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THEORETICAL & APPLIED MECHANICS LETTERS 3, 013006 (2013)
Optimization of high-speed railway pantographs for improvingpantograph-catenary contact
Jorge Ambrosio,a) Joao Pombo,b) and Manuel Pereirac)
IDMEC-IST Technical Univ. of Lisbon Av. Rovisco Pais, Lisbon 1049-001, Portugal
(Received 25 September 2012; accepted 30 October 2012; published online 10 January 2013)
Abstract A crucial system for the operation of high-speed trains is the pantograph catenary interfaceas it is the sole responsible to deliver electrical power to the train. Being the catenary a stationarysystem with a long lifespan it is also less likely to be redesigned and upgraded than the pantographsthat fit the train vehicles. This letter proposes an optimization procedure for the improvement ofthe contact quality between the pantograph and the catenary solely based on the redesign of thepantograph head suspension characteristics. A pantograph model is defined and validated againstexperimental dynamic characteristics of existing pantographs. An optimization strategy based on theuse of a global optimization method, to find the vicinity of the optimal solution, followed by the use ofa deterministic optimization algorithm, to fine tune the optimal solution, is applied here. The springstiffness, damping characteristics and bow mass are the design variables used for the pantographoptimization. The objective of the optimal problem is the minimization of the standard deviation ofthe contact force history, which is the most important quantity to define the contact quality. Thepantograph head suspension characteristics are allowed to vary within technological realistic limits.It is found that current high-speed railway pantographs have a limited potential for mechanicalimprovements, not exceeding 10%–15% on the decrease of the standard deviation of the contact force.c© 2013 The Chinese Society of Theoretical and Applied Mechanics. [doi:10.1063/2.1301306]
Keywords high-speed, railway, pantograph
A limitation on the velocity of high-speed trainsconcerns the ability to supply the proper amount of en-ergy required to run the engines, through the catenary-pantograph interface.1 Due to the loss of contact notonly the energy supply is interrupted but also arcingbetween the collector bow of the pantograph and thecontact wire of the catenary occurs leading to their dete-rioration of the functional conditions of the two systems.The increase of the average contact force would improvethe energy collecting capabilities but would also lead toa rapid wear of the registration strip of the pantographand of the contact wire.2 The topology of a pantographmust address three stages of its operation: lift the panhead to contact wire height and compensate for spanswith lower catenary heights, generally with frequenciesof 1–2 Hz; handle the displacements with middle rangefrequencies associated to steady-arms passage, i.e., upto 10 Hz; deal with the higher frequency but low ampli-tude events.3 Typically the pantograph head, with itssuspension, is responsible to handle the high-frequencyexcitations while the lower stage, with the pneumaticbellow, deal with the low frequency.
A large majority of the pantographs in operationhave been developed with particular catenary systemsin mind, forming national pantograph-catenary pairssuch as the CX–LN2, which prevails in the French net-works, DSA380–Re330, common in the German high-speed lines, and ATR95-C270, used in Italy. However,
a)Corresponding author. Email: [email protected])Email: [email protected])Email: [email protected].
present trends for interoperability result in new panto-graph design requirements allowing operations in dif-ferent catenary systems. It is accepted that the im-provement of the current collection capabilities requireslighter bows, which in turn suggests the use of new ma-terials and construction concepts and that the head sus-pension is adjusted accordingly.4 The need for the pan-tographs to have low aerodynamic drag and noise emis-sion and to be compatible for cross-border operationsets some of the development directions for improvedcollectors.3
The norm EN503675 specifies the technical criteriafor the interaction between pantograph and overheadline. The experimental data on the contact force allowsobtaining the most important parameters required toevaluate the quality of the contact. The norm EN503675
specifies the following thresholds for pantograph accep-tance.
(1) Mean contact force Fm
(2) Standard deviation σmax < 0.3Fm
(3) Max contact force Fmax < 350 N(4) Max CW uplift at steady-arm dup � 120 mm(5) Max pantograph vertical amplitude Δz �
80 mm(6) Percentage of real arcing NQ � 0.2%Different numerical approaches exist in the liter-
ature to handle the pantograph-catenary interaction.References 6 and 7 propose the use of a finite-differencesmethod to describe the catenary and a multibody ap-proach for the pantograph. The catenary used is pla-nar and a co-simulation procedure controls the coordi-nation between the dynamic simulation of the panto-graph and catenary dynamics. More recently other au-
013006-2 J. Ambrosio, J. Pombo, and M. Pereira Theor. Appl. Mech. Lett. 3, 013006 (2013)
thors proposed the use of the finite element method todeal with the pantograph and catenary dynamics.1,8 Al-though the catenary models used are fully spatial, onlylumped mass pantograph models can be used with theseapproaches. Ambrosio et al.9 proposed the representa-tion of the catenary spatial dynamics using the finiteelement method while the pantograph is handled by ageneral multibody dynamics approach. A co-simulationprocedure is used to ensure the synchronization betweenthe finite elements and the multibody part of the code,pretty much in line with other co-simulation approachesused in different areas of computational mechanics.10,11
In this work a realistic model for a high-speed pan-tograph model is presented, being its identification dis-cussed. An optimization procedure, based on a ge-netic algorithm, is used to study the improvement ofthe pantograph dynamics to improve the pantograph-catenary interaction, described in terms of the quan-tities identified in the norm EN50367. The catenarymodel is developed in finite elements12 and the panto-graph model dynamics is studied with a multibody dy-namics code,1,13 being their interaction done through aco-simulation approach9 using a proper penalty contactformulation.14,15 All analysis are developed accountingonly for the contact interaction and disregarding theaerodynamic forces16 according to the European normfor experimental acquisition of the contact force. Thepotential for pantograph enhancement is, finally, dis-cussed.
The motion of the catenary is characterized by smallrotations and small deformations, in which the onlynonlinear effect is the slacking of the droppers. Theaxial tension on the contact, stitch and messenger wireis constant and cannot be neglected in the analysis.All catenary elements, contact and messenger wires aremodeled by using Euler–Bernoulli beam elements thatinclude axial tension, in particular to represent the mes-senger, stitch and contact wires.3 Using the finite ele-ment method, the equilibrium equations for the cate-nary structural system are assembled as
Ma+Cv +Kx = f , (1)
where M , C and K are the finite element global mass,damping and stiffness matrices. Proportional dampingis used to evaluate the global damping matrix, i.e, C =αK + βM with α and β being suitable proportionalityfactors.12 The nodal displacements vector is x while vis the vector of nodal velocities, a is the vector of nodalaccelerations and f is the force vector given by
f = fc + fa + fd, (2)
being fc the pantograph contact forces, fa the aero-dynamic forces, and fd the dropper slacking compen-sating terms. For typical catenary finite element mod-els the Newmark family of integration algorithms aresuitable methods for the integration of the equations ofmotion.17
The mechanical system that guarantees the requiredcharacteristics of the trajectory of the pantograph head
during rising is generally made up by a four-bar link-age for the lower stage and another four-bar linkage forthe upper stage. Another linkage between the head andthe upper stage of the pantograph ensures that the bowis always leveled. In order to control the raise of thepantograph one bar of the lower four-bar linkage is ac-tuated upon by a pneumatic actuator. Regardless ofusing multibody or lumped mass pantograph models,the equations of motion for a constrained multibodysystem (MBS) of rigid bodies, such as a pantograph,are written as[
M ΦTq
Φq 0
][qrλ
]=
[g
γ
], (3)
where M is the system mass matrix, q is the vectorwith the state accelerations, g is the generalized forcevector, which contains all external forces and moments,including the contact force that also appears in the cate-nary loading vector depicted by Eq. (2). λ is the vectorthat contains m unknown Lagrange multipliers associ-ated with m constraints, for which Φq is the Jacobianmatrix and γ is the right side of acceleration equationscontaining the terms that are function of velocity, posi-tion and time.
In dynamic analysis, a unique solution is obtainedwhen the constraint equations are considered simulta-neously with the differential equations of motion withproper set of initial conditions, i.e., a set of initial con-ditions that fulfils the position and velocity constraintequations. In each integration time step, the accelera-tions vector, q, together with velocities vector, q, areintegrated in order to obtain the system velocities andpositions at the next time step.18 This procedure is re-peated up to final time will be reached. Due to thelong simulations time typically required for pantograph-catenary interaction analysis, it is also necessary toimplement constraint violations correction methods, oreven the use of the coordinate partition method for suchpurpose.18
The contact between the registration strip of thepantograph and the contact wire of the catenary, fromthe contact mechanics point of view, consists in the con-tact of a cylinder made of copper with a flat surfacemade of carbon. The contact problem can be treatedby a penalty formulation where the contact force is de-fined in function of the relative penetration between thetwo cylinders.19,20 In this work, the Hertzian type con-tact force including internal damping is used. A suitablerepresentation of the contact model is written as14
fc = Kδn
[1 +
3(1− e2)
4
δ
δ(−)
], (4)
where fc is the magnitude of vector fc in Eq. (2), thepenalty term K is the generalized contact stiffness, e isthe restitution coefficient, δ is the relative penetrationvelocity and δ(−) is the relative impact velocity. Theproportionality factorK is obtained from the Hertz con-tact theory as the external contact between two cylin-ders having their axis perpendicular to each other.
013006-3 Optimization of high-speed railway pantographs Theor. Appl. Mech. Lett. 3, 013006 (2013)
m0
m2
m1
k2-3
c2-3
k1-2
c1-2
k0-1
c0-1
m3
FStatic
Fig. 1. Minimal topology for a pantograph lumped massmodel to be used in high-speed railway applications.
Z↼t↽Excitation bar with
prescribed motion
Fig. 2. Multibody model of the virtual test rig.
When using the same code to simulate both cate-nary and pantograph models the type of dynamic anal-ysis must be the same for both, i.e., if the catenaryis modeled with linear finite elements the pantographdynamics must also be linear. This is not compatiblewith the use of more general, and realistic, multibodypantograph models for which large rotations may ex-ist or that may follow curved tracks.21 In order to takeadvantage of the more adequate type of dynamic for-mulations to be used for catenary and for pantographa co-simulation between finite elements and multibodyapproaches is used here.9
Pantograph models used for the study of thepantograph-catenary interaction can be of two types:lumped mass or multibody models. Due to the factthat they can be studied in common linear finite ele-ment codes, also used for the analysis of the catenary,the lumped mass models are the most common appear-ing in the literature. However, the lumped mass panto-graph models are mathematical abstractions that needto be identified in laboratory tests and, for which, al-most none of the model parameters have any physicalsignificance. For a typical three-stage lumped mass pan-tograph model used for high-speed applications, as theone depicted in Fig. 1, only the top mass, stiffness anddamping coefficients can be kept similar to the panto-graph head mass and suspension characteristics.
Physical laboratories, as that one existing at Po-
Virtual testing
laboratory control
Read the
system data
Read the test data
For each excitation frequency
tested build the FRF
For each
experimental test
Pantograph
model
FRF of the pantograph
model
Test setup &
experimental
data
Generate the force
excitation time history
Perform the virtual test
on the pantograph
Fig. 3. Flowchart of the virtual laboratory for testing thepantograph models.
litecnico di Milano,1 can be used to measure the ex-perimental response of a physical prototype of a panto-graph in terms of frequency response functions (FRF).With such FRF available, the lumped mass model ofthe pantograph is identified by setting a virtual test-ing laboratory, as the one depicted in Fig. 2, in whichthe pantograph model is simulated. The model param-eters are tuned such that the model FRF match thosemeasured experimentally.
The multibody model of the test rig is composed ofthe excitation system and the pantograph model, con-nected via a rigid joint. This setup allows to excite thehead of the pantograph with the same contact force,frequency and amplitude used in the physical labora-tories and to collect the accelerations in any point ofthe pantographs to build the FRF. The time history ofthe prescribed displacement of the excitation bar, z(t),generated from the test data supplied by the physicaltesting laboratories, starts with the bar being raisedor lowered to the testing position followed by a sinu-soidal excitation with prescribed frequency and ampli-tude. The flowchart of the virtual laboratory is shownin Fig. 3.
To ensure that the virtual laboratory tests are per-formed in a suitable manner, the number of cycles ofexcitation at each frequency must be such that a sta-tionary behavior of the system is obtained before its dy-namic response is considered for the evaluation of theFRF. The process is shown in Fig. 4.
By setting the head mass and suspension character-istics of the lumped mass pantograph model to the val-ues obtained from measuring the component and iden-tifying all other parameters of the model, the validatedthree-stage pantograph model depicted in Table 1 is ob-tained.
The FRF corresponding to the pantograph headmass, m3, upper frame, mass m2, and pantograph knee,
013006-4 J. Ambrosio, J. Pombo, and M. Pereira Theor. Appl. Mech. Lett. 3, 013006 (2013)
4.900 5
4.901 0
4.901 5
4.902 0
Dis
pla
cem
ent/m
10 cycles of excitation
with ith frequency
10 cycles of excitation with
↼i⇁1↽th frequency
4 cycles with ithfrequency for FRF
4 cycles with ↼i⇁1↽thfrequency for FRF
5
0 5 10
Frequency/Hz
4.900 0
4.899 5
4.899 00 5 10 15 20 25 30 35 40 45 50
Time/s
0.20
0.15
0.10
0.05
015 20
|(a
2�ft↽|
Fig. 4. Process for the construction of the FRF from thedynamic response of the harmonic excitation. The experi-mental results are those supplied by the test house.
Table 1. Properties of the lumped mass model.
Property Value Units
m1 5.58 kg
k0-1 178.45 N/m
c0-1 108.39 Ns/m
m2 8.78 kg
k1-2 154.88 N/m
c1-2 0.09 Ns/m
m3 7.75 kg
k2-3 7 000 N/m
c2-3 45.85 Ns/m
mass m1, are displayed in Fig. 5. It is clear that the re-sponse of the lumped mass model and the pantographexperimental prototype are similar.
The optimization problem associated to the im-provement of the pantograph for the catenary-pantograph interaction is defined asminimize F0(ui)
subject to:
⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩
fj(ui) = 0,
j = 1, 2, · · · , nec,
fj(ui) � 0,
j = (nec + 1) , (nec + 2) , · · · , ntc,
uloweri � ui � uupper
i ,
i = 1, 2, · · · , nsv,
(5)
where the objective function is denoted by F0, ui arethe design variables, nec, ntc and nsv are the num-ber of equality constraints, inequality constraints andbounded design variables. In this work the standarddeviation of the contact force is used as the measureto minimize. The rationale for this choice is that the
0
0.1
0.2
0.3
0.4
0 5 10 15 20
Experimental
Model
0 5 10 15 20
0 5 10 15 20
0.20
0.15
0.10
0.05
0
0.20
0.15
0.10
0.05
0
|(a
3�ft↽|
|(a
2�ft↽|
|(a
1�ft↽|
Experimental
Model
Experimental
Model
Frequency/Hz
Frequency/Hz
Frequency/Hz
(a) m3
(b) m2
(c) m1
Fig. 5. FRF of the generic pantograph lumped mass modelvs the physical prototype.
ultimate goal is to minimize the mean pantograph con-tact force, which is nowadays defined by the TSI. Suchmean contact force can be controlled using the lowerstage pneumatic actuator provided that the standarddeviation of the contact force remains equal or below30% of the mean force. Therefore, minimizing the stan-dard deviation allows reducing the mean contact force,if permitted by regulations, and ensures that the pan-tograph performs with a better safety margin, as thehomologation limit of σstd < 0.3Fmean is more easilyachieved. Therefore,
F0(ui) = σstd. (6)
The design variables used in the optimization pro-cess are the only ones that have physical meaning in thelumped mass pantograph models, i.e., the mass, stiff-ness and damping of the upper stage of the three-stagemodels used. With reference to Fig. 1, the vector ofdesign variables is, therefore,
u =[m3 k2−3 c2−3
]T. (7)
The constraints used in the optimization process arethe side constraints of the design variables and any func-tional constraint. The technological constraints are ex-pressed here as side constraints being
0.8mref � m3 � 1.2mref ,
013006-5 Optimization of high-speed railway pantographs Theor. Appl. Mech. Lett. 3, 013006 (2013)
0.8 kref � k2−3 � 1.2 kref ,
0.1 cref � c2−3 � 100 cref . (8)
A constraint maintained in the optimization pro-cess is the mean contact force Fmean = 150 N. This con-straint recognizes the fact that by reducing the meancontact force there are variations on the standard de-viation, which is generally reduced. Therefore the con-straint ensures that the reduction of the standard devi-ation is at the cost of the functional characteristics ofthe pantograph and not of the operational conditions.
The choice of the optimization methods suitable forthe solution of the optimal problem is of crucial impor-tance. The problem of the characterization of the op-timal pantograph is non-convex and subjected to largenumerical noise. Besides these factors, there is no as-surance that the design space is continuous. All thesefactors lead to the existence not only of local minima butalso to convergence difficulties when using determinis-tic optimization algorithms, i.e., algorithms in which thesearch is based on the definition of gradients. Therefore,global optimization algorithms are selected to addressthe problem, among which a genetic algorithm is thechoice.
The genetic optimization algorithm used here isavailable in Matlab. All default values of the ge functionare selected with the exception of the number of indi-viduals (i.e., the number of sets u for which the functionevaluation is done), which is Nindividuals = 60, the max-imum number of generations, which is Ngenerations = 10and the maximum number of generations to run withouthaving an improvement above the default limit, whichis StallLimit = 5. In order to ensure any feasible im-provement of the best design detected by the geneticalgorithm a deterministic algorithm is used to find theminimum close to such point. The sequential quadraticprogramming (SQP) method is used for the purpose.
The lumped mass pantograph model, depicted inFig. 1, and with the data provided in Table 1, is thereference model for the optimization. The pantographruns on a high-seed catenary of the simple type.22
The design space of the pantograph is searched by thegenetic optimization algorithm with 7 generations, inwhich 420 analysis are performed. The result is furtherimproved by the SQP optimization taking 4 iterationsmore to reach the best pantograph design.
The search of the design space is depicted in Fig. 6,where each point is a different pantograph design. Itcan be observed that the optimal pantograph has a bowmass and head suspension damping as low as the de-sign space allows while the suspension stiffness increasesreaching its maximum allowed limit.
The performance of the genetic algorithm is de-picted in Fig. 7, in which the values of the worst, meanand best standard deviations for each generation arerepresented. It is noticed that the best design is ob-tained after 5 generations while the average values of theobjective function decrease in every generation. This in-dicates that the best individuals are already included in
m3/kg k3/(N . m-1)
c3/(N
s .m
-1)
c3/(N
s .m
-1)
m3/kgk
3/(N
.m
-1)
m3/kg
4.0
3.0
2.0
1.0
0
6.06.5
7.07.5
8.06.57.0
7.58.0
8.59.0
Τ103
Τ103
Τ103
Τ103
4.5
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
06.5 7.0 7.5 8.0 8.5 9.0
8.0
7.5
7.0
6.5
6.0
6.5 7.0 7.5 8.0 8.5 9.0
Fig. 6. Design space of the pantograph. The optimal designis identified as * (star).
Generation
Scores
Fig. 7. Best, mean and worst standard deviation of thecontact force for each generation.
013006-6 J. Ambrosio, J. Pombo, and M. Pereira Theor. Appl. Mech. Lett. 3, 013006 (2013)
m0
m2
m1
k2-3
c2-3
k1-2
c1-2
k0-1
c0-1
m3
FStatic
Lumped Mass Original Optimal
m1 5�58 5�58
m2 8�78 8�78
m3 7�75 6�2
k0-1 178�45 178�45
k1-2 15 488 15 488
k2-3 7 000 7 860�5
c0-1 108�39 108�39
c1-2 0�09 0�09
c2-3 45�85 64�12
Std Dev 41�50 36�93
Fig. 8. Reference (or original) and optimal lumped masspantographs.
0
50
100
150
200
250
300
454 508 562 616 670
Contact fo
rce/N
(filtered 0-
20 H
z)
Track lenght/m
Reference
Optimal
Fig. 9. Contact force time history of the reference andoptimal pantographs.
the last generation and that, most likely, they can notbe improved.
The resulting model for the optimal pantograph isdepicted in Fig. 8, together with the model of the ref-erence pantograph, i.e., the model that results fromthe experimental identification of an existing prototype.The optimal pantograph performance on the referencecatenary exhibits a standard deviation of the contactforce σstd = 37 N, which is an improvement of 11% ofits contact quality.
To have a better understanding of the behavior theoptimal pantograph in the actual interaction with thecatenary it is relevant to analyze the contact force char-acteristics that are required by the regulation. Figure 9presents the time history of the contact force in selectedspans of the catenary. It is observed that the generalbehavior of the reference and optimized pantographs issimilar, except for the peak forces, in which the opti-mized pantograph exhibits lower maxima. The statisti-cal characteristics of the contact forces are depicted inFig. 10 in terms of the maximum and minimum con-tact forces, amplitude, standard deviation and statis-tical minimum force. The improvement on the pan-tograph performance can be observed not only on thedecrease of the standard deviation but also on the reduc-tion of the maximum force and increase of the minimumcontact force.
Another relevant aspect of the pantograph perfor-mance concerns the RMS-value of the contact forces at
Maximum MinimumAmplitude MeanStandard
Deviation
Statistical
Maximum
Statistical
Minimum
Optimal 233.8 92.4 141.4 150.0 37.0 260.9 39.2
Reference 238.9 81.7 157.2 150.0 41.5 274.6 25.4
Variation -2% 13% -10% 0% -11% -5% 54%
0
0.5
1.0
1.5
2.0
2.5
3.0
Maximum Minimum Amplitude Mean Standard
Deviation
Statistical
Minimum
Contact force/N
Reference
Optimal
Τ102
Fig. 10. Contact force statistical characteristics of the ref-erence and optimal pantographs.
(0-2)Hz RMS/N
First Harmonic
(0-5)Hz RMS/N
Span passing
(7.5-13) Hz RMS/N
Dropper passing
Optimal 26.8 29.9 17.4
Reference 28.2 31.2 22.9
Variation -5% -4% -24%
Fig. 11. RMS values, for selected frequency ranges for thereference and optimal pantographs.
relevant frequency ranges, in particular for the first har-monic, span passing frequencies and dropper passingfrequencies.. It is observed in Fig. 11 that the RMS-values of the optimal pantograph are reduced in all fre-quency ranges for the optimized pantograph.
Another aspect of the pantograph catenary interac-tion that requires attention is the maximum uplift onthe catenary caused by the pantograph contact. Figure12 shows the uplift in a selected support of an inter-mediate span of the catenary for the reference and op-timal pantographs. It is observed that the changes inthe pantograph have no effect on the steady-arm uplift.The same trend is observed in all other steady arms ofthe catenary, being 6 cm the maximum uplift value ob-served anywhere in the catenary, which is far from the10 cm limit imposed by the infrastructure owner.
0 1 2 3 4 5 6 7
Time/s
0.07
0.05
0.03
0.01
-0.01
-0.03
Support uplift
/m
Reference
Optimal
Fig. 12. Catenary uplift at a selected support for the refer-ence and optimal pantographs.
013006-7 Optimization of high-speed railway pantographs Theor. Appl. Mech. Lett. 3, 013006 (2013)
The results of the pantograph optimization showthat the room for improvement of the existing equip-ment performance on the standard catenaries in whichthey are operated is rather limited. The potential im-provement on a standard high-speed pantograph perfor-mance is of about 11%. The results not only show thatpantograph developers do a background work to find thebest tuning for their performance to the national cate-naries where they are expected to be operated. How-ever, no conclusions can be drawn on the cross-borderoperation of the pantographs as they have not been ob-ject of this study. The use of the optimization tools canprovide valuable information on the changes requiredin the pantographs, from the mechanical point of view,for improved quality of operation across border in thesame form that the performance of the current catenary-pantograph pair was demonstrated. Further work is re-quired to demonstrate that the optimal pantograph sys-tems are in fact improved for different operating condi-tions in the networks in which they are accepted, usingexperimental geometric data of the catenaries in partic-ular.
This work was supported by the projects SMAR-
TRACK, funded by FCT with the contract PTDC/EME-
PME/101419/2008 and PANTOTRAIN, funded by the EC
with the contract SC8-GA-2009-234015, led by UNIFE.
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