 1a Model 1b Model 1c

Coef. Rob. Std. Err. Coef. Rob. Std. Err. Coef. Rob. Std. Err.

C**on**s. 16.5063*** 0.1124 16.3996*** 0.0763 16.5134*** 0.0826

Wear_den. 0.3616** 0.1495 0.0137 0.0974 0.2845** 0.1151

Wear_den.^2 -0.4863** 0.2443 - - -0.3613*** 0.1183

RCF_den. 0.0885 0.1520 0.3544*** 0.1195 - -

Settl. _den. 0.6205** 0.3091 -0.0359 0.1230 0.1178 0.1080

Comp. _den. -0.6328** 0.2965 - - - -

Comp. _den.^2 -0.1029 0.2961 - - - -

Wear_den.Settl. _den. - - - - 0.2189** 0.0929

Wear_den.Comp. _den. 0.3396 0.2479 - - - -

Mean VIF 15.58 3.01 3.06

R^2 0.22 0.14 0.16

Adj. R^2 0.18 0.12 0.14

We transform all data by dividing by

coefficients can be interpreted as **on**s

Note: ***, **, *: Significance at 1 %, 5 %, and 10 % level, respectively

Leaving out

omitted variable bias. If that is

underestimated. Hence, in Model 2, we include a set **on**trol variables that we believe to be

important in this c**on**text. The average quality class **on** a track secti**on** (Qual_ave, where low

values indicate high speeds allowed) can be important to include as higher speeds imply

stricter requirements **on** track quality (track geometry). This may increase

rectify **on** term between

Settlement and Qual_ave is negative, which suggests that

lower for low linespeeds compared to high linespeeds. The first order coefficient for

Qual_ave is negative, yet not significant. Here it should be noted that differences in track

irregularities, curvature, linespeeds and traffic volume have been (at least substantially)

23

normalized, as **on**s and

damage measures.

Rail age is also included in **on**. Older **rail**s seem to be more

C**on**sidering that **rail** age is a proxy for track standard due to accumulated use (yet, not a

perfect proxy), a positive and significant coefficient is intuitive as high maintenance **on**

old and heavily used track is expected, which eventually makes a renewal ec**on**omically

justified.

Table 8: Estimati**on** results, Model 2

Model 2a

Model 2b

Coef. Rob. Std. Err. Coef. Rob. Std. Err.

C**on**s. 16.4675*** 0.1030 16.4779*** 0.1024

Wear_den. 0.1079 0.0718 0.1182* 0.0714

RCF_den. 0.0485 0.0805 - -

Settl._den. 0.0996 0.0983 0.1345* 0.0719

Track_length 0.9303*** 0.0582 0.9385*** 0.0588

Qual_ave -0.0428 0.2185 -0.0237 0.2113

Qual_ave^2 -0.9850* 0.5132 -1.0099* 0.5205

Rail_age 0.2575* 0.1337 0.2699** 0.1340

Settl._den.Qual_ave -0.5618*** 0.1189 -0.5685*** 0.1188

Regi**on**_West 0.3264** 0.1429 0.3254** 0.1431

Regi**on**_North 0.0442 0.1903 0.0395 0.1886

Regi**on** _Central -0.2981** 0.1500 -0.2933* 0.1491

Regi**on** _South -0.2179 0.1395 -0.2173 0.1395

Mean VIF 2.73 2.31

R^2 0.70 0.70

Adj. R^2 0.67 0.67

We transform all data by dividing by

coefficients can be interpreted as **on**s

Note: ***, **, *: Significance at 1 %, 5 %, and 10 % level, respectively

Turning to

**on**e

24

correlati**on** coefficient with settlement, which generates a slightly higher estimate for

settlement. The coefficients for wear and settlement are now statistically significant at

per cent level. The sum

Model 2b, respectively, indicating that

by

Settlement has a significant interacti**on** term with Qualave. To produce an estimate for

settlement with respect to **on**, we calculate

**on** equati**on** (6).

γ̂iSettl. = β 1 + 2 ∙ β 2 lnQualave i , (6)

This results in a

6.1 Marginal

To calculate

**on** **on**-km each **on**s in

**on** approach. In that way, we produce a **on**-km

which is **on** results in

previous secti**on**, **on**g correlati**on** between

difficult to isolate **on**gly correlated with

track comp**on**ent fatigue and RCF, we expect

**on** results in

Model 2b. This generated damage estimates that are statistically significant at

level.

25

In **on** presented below, we use

for wear and settlement (evaluated at **on**

n**on**-significant

The

MC j = ∂C

= D j ∂C C

= ∂lnC C

∂D j C ∂D j D j ∂lnD j

D j

, (7)

where D is damage. Hence, from equati**on** (7) we can be express

MC j = γ̂j ∙ AC ̂j, (8)

where γ̂j is

D j

where C is predicted

C i = exp [ln C i − v̂i + 0.5σ̂2], (9)

Equati**on** (9) derives from **on** and **on**

distributed residuals (see for example Munduch et al. 2002).

We use a weighted **on**s in this study, according to

equati**on** (10) below. This implies that we use **on**, which

produces a

**on**-km - as if each secti**on**’s

MC ij W = MC ij ∙

D ij

∑ D ij

i /N , (10)

26

The average

become quite low as

Table 9: Average and

Variable Mean Std. Err. [95% C**on**f. Interval]

Average

Settlement 2.56E-07 5.08E-08 1.56E-07 3.57E-07

Marginal

Settlement 3.45E-08 6.84E-09 2.10E-08 4.80E-08

Weighted

Settlement 9.37E-09 5.97E-10 8.19E-09 1.05E-08

The

**on** is that

average

estimates that are comparable with **on** **rail** **infrastructure**

caused by a t**on**-km **rail** **infrastructure**

**on**-km in

damage type presented in Table 10 do not differ at

Table 9. In o

normalized in Table 10.

Before we estimate a

**on** with previous estimates **on** Swedish data.

More specifically, we use **on**-km and mean settlement per t**on**-km for all

traffic **on** **on**s and multiply with

**on**-km for wear and settlement

4 Note that **on** (10) is

D ij

**on** (2008).

∑i

D ij

27

0

.05

.1

.15

.2

are illustrated in Figure 6, showing that **on** density. Similar shapes

were found for a number

The weighted **on**-km is 0.0148 SEK, which is

mechanisms’ weighted

t**on**-km, we get a

estimates **on** Swedish data in Anderss**on** (2008) and Odolinski and Nilss**on** (2015), which are

0.0081 SEK and 0.0083 SEK respectively (in 2013 prices).

0 10 20 30 40 50

Milli**on** t**on** density

MC settlement MC wear

Figure 6: Marginal

Similar to Smith et al. (2014), we use **on** results to calculate a

t**on**-km and **on**s for **on**

**on**s (as described in secti**on** 3), producing a damage value per t**on**-km for a

5 The share **on**-km **on** a secti**on** with respect to **on**-km is used as weights.

28

**on** a certain track secti**on**. We use a mean value **on**-km a

**on** **on**s it ran **on** during 2014. 6 The product

value (damage per t**on**-km) and **on**ding damage mechanism is

**on**-km for that

km/h, has a mean wear per t**on**-km at 32.4 and a mean settlement per t**on**-km at 743 564. Its

The damage per t**on**-km for each

values partly depend **on** which track secti**on**s **on** during 2014,

c**on**sidering that track quality differs between **on**s. More specifically, we used

measurements **on** track geometry and track irregularities as input in **on**, as this will

affect **on**sidering that each **on**

all **on**s,

measures for RCF and track comp**on**ent fatigue are presented in Table 13 and Table 15 in

appendix.

The calculated

Table 10 al**on**g with

high center

**on**d highest estimate. The o

0.0186 SEK, indicating ra

Interestingly, a laden freight wag**on** 2x22t, V max 100km, has

(0.0096 SEK), while its tare counterpart has a **on** for

this relati**on**ship is that **on** has a factor 1.81 higher wear per t**on**-km than a

laden freight wag**on**, while **on** **on**ly has a factor 1.17 higher settlement per t**on**-

6 We prefer **on** which

track secti**on**s **on** during 2014.

29

km than **on** (cf. Table 10). C**on**sidering that

damage

Table 10: Marginal **on**-km and

Vehicle type

Mean wear

per t**on**-km

Mean settlement

per t**on**-km

Marginal

per t**on**-km, SEK

Motor coach 4x21 t, V max 200 km/h, high center

Three-piece bogie 4x30 t, V max 60 km/h 87.1 867 067 0.0204

Passenger car 4x14 t, V max 160 km/h 81.5 756 207 0.0186

Motor coach with Jacob bogie 3x16.5 t, V max 160 km/h** 93.9 502 314 0.0179

Freight loco 6x20 t, V max 120 km/h 60.3 967 401 0.0176

Passenger Loco 4x19 t, V max 140 km/h 73.0 762 578 0.0174

Y25 bogie 4x22 t, V max 100 km/h 58.3 815 878 0.0158

Passenger Loco 4x19 t, V max 175 km/h 60.2 770 689 0.0157

Motor coach 4x16 t, V max 200 km/h** 40.8 877 980 0.0140

Freight wag**on** 2x6.5, V max 100 km/h 67.2 394 093 0.0132

Freight loco 6x30 t, V max 70 km/h 21.8 995 613 0.0124

Motor coach 4x16 t, V max 200 km/h* 36.8 696 335 0.0117

Freight loco 4x20 t, V max 120 km/h 32.4 743 564 0.0115

Motor coach with Jacob bogie 3x12.5 t, V max 200 km/h* 43.8 571 067 0.0115

Motor coach 4x12 t, V max 140 km/h* 33.9 682 236 0.0112

Three-piece bogie 4x6.5 t, V max 60 km/h 32.1 603 891 0.0102

Freight wag**on** 2x22 t, V max 100 km/h 37.1 462 377 0.0096

* Flexible wheelset guidance, ** Stiff wheelset guidance

Finally, it should be pointed out that

estimati**on** are c**on**sidered to also capture effects **on**ent fatigue.

However, **on** between **on**-km is quite low compared to

**on** coefficients in Table 6 that are calculated for track secti**on**s. For example,

correlati**on** coefficient between wear and RCF for

between settlement and track comp**on**ent fatigue. This implies that **on**ship between

30

all damage mechanisms. Hence,

The

**on**ship between

**on**

statistically significant.

7.0 C**on**clusi**on**

This paper c**on**tributes to

Smith et al. (2014) can produce estimates **on**

be informative for **infrastructure** managers in Europe. Specifically, by combining engineering

and ec**on**ometric approaches, we have estimated

**on** **rail**way network. We have developed previous work **on** this method by using

a larger set **on**s

that c**on**stitutes a major part **rail**way network.

The

secti**on**s, making it difficult to isolate

was able to provide significant

be used in **on**

capture **on**ent fatigue to a large extent. However,

downside is that

t**on**-km and track comp**on**ent fatigue per t**on**-km between

The results in this paper indicate a substantial variati**on** in **on**km

for **on** **rail**way, which is due to differences in

**on**e by

31

access charges with similar relative differences between **on**g

incentives for using more track friendly

More observati**on**s over time can be valuable for future research in order to generate

more reliable and robust estimates. The results from our approach can also be used to

charges that Switzerland has proposed to implement in 2017. More specifically, future work

can use **on** results **on** how line speed adds to

relative

Acknowledgements

The authors are grateful to

Administrati**on** for funding this research. We also wish to thank Aniruddha Kaushal for

helping us with our questi**on**s c**on**cerning **on**s, as well as Vivianne Karlss**on**,

Anders Nilss**on** and Martin Li at Trafikverket for providing data. Helpful comments have

been provided by Jan-Erik Swärdh, as well as by participants in a seminar **on** this paper,

organized at VTI in Stockholm, September 7, 2016. All remaining errors are **on**sibility

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Appendix

Table 11: Definiti**on**

Wear_den. = ln(wear density)

Wear_den.^2 = ln(wear density)*ln(wear density)

RCF_den = ln(RCF density)

Settl._den = ln(Settlement density)

Comp._den, = ln(Track comp**on**ent fatigue density)

Comp._den.^2 = ln(Track comp**on**ent fatigue density)*ln(Track comp**on**ent fatigue density)

Wear_den.Settl._den = ln(wear density)*ln(settlement density)

Wear_den.Com._den. = ln(wear density)*ln(Track comp**on**ent fatigue density)

Track_length = ln(track length)

Qual_ave = ln(average quality class)

Qual_ave^2 = ln(average quality class)* ln(average quality class)

Rail_age = ln(average age **rail**s)

Settl._den.Qual_ave = ln(settlement density)*ln(average quality class)

Regi**on**_West = Dummy for regi**on** West

Regi**on**_North = Dummy for regi**on** North

Regi**on**_Central = Dummy for regi**on** Central

Regi**on**_South = Dummy for regi**on** South

Regi**on**_East = Dummy for regi**on** East

Table 12: Model 2a - Average and

(based **on** n**on**-statistically significant

Variable Mean Std. Err. [95% C**on**f. Interval]

Average

RCF 1.65E+00 2.12E-01 1.23E+00 2.07E+00

Settlement 2.58E-07 5.27E-08 1.54E-07 3.63E-07

Marginal

RCF 8.02E-02 1.03E-02 5.99E-02 1.01E-01

Settlement 2.57E-08 5.25E-09 1.53E-08 3.61E-08

Weighted

RCF 2.34E-02 1.49E-03 2.04E-02 2.63E-02

Settlement 6.94E-09 4.43E-10 6.07E-09 7.82E-09

35

Table 13: Model 2a - Marginal **on**-km and

(based **on** n**on**-statistically significant

Mean

Mean

Mean

Marginal

Vehicle type

wear per

settlement

RCF per

per

t**on**-km

per t**on**-km

t**on**-km

t**on**-km, SEK

Motor coach 4x21 t, V max 200 km/h, high center

Three-piece bogie 4x30 t, V max 60 km/h 87.1 867 067 0.25 0.0231

Freight wag**on** 2x6.5, V max 100 km/h 67.2 394 093 0.37 0.0200

Passenger car 4x14 t, V max 160 km/h 81.5 756 207 0.16 0.0195

Motor coach with Jacob bogie 3x16.5 t, V max 160 km/h** 93.9 502 314 0.11 0.0181

Freight loco 6x20 t, V max 120 km/h 60.3 967 401 0.10 0.0167

Passenger Loco 4x19 t, V max 140 km/h 73.0 762 578 0.07 0.0164

Y25 bogie 4x22 t, V max 100 km/h 58.3 815 878 0.10 0.0155

Passenger Loco 4x19 t, V max 175 km/h 60.2 770 689 0.10 0.0154

Motor coach 4x16 t, V max 200 km/h** 40.8 877 980 0.14 0.0146

Motor coach with Jacob bogie 3x12.5 t, V max 200 km/h* 43.8 571 067 0.17 0.0135

Motor coach 4x16 t, V max 200 km/h* 36.8 696 335 0.10 0.0118

Freight wag**on** 2x22 t, V max 100 km/h 37.1 462 377 0.12 0.0108

Freight loco 4x20 t, V max 120 km/h 32.4 743 564 0.06 0.0108

Freight loco 6x30 t, V max 70 km/h 21.8 995 613 0.04 0.0107

Three-piece bogie 4x6.5 t, V max 60 km/h 32.1 603 891 0.09 0.0105

Motor coach 4x12 t, V max 140 km/h* 33.9 682 236 0.06 0.0104

* Flexible wheelset guidance, ** Stiff wheelset guidance

Table 14: Descriptive statistics, obs. from 169 track secti**on**s

Median Mean St. dev. Min Max

Maintenance **on** SEK 13.71 19.71 22.42 0.53 209.22

Track length, km 58.71 72.51 51.85 4.52 251.39

Average quality class 2.89 2.94 1.15 1.00 5.17

Average age **rail**s 21.2 22.3 9.6 2.3 53.1

Milli**on** t**on** density 4.52 8.15 9.39 0.00 61.98

Regi**on** West 0 0.20 0.40 0 1

Regi**on** North 0 0.14 0.34 0 1

Regi**on** Central 0 0.18 0.39 0 1

Regi**on** South 0 0.25 0.43 0 1

Regi**on** East 0 0.24 0.43 0 1

36