Original Ar ticle
Ne w contr ol appr oaches to impr o v e
contact quality in the con v entional
spans and o v erlap section in a
high-speed catenar y system
Farzad V esali
1
, Habibollah Molatefi
1
, Mohammad A Rezvani
1
,
Bijan Moa veni
2
and Markus Hecht
3
Abstract
Continu ous and quality contact between the pantograph and the catenar y s ystem is one of the major challenges for
increasing the s peed of elec tric trains. Compared to oth er studies, th is paper has con sidered the catenar y s ystem as the
main system an d has assumed the pantograph as an ex citation factor . Based on this, a con tr oll er has been applied on the
contact wire , once at the last span and another time near t he contact point of the pan tograph. Results were compared with
the con ven tional contr ollers that ex er t contr ol f o r ce on a colle ctor’ s head. Based on th is, tw o differ ent objectiv es were
consider e d for the contr oller inc l uding ‘impr ovement of contac t quality’ and ‘minimis ation of the vertical v elocity of the
overlap poin t’. F or this pu rpose, a f ull analytical mo del of the catenar y sy stem was presented and was verified usi ng the
relevant standards, and then thre e types of linear quadratic optimal controllers wer e add ed to the model with the two
objectives mention ed above. The res ults of th e study show that if th e model aims to reduce the overlap point vertical
velo city , contact quality will be impr oved. How ev er , in case it aims to enhance c ontact quality , the v elocity of the ov erlap
poin t will not nece ssarily be reduced. Moreover , the contact point con tr oll er aiming at reducing the overlap point velocity
outperforms o ther contr ollers and mak es 71% impr ove ment in contact qua lity in comparison with th e no-c ontr o ller case.
K e ywor ds
Linear quadratic regulator , contact wire, catenar y system, analytical model, pantograph, contact quality , overlap section
Date received: 14 August 2018; accepted: 9 December 2018
Intr oduction
Today, all of the world’s high-speed fleets are electric,
as diesel locomotives are essentially unable to gener-
ate power to reach high speeds.
1
As contact force
oscillations amplify at high speeds, which may lead
to mechanical wear or the electrical wear of panto-
graph collector head and/or contact wire, the dynamic
interaction of pantograph and catenary system is one
of the most critical limiting factors at the maximum
speed of a fleet.
2
The pantograph is a device over the
roof of locomotive collecting power from catenary. A
catenary system usually consists of two horizontal
wires (a messenger cable and a contact wire) and
many vertical cables (droppers). The contact wire is
in contact with the pantograph and the messenger
cable bears contact wire weight using droppers and
prevents them from sagging. Figure 1 shows a dia-
gram of the pantograph and catenary system. The
most important function of the pantograph is to
maintain the contact force of the collector’s head by
changing the height of the contact wire. It is favour-
able to reduce dynamic force oscillations in the inter-
action of the pantograph and catenary system, which
is called contact quality. To study the effect of differ-
ent parameters on contact quality, various software or
models have been developed. Some of them use FEM
and have made a 3D model of catenary,
3
some others
use FD for 2D model of catenary
4
and Seo et al.
5
have
generalised the catenary model to large deformation
analysis.
1
School of Railwa y Engineering, Iran University of Science and
T echnolog y , T ehran, Iran
2
Faculty of Electrical Engineering, K. N. T oosi University of T echnolog y ,
T ehran, Iran
3
Department of Rail V ehicle, T echnical University of Berlin, Berlin,
Germany
Corresponding author :
Habibollah Molatefi, Iran University of Science and T echnology Narmak,
T ehran 1684613114, Iran.
Email: [email protected]
Proc IMechE P ar t F:
J Rail and Rapid T ransit
2019, V ol. 233(9) 988–999
! IMechE 2019
Article reuse guidelines:
sagepub.com/journals-permissions
DOI: 10.1177/0954409718822861
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The primary challenge is to improve the contact
quality, in particular for high-speed trains either by
active controlling or optimised design of the panto-
graph. There are many studies on the optimisation of
geometric and dynamic characteristics of panto-
graphs.
6,7
In the next step, some researchers followed
the use of an active controller in a pantograph.
8–12
In 2005–2015, many researchers concentrated on the
pantographs with active controllers; however, the stu-
dies had insufficient outputs in the industry, and few
of them reached the prototyping step,
13
as they pro-
vided no considerable improvement in contact quality
as compared with the increase of initial cost. The rela-
tive failure in the performance of pantographs with an
active controller allowed the problem to be solved in
this study using a change in attitude. In all the reviews
on the use of an active controller, a pantograph has
been introduced as the main system and contact wire
in a catenary system as an excitation factor, but this
study considers a catenary system as the main system
whose oscillations should be controlled. A panto-
graph has been considered as the factor causing oscil-
lations. Therefore, the control force has been applied
to the contact wire for the first time in this study.
The contact quality between the pantograph and
catenary system along the whole route is not fixed
and it is lower at some points, which are called ‘crit-
ical points’ in Harell et al.
14
Contact wire replacement
place is one of these points (overlap section). The ten-
sioning device,
15
which commonly consists of a wheel
and some suspended weights, is used to make the
expansion and contraction of the contact wire and
messenger cable has no impact on its tension. It is
designed for a limited length of contact wire (about
1200–1500 m). After that, there should be another
contact wire with an independent tension system.
The overlap section of contact wires is recognised as
a critical point, and one of the bottlenecks of contact
quality is between the pantograph and catenary
system. In the static analysis of the catenary system,
a parameter is defined as ‘elasticity’. Elasticity means
the vertical deflection of the contact wire against a
unit vertical load. Presence of droppers and masts in
the catenary system makes the elasticity of the caten-
ary system variable. Elasticity variation is one of the
most important reasons for oscillations in contact
force. In Harell et al.,
14
the researcher introduces
the section overlaps and section insulator as the crit-
ical section of the catenary–pantograph system based
on experimental results and interviews with personnel
at Swedish National Administration. Elasticity for
catenary is defined as the displacement caused by
the vertical unit force. In the overlap section, the
applied load must raise the contact wire of both sec-
tions. Therefore, a considerable reduction of elasticity
occurs at the overlap point; for this reason, this point
is considered as a bottleneck. It is clear that contact
quality improvement should be at bottleneck points to
lead to the overall increase of course speed. Based on
this, the controller in this study aims at improving
contact quality and reducing impact in the overlap
section of the contact wire (cost function).
It can be mentioned that this study considered two
new issues. The controller aimed at reducing contact
force oscillations and reducing impact at the overlap
section of the wire. The control force was once
imposed on the pantograph and another time on the
catenary system and the results were compared.
Standard EN 50318 has introduced a reference
model (pantograph and catenary) and has given all
required parameters, to be considered for simulation
by any software. Then the standard defined seven par-
ameters for comparing and validating the results of
software, which include mean values of contact force,
Figure 1. Components of the pantograph and catenar y system.
V esali et al. 989

the standard deviation of contact force, the statistical
maximum of contact force, the statistical minimum of
contact force, the actual maximum of contact force,
the actual minimum of contact force and the max-
imum uplift at support. In the next step, the software
should be validated via the results of the field test.
The considered software for this study is verified by
EN 50318 standard.
In the following section, the controller was
imposed on a tensile beam to study its performance.
Then, three types of actuators were considered on the
complete model of catenary system and pantograph,
one on the collector head, other on the last span of the
catenary system near the overlap span and the last one
on the catenary system and near the contact point
with the pantograph. Two separate cost functions
were considered including contact force oscillations
and overlap point vertical velocity. Therefore, six con-
trol problems were generally considered and the
results were compared.
Analytical model
Concerning the complexity and variety of pantograph
and catenary system components, the models formu-
lated for the analysis of the dynamic interaction of
pantograph and catenary system are considered as
software. A variety of software are available all over
the world for this task, and Bruni et al.
16
compared
them in 2015. The software, which was formulated in
Iran University of Science and Technology, is known
as CatAna whose major unique feature is the analyt-
ical solution. In other words, it never uses the finite
element methods and calculates catenary system
motion using the Galerkin method and eigenfunction
expansion. Two options are available for modelling
the pantograph: multibody model and low-order
model.
17
A multibody model involves many details
such as kinematic constraints of joints and geometric
features of components, while the low-order model
considers equivalent mass, stiffness and damping of
each degree of freedom. Low-order models are
commonly used with numerator translation degree
of freedom. Following the reference model of EN
50318, the pantograph was considered as a 2-DOF
system. The contact wire and messenger cable were
modelled in CatAna as a tensile Euler–Bernoulli
beam; the droppers and supports were considered as
the springs with a specified stiffness and the mass of
clamps of droppers was considered. In the leading
software, the droppers are subject to buckling at
compression and bear no force; however, this
study overlooked the non-linear behaviour of drop-
pers (buckling) concerning the use of the linear con-
troller. Figure 2 shows a schematic diagram of the
analytical model.
Equation (1) expresses the governing equations for
contact wire and messenger cable
 Aw , tt þ Cw , t  Pw , xx þ EIw , xxxx
¼ F dr x dr , t ðÞ þ F s x s , t ðÞ
þ F p ð x p , t Þþ F c ð x c , t Þ ,0 4 x 4 l , t 5 0
ð 1 Þ
where q A is the mass per unit length, C is the damping
of each wire, P is the tension applied to the wire and
EI is the bending stiffness of wire. F p is the force
imposed by the pantograph; x p is the place to use
the force, which is obtained by multiplying velocity
by time as the pantograph velocity is fixed; and F c is
the control force applied to the wire in position x cont .
F s and F dr are the forces of supports and droppers;
each is applied to its relevant position ( x s and x dr ).
w ð x , t Þ is the vertical displacement of the wire at the
location of x and time of t. t and x subscripts, respect-
ively, indicate the partial differentiation from time
and displacement. Based on the variation separation
method, the answer to the differential equation of
equation (1) can be considered as wx , t ðÞ ¼ Xx ðÞ T ð t Þ .
The natural frequencies and mode shapes of the con-
tact wire and the messenger cable can be calculated by
inserting the answer in the homogeneous equation
proportional to equation (1) and concerning the
boundary condition and relevant initial conditions.
Figure 2. A schematic diagram of the analytical model for the pantograph and catenar y system interaction.
990 Proc IMechE P ar t F: J Rail and Rapid T ransit 233(9)

It is possible to convert the partial differential equa-
tion into an ordinary differential equation (equation
(3)) based on the orthogonality of mode shapes using
eigenfunction expansion and inserting equation (2) in
equation (1)
w ð x , t Þ¼ X ’ i x ðÞ q i ð t Þð 2 Þ
In thi s re la tio n, ’ i x ðÞ is th e i th mode sh ap e, an d q i ð t Þ
is th e i th el em en t o f the vect or of mo dal co ord ina tes
 i €
q i t ðÞ þ  i c i _
q i t ðÞ þ  i ! i 2 q i t ðÞ ¼
 Z l
0
’ i x ðÞ F p Vt , t ðÞ  x  Vt ðÞ þ F dr x dr , t ðÞ  x  x dr
ðÞ

þ F s x s , t ðÞ  x  x s
ðÞ þ F c t ðÞ  x  x c
ðÞ Þ d x , t 5 0
ð 3 Þ
 i ¼  A Z l
0
’ i 2 x ðÞ d x ð 4 Þ
Equation (3) is written for the messenger cable and
the contact wire. Each element of modal coordinates
of wire can be considered as one degree of freedom of
that wire. Based on this, if the contact wire is con-
sidered having n mode shape and messenger cable
considered to have m mode shape, the whole system
has n þ m þ 2 degrees of freedom considering the
pantograph. M, C and K matrices can be achieved,
and the entire equation governing the system can be
written as equation (5) through integrating contact
wire, messenger cable and pantograph equations
M €
Q þ C ð t Þ _
Q þ K ð t Þ Q ¼ DFp ð t Þþ EFc ð t Þð 5 Þ
In th is ma tr ix equ atio n, Q is a ve cto r of degr ees o f
fr ee dom , whic h enco mpas ses th e mo dal co ord inat es of
co nt ac t wir e and m esse nge r cab le , an d panto grap h
DOF s. M is the ma ss matr ix of the who le syst em , whic h
defi nes the ma ss of leng th unit of wi re s, th e mass o f
cl am ps o f ea ch dro pp er and ma ss of p an togr ap h co m-
pone nts. Droppe r stiffnes s, natural f requencies of
wi re s, st iff nes s of pant ogra ph co mpon en ts, and co ntact
st iff nes s bet we en th e pan tog rap h an d co ntact wir e are
sh ow n in matr ix K elem en t s. C expr esses the da mping
pro port ional to an y stiff ness of ma tri x K . Sin ce th e
pla ce of co ntact bet ween p an togr ap h an d cont ac t
wi re chan ges, so me of the el emen ts of K and C ma tric es
ch an ge ove r tim e. The el emen ts wh ic h det er mine the
re la ti on o f co nta ct wire and th e first ma ss of p an to -
gr ap hs ( m p 1 in Fi gu re 2) ar e no t cons t an t any mo re .
The up lift fo rce of panto grap h and th e cont rol fo rce
ar e pla ced o n th e le ft si de of th e equa t ion an d D an d E
co effi ci en ts sp eci fy th e pla ce t o appl y th e forc es.
Contr oller
Concerning t he analytical model and linearity of t he
whole system, a linear quadratic optim um controller
was used to contr ol the contact wi re and pantogra ph.
18
A state-s pace model is re quired for th e design of thi s type
of cont roller ; theref ore, by usi ng equat ions (6 ) to (9),
equation ( 5) can be convert ed to a state-spa ce equation
A ¼ 0 I
 M  1 K  M  1 C
 ð 6 Þ
B ¼ 0
M  1 D
 ð 7 Þ
H ¼ I
M  1 E
 ð 8 Þ
Xt ðÞ ¼ q ð t Þ
_
q ð t Þ
 ð 9 Þ
The state-space variables (vector X ) are defined as
the amount and time derivative of the variables of
system DOFs. After obtaining A, B and H values, it
is possible to write equations similar to equations (10)
and (11) as state-space equations
_
X ð t Þ¼ AX t ðÞ þ BU t ðÞ þ Hf ð t Þð 10 Þ
Ut ðÞ ¼ G ð t Þ Xt ðÞ ð 11 Þ
Ut ðÞ is the control input and f ð t Þ is pantograph
excitation (uplift force). In the linear control, the con-
trol force is assumed as the linear coefficient of state-
space variables and controller design is summarised in
finding the appropriate G value. To design optimal
control, the following integral should be minimised
J ¼ Z t f
0
X T t ðÞ QX t ðÞ þ U T t ðÞ RU ð t Þ

d t ð 12 Þ
Matrices Q and R are of paramount importance in
controller design. Q determines the importance of
state-space variables in controller target. The appro-
priate selection of this matrix can select controller
target or cost function. For instance, the control
target can be defined by zeroing the displacement of
a point or zeroing the speed of a few meters of the
contact wire. This problem, which aims at zeroing the
speed of wire overlap point in the middle of overlap
span, matrix Q is defined similar to equation (13)
Q ¼ q c T q c ð 13 Þ
q c ¼ ’ c
½
1  n 0 ½
1  m 0 ½
1  2 0 ½
1  n 0 ½
1  m 0 ½
1  2

1  2 ð n þ m þ 2 Þ
ð 14 Þ
’ c
½
1  n ¼ ’ c 1 ð x pt Þ ’ c 2 ð x pt Þ ... ’ cn ð x pt Þ

ð 15 Þ
q c (equation (14)) is the vector whose elements are as
many as the state-space variables of the whole system
and defines cost function. The arrays of this vector
V esali et al. 991

consist of six main sections. Mode shapes of contact
wire, messenger cable and the displacement of panto-
graph DOFs form the three initial parts of the vector
and their derivation relative to time forms the second
three parts of the vector. To make the displacement
minimisation of a point on the contact wire be defined
as a cost function, vector ’ c should be located in the
first section of the vector q c and other sections become
zero. The elements of the vector ’ c and the mode
shapes of contact wire are equal. The i th element of
the value of i th mode shape is on the target point.
Ma tr ix R de te rm ine s a co ntro l force in th e cost fu nc-
ti on . In this p robl em, in whic h th ere is onl y one con tr ol
fo rc e, th is ma trix is jus t a numb er . The sm al le r th e
num ber is , the hi gh er th e cont rol fo rc e w ill b e, and
th e bet te r the co ntr oll er per form an ce will be. R is us u-
al ly de term i ne d wit h resp ect to th e limi tat io ns im pose d
by the a ctuator on impos ing force. To find th e optimal
va lu e of G as the co ntro ller ga in fa cto r, th e S value
sh ou ld b e fo und u si ng Ri ccat i equ at io n (e qu atio n (1 6) )
an d th en G va lu e sh oul d be fo un d usi ng eq uat ion (1 7)
_
St ðÞ þ St ðÞ A  1
2 St ðÞ BR  1 St ðÞ þ A T St ðÞ þ 2 Q ¼ 0, St
f
 ¼ 0
ð 16 Þ
Gt ðÞ ¼ 
1
2 R  1 B T St ðÞ ð 17 Þ
The gain factor was considered fixed and its
changes versus time were overlooked in this problem.
The results of using a contr oller in a
simple span
As stated earlier, one of the reasons to design a con-
troller is to reduce the vertical velocity of the contact
wire in the overlap section. A controller has been
designed for the governing equations of a simple ten-
sile beam, and the results have been studied to observe
the performance of an optimal linear controller.
Figure 3(a) shows the model schematic diagram.
It shows that the control force is imposed at the
distance of X c from the support and the controller
aims at reducing the vertical oscillations within the
range of X 1  X 2 from the support (target span).
Under the initial conditions of the problem, an initial
displacement wave was considered in the middle of
the span, and the problem is solved from 0 to 3 s.
Table 1 shows the parameters considered for the
problem.
Figure 4 shows the effect of the use of the optimal
linear controller for the problem of Figure 3(a) on the
dynamic response of the whole system.
The displacement wave considered for the initial
condition moves towards the two sides of the span.
In the side with no controller, the wave hits the sup-
port and reflects with reverse amplitude. However, the
wave moving where the control force is applied is
dissipated by the controller. The controller shows no
resistance against the first pass of wave and it traps it
between the support and controller connection place.
Each time that the wave reflected from the support
hits the controller, some of it reflects and the other
part is dissipated. The same thing happens to the
second wave reflected from the other side of the
span after passing under the controller. Figure 4(c)
shows that point P t (the point in the middle of the
target span) senses only two waves passing under it.
The two waves pass through the target span before
passing through the controller. This figure shows that
the optimal linear controller is capable of managing
the traversal waves of a beam under tension in a way
that it makes minimum oscillations in the target span.
Figure 3. (a) A simple problem to examine the controller performance in a beam under tension and (b) the schematic diagram of the
spans and the place to impose contr ol for ce to contr ol the overlap point.
992 Proc IMechE P ar t F: J Rail and Rapid T ransit 233(9)

T able 1. The geometric and dynamic specifications of the tensile beam, catenar y and pantograph used in the simulation.
T ensile beam (Figure 3(a))
Parameter V alue Parameter V alue Parameter V alue
x 1 m ðÞ 20 Z 0 mm ðÞ 5 Bending stiffness N m 2
 195
x 2 m ðÞ 30  Z 0 mm ðÞ 10 Mass per unit length kg = m ðÞ 1.35
x c m ðÞ 10 T ension N ðÞ 2000 Length of beam (m) 100
Catenar y
Parameter V alue Parameter V alue Parameter V alue
Span length (m) 60 Mass/unit length of CW (kg/m) 1.35 Mass/unit length of MC (kg/m) 1.07
Encumbrance (m) 1.2 T ension of CW (kN) 20 T ension of MC (kN) 16
Pre-sag at mid-span (mm) 0 Bending stiffness of CW (N m
2
) 195.0 Bending stiffness of MC (N m
2
) 131.7
Stagger (mm)  200 Mass of dropper clamps on CW (g) 0 Mass of dropper clamps on MC (g) 0
No. of spans: 10 Mass of stager (g) 400 No. of dr oppers per span 9
Stiffness of dropper (kN/m) 10 Stiffness of registration arm (N/m) 340 Mass of droppers (kg) 0
Dr opper position in each span (m) 5 10.5 17 23.5 30 36.5 43 49.5 55
Pantograph
0 Effective dynamic mass (kg) Stiffness (N/m) Damping (N s/m)
Contact spring – 50,000 120
Collector head 7.2 4200 10
Articulation frame 15 50 90
Uplift force (N) 100 Speed (km/h) 200
CW: Contact wire; MC: Messenger Cable.
(a) (b)
(c) (d)
Figure 4. The results obtained from the effect of the contr oller on a beam under simple tension in the initial excitation of
displacement wa ve (a) contour , position, time, vertical velocity for the beam with no controller ; (b) position–time–velocity contour for
the beam with the controller ; (c) vertical velocity of point P t in Figure 3; and (d) requir ed control for ce for the controller .
V esali et al. 993

Different types of contr ollers on
catenar y system
Three types of controllers were considered with two
various targets to compare the performance of differ-
ent controllers (Figure 3(b)). One time, the control
force of F ce ð t Þ was considered in the last span of the
contact wire near the overlap point of contact wire.
Other time, the control force was considered on the
contact wire near the contact point with the panto-
graph ( F cm ð t Þ ). Finally, similar to other studies, the
control force was imposed on the controller head of
pantograph ( F cp ð t Þ ). Ten spans of a catenary system
were considered in this problem. Table 1 shows the
pantograph and catenary system specifications.
In the modal analysis method, the shape of the
catenary is described by linear summation of mode
shapes and the mode shapes related to higher frequen-
cies have a shorter wavelength. Therefore, it can be
concluded that the least number for the modes should
be selected in a way that in the distance between the
two concentrated loads (or two droppers), more than
a pair of nods can be observed in highest mode shape.
Such a criterion can be presented as in equation (18)
m  L
Ddr ð 18 Þ
In equation (18), L is the whole length of wire or
cable, Ddr is the minimum distance between two adja-
cent droppers and m is the required natural frequency
to be considered. In this simulation, the first 373 mode
shapes of contact wire and the first 310 mode shapes
of the messenger wire were considered.
In order to apply a control force to the contact wire
near the tension wheel, a linear actuator could be
attached to the mass, and by accelerating the mass,
the reaction force will be applied to the contact wire
as the controlling force. Figure 5(a) shows the sche-
matic diagram of this idea. Applying control force to
the contact wire near pantograph would be possible
by considering magnetic field near pantograph. Since
the contact wire is carrying considerable current near
pantograph, it is possible to induct force to a wire
with have current (Figure 5(b)). Imposing control
force on the pantograph collector head may also be
performed by different methods such as an aero-
dynamic flap, wire, electromagnetic actuator, etc.
This study does not aim at examining the source of
control force; it mainly focuses on its effect on panto-
graph interaction and catenary.
As stated earlier, the controller was designed in all
three modes using two different attitudes. Initially,
attempts were made to maintain the contact force
fixed, which is called ‘contact force controller’ in
this study. The controller aims at minimising contact
wire impact (related to the following mechanical sec-
tion) on the pantograph in the overlap section, which
is called ‘overlap point controller’ in this study. In the
overlap point controller, the controller aimed at redu-
cing the vertical velocity of contact wire in the overlap
point. This study assumed that the pantograph con-
tacts with the contact wire of the following mechan-
ical section from the middle of the 10th span. In the
contact force controller, the standard deviation of
contact force (contact force SD) is the criterion of
controller efficiency; in the overlap point controller,
the vertical velocity of the overlap point may indicate
controller performance. Figure 6 shows the contact
force SD while simulating the problem. The horizon-
tal axis shows the longitudinal position of the contact
wire and the vertical axis shows the contact force SD
from  30 to þ 30 m at any point. For example, the
vertical axis at X ¼ 30 m indicates the contact force
SD from X ¼ 0 m to X ¼ 60 m. Therefore, it is clear
that it is impossible to define standard deviation for
X < 30 m and 570 m < X < 600 m.
Each diagram of Figure 6 shows the contact force
SD in the three modes. The black line indicates the
contact force SD with no controller. The contact force
SD in the last spans exceeds other spans with respect
to the reflection of the propagated wave. The cost
function of contact force oscillations was once con-
sidered for controllers (grey line); another time, over-
lay point velocity oscillations were considered (a grey
dashed line). As Figure 6(a) shows, the effect of the
Figure 5. Suggestions on placing the actuator on the contact wire. (a) Last span and (b) contact point vicinity .
994 Proc IMechE P ar t F: J Rail and Rapid T ransit 233(9)

controller is more evident in the last span because the
actuator is at the end of the span. When the panto-
graph is on the first few spans, no wave reaches the
overlap point; therefore, as expected, there would be
no difference between the mode in which the overlap
point controller is on the first spans and the no-con-
troller mode. Contact force oscillations are equal to
contact force SD. Therefore, it is predictable that the
contact force controller would reduce contact force
SD more than the overlap point controller does.
Attempts were made in all the six control modes to
maintain the maximum control force at about 200 N
through regulating R in equation (12). For instance,
Figure 7 shows the schematic diagram of the control
force required by the controller in the last span.
Figure 7 shows that when the actuator is on the last
span and it aims to control overlap point oscillations,
the control force is initially negligible; however, when
the pantograph approaches the overlap point, it
increases rapidly. In Figure 8 the effect of using a
controller at end span on the contact force between
pantograph and contact wire is illustrated.
As the actuator is located at the end span, the effect
of the controller is only observable in last spans. The
contact between pantograph and contact wire is mod-
elled with a stiff sparing ( K c ¼ 50 kN = m in Figure 2).
Figure 6. The contact force SD for any designed controllers: (a) actuator in the last span on the contact wire, (b) actuator in the
vicinity of the contact point on the contact wire and (c) actuator on the pantograph collector head.
Figure 7. The magnitude of control for ce for two contact force controllers and time ov erlap point of the actuator on the last span.
V esali et al. 995

Therefore, the amplitude of contact force can be cal-
culated according to equation (19)
19,20
In this equation, F
c
is the contact force; K c and C c
are the stiffness and damping of contact point,
respectively; w c and y 1 are the deflections of contact
wire and displacement of collector head of panto-
graph as well.
In Figure 6(b), the actuator was applied near the
contact point and on the contact wire. In this mode,
despite the earlier mode, both controllers were able to
make a considerable reduction in the contact force
oscillations. Interestingly, the overlap point controller
outperformed in reducing oscillations. The contact
force is a function of the relative displacement
between the head of pantograph and contact wire at
the contact point. In other words, the contact force is
affected by contact wire displacement and head of the
pantograph. If the whole system is divided into panto-
graph and catenary sections, controller performance
will be more favourable when both the cost function
and the actuator are located in the same section.
In Figure 6(b) the actuator is located in the contact
point in the catenary. The overlap point controller is
located on the contact wire as well, while for contact
force controller, one parameter which describes con-
tact force (equation (19)) is a deflection of contact
wire which belongs to catenary and the other param-
eter is vertical displacement (or velocity) of collector
head, which belongs to the pantograph. Therefore, it
can be expected that for the contact point actuator,
the overlap point controller has better performance
than contact force controller.
In the last mode (Figure 6(c)), it is assumed that the
actuator is on the pantograph collector and it imposes
the control force on the head of the pantograph dir-
ectly. In this mode, the control force is immediately
imposed on one of the variables causing contact force
(head of the pantograph displacement). If the control-
ler aims at reducing contact force oscillations, it may
prevent to change the distance of head of pantograph
and contact wire and it may reduce contact force SD
considerably. Overlap point controller could reduce
contact force SD. In this controller, the controller
effect becomes more tangible with the pantograph
approaching the overlap point.
As mentioned earlier, two separate targets were
considered in the simulations, and Figure 6 shows
only the controller effect on one of the targets (reduc-
tion of contact force oscillations). Figure 9 shows the
performance of controllers to achieve another target.
Since the pantograph touch the second contact
wire at point x ¼ 570 m, this point has been zoomed
in the diagrams. Figure 6 shows that even if it aims
to reduce overlap point oscillations, contact quality
improvement can be guaranteed as one of the side
advantages of the controller. This is due to the cre-
ation of damping in the contact wire and prevention
of the return of wave from the supports. However,
Figure 9 proves that the opposite is not true. In
other words, when the controller focuses on contact
quality improvement, this will not reduce the overlap
point oscillations of the contact wire. Figure 9(a)
shows that if the control actuator is on the last span
and it is aimed to reduce overlap point oscillations,
the vertical velocity of the point will generally reduce.
However, when the vertical pantograph is crossing,
the overlap point is similar to the time when no con-
troller is applied. If the contact force controller is
used, overlap point velocity oscillations will remain
from the beginning of the simulation, as the actuator
is near the overlap point and it affects its vertical vel-
ocity considerably.
F c t ðÞ ¼ K c w c V p t , t

 y 1

þ C c
dw V p t , t
ðÞ
dt þ V p dw x , t ðÞ
dx j x ¼ V p t

 _
y 1

F 4 0
0 F 4 0
()
ð 19 Þ
Figure 8. Comparison of the contact without a controller , with contact force contr oller and overlap point contr oller . The actuator
is located in end span.
996 Proc IMechE P ar t F: J Rail and Rapid T ransit 233(9)

Figure 9(b) shows that the overlap point velocity
will reduce if the actuator is on the contact wire and
the overlap point controller is used. However, in case
it aims to improve the contact quality, it will not have
a noticeable effect on reducing overlap point velocity.
Figure 9(c) shows that when the actuator is on the
pantograph collector head, both the overlap point
controller and contact force controller may reduce
overlap point oscillations. In this mode, the mean
value of the contact force reduces considerably. In
other words, when the actuator is placed on the col-
lector’s head, the controller reduces the mean of con-
tact force to lower contact force oscillations, and it
increases the probability of detachment of pantograph
from the contact wire. Table 2 shows a summary of
the efficiency of each controller.
It is expected that the controller, which was
designed aiming at improving contact quality, would
further reduce contact force SD. However, the con-
troller used to reduce overlap point oscillations may
not necessarily lead to the reduction of the vertical
velocity of the overlap point when the pantograph is
crossing. In other words, the vertical velocity of the
overlap point is a vector in which the controller
attempts to reduce all its values. However, the overlap
point vertical velocity during the cross of pantograph
is only one point of the vector whose value does not
necessarily reduce as compared with the no-controller
mode. Table 2 shows that if the actuator is on the last
span, the overlap point velocity will increase at the
time of crossing in both the contact force controller
and the controller of overlap point velocity. However,
Figure 9. Performance of contr ollers in reducing the velocity of the pantograph to hit the following contact wire in the overlap
point: (a) the actuator in the last span on the contact wire, (b) the actuator near the contact point on the contact wire and (c) the
actuator on the pantograph collector head.
T able 2. A summar y of the results obtained fr om different contr ol actuators with varied contr ol targets.
Location of actuator Aim of controller
Standard deviation
of overlap
point’ s vertical
velocity (m/s)
Overlap point
vertical velocity
while pantograph
is passing (m/s)
Standard deviation
of the contact
force (N)
Mean value
of contact
for ce (N)
Last span Contact force 0.091 0.19 50.73 91.04
V elocity of overlap point 0.057 0.21 51.68 90.65
Contact point Contact force 0.087 0.06 34.51 79.93
V elocity of overlap point 0.010  0.01 15.30 112.62
Collector head Contact force 0.019 0.04 9.76 25.37
V elocity of overlap point 0.015 0.02 30.01 35.62
No contr oller 0.067 0.18 54.50 91.22
V esali et al. 997

the overlap point controller was able to reduce the SD
of overlap point velocity by about 20%.
Me an wh il e, th e ove rlap po in t cont roll er has a very
fa vo ur ab le pe rf orma nce w hen th e ac tu at or is nea r the
co nt ac t poi nt . The co nt ro ll er co uld redu ce the co nta ct
f o r c eS Df r o m5 4 t o1 5 N .O nt h eo t h e rh a n d ,i t
re duc ed the ove rl ap po int ve lo ci ty co nsi der ab ly , as
th e vel ocit y re du ce d fro m 18 to 1 cm/s at the tim e of
pan togra ph cro ss in g. Al l the imp ro veme nts we re made
wh il e th e av erag e co nta ct fo rce w as n ot redu ce d and it
wa s incr ease d slig htl y (112 N of aver age co nta ct for ce
wa s obta ined fr om 100 N up lift fo rc e) . Th is is th e high -
es t su per iori ty of co nta ct p oin t actu ato r to the ac tuato r
on th e pan to gr ap h coll ecto r hea d. If th e actu ator is o n
th e pan togr aph h ea d co lle ctor , it redu ces it s aver age
va lu e t o re du ce co nta ct fo rc e osc illa tion s. Tabl e 2
sh ow s th at al thou gh th e actu ato r in stal le d on th e con -
tr oll er’s he ad wi th con tact for ce con tr oll er coul d
re duc e conta ct fo rc e SD an d over lap po int ve loc ity
co ns ide rab ly , it re duce d th e aver age co nta ct fo rce
an d it cr eate d 25 N of t he aver age co nta ct fo rce fr om
th e up lif t forc e of 10 0 N. As th e resu lts of T ab le 2
sh ow , am o ng th e sele cted act uato rs and the t wo co st
fu nct ions , th e u se o f th e ac tu at or at the co nta ct po int
an d th e co st fu nct io n of ov erla p poin t ve lo ci ty w il l le ad
to the mo st favo ura ble re sult .
Conclusions
This report considered an analytical model for the
interaction of the pantograph and the catenary
system of an electric fleet. To prevent pantograph
wear and to improve contact quality, despite other
studies, two types of novel controllers were designed
for the catenary system, and the results were com-
pared with the time the controller is placed on the
controller head of pantograph. Regarding high tensile
load in catenary cables and small deflection of caten-
ary due to pantograph motion, most of the studies in
this field consider linear equation for catenary and
pantograph motion. Performance of the designed
linear quadratic regulator was first examined on a
tensile beam, and the results showed that the control-
ler might trap and dissipate the propagated wave. The
controller was then added to complete the catenary
system model with three different actuators including
collector head actuator, last span actuator of contact
wire and contact point actuator on contact wire. Two
cost functions were considered for any actuator.
Contact quality improvement was the traditional
objective of all the studies in this field. In addition
to contact quality, elimination of the overlap point
of the contact wire was considered as the other object-
ive of the study, as most studies discuss the overlap
point of contact wire as the bottleneck of pantograph
contact quality and contact wire. The results for the
simulation of the designed linear optimal controller
show that if the controller is designed to reduce over-
lap point oscillations, it will improve contact quality,
as it dissipates the wave returning from the support.
However, if the controller aims at improving contact
quality, it will not necessarily reduce the vertical vel-
ocity of the overlap point. The simulation results
showed that the actuators installed on the collector’s
head and the contact point outperformed the last span
actuator; however, if the actuator is installed on the
collector, it reduces the average contact force, which
may lead to pantograph de-wirement. Finally, it can
be mentioned that the use of the actuator on the con-
tact wire near the contact point with the pantograph
with the cost function of overlap point oscillations
may reduce the overlap point velocity considerably
and improve contact quality.
Ackno wledgements
The authors would like to acknowledge Mr Mohrich Joerg
from Balfour Beatty Rail Company and Prof. Giorgio
Diana from Polytechnic University of Milan for sharing
experiences and presenting challenges.
Declaration of Conflicting Interests
The author(s) declared no potential conflicts of interest with
respect to the research, authorship, and/or publication of
this article.
Funding
The author(s) disclosed receipt of the following financial
support for the research, authorship, and/or publication
of this article: This research was supported by the office
for ‘National Master Plan for High-Speed Trains’ at the
Iran University of Science and Technology.
ORCID iD
Habibollah Molatefi http://orcid.org/0000-0002-7584-
9637
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Appendix
Notation
c i damping of the i th mode shape
c p 1 damping of the collector head of the
pantograph
c p 2 damping articulation frame of the
pantograph
C c damping of the contact point
C dr i , j damping of the j th dropper in the i th
span
C s damping of support at masts
EI bending stiffness
F c controlling force
F ce end span controlling force
F cm contact point controlling force
F cp collector head controlling force
F dr dropper force
F s support force
F u uplift force of the pantograph
k dr i , j stiffness of j th dropper in the i th span
k p 1 stiffness of the collector head of the
pantograph
k p 2 stiffness articulation frame of the
pantograph
k star stiffness of the steady arm
K c stiffness of the contact point
K s stiffness of the support at masts
m cd mass of the dropper clamps on the
contact wire
m cu mass of the dropper clamps on the
messenger cable
m p 1 mass of the collector head of the
pantograph
m p 2 mass of the articulation frame of the
pantograph
m star equivalent mass of the steady arm on
the contact wire
T d tension of the contact wire
T u tension of the messenger cable
wx , t ðÞ vertical deflection of the wire
x c location of the controlling force
x dr location of the dropper
x s location of support at masts
y 1 vertical displacement of the collector
head of the pantograph
y 2 vertical displacement of the articulation
frame of the pantograph
 A mass per unit length of the wire
’ i x ðÞ i th mode shape of the wire
! i i th natural frequency of the wire
V esali et al. 999

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