TECHNICAL TRANSACTIONS ELECTRICAL ENGINEERING CZASOPISMO TECHNICZNE ELEKTROTECHNIKA DOI: 0.4467/353737XCT.6.46.6045 -E/06 MAREK DUDZIK, ADAM S. JAGIEŁŁO * DYNAMIC ASPECTS OF INTERACTION: PANTOGRAPH CATENARY DYNAMICZNE ASPEKTY WSPÓŁPRACY PANTOGRAFU Z SIECIĄ TRAKCYJNĄ Absrac There are many exising moels for he ineracion of a panograph caenary sysem. The auhors propose a specific moel, which is amene by aking ino accoun he ynamic par of he moion equaion. This moel was compare o anoher aemp an conclusions were rawn. A numerical suy was performe using he Malab Simulink Environmen. Keywors: panograph, caenary, panograph-caenary moel, racion sysem Sreszczenie W arykule przesawiono zaganienie moelowania ukłau rgającego obierak prąu sieć rakcyjna jako moelu fizycznego z masą skupioną sieci rakcyjnej. Do celów symulacyjnych spośró wielu isniejących moeli panografów wybrano moel WBL 85 3 kv, w kóry m.in. wyposażone są lokomoywy EU oraz EU43. Zaprezenowano wyniki symulacji la ego moelu obieraka prąu przy ruchu posępowym wzłuż sieci jeznej. Zaprezenowane wyniki pochozą z symulacji wykonanej la wóch moeli maemaycznych sieci rakcyjnej. Moele e różniły się o siebie jenym czynnikiem, kóry zazwyczaj jes pomijany w baaniach na opisywanym ukłaem. Nasępnie wyniki e porównano. Słowa kluczowe: rakcja elekryczna, unele, profil rasy kolejowej, zużycie energii * Ph.D. Eng. Marek Duzik, Prof. D.Sc. Ph.D. Eng. Aam S. Jagiełło, Insiue of Elecrical Engineering an Compuer Science, Faculy of Elecrical an Compuer Engineering, Cracow Universiy of Technology.
4. Inroucion Wih an increase in he spee poenial of racion vehicles, here is also an increase in he qualiy requiremens relae o heir elecrical energy consumpion. The energy is ransferre by a moving conac poin beween he panograph an he caenary []. I is ifficul o mainain a consan clamping force of he panograph o he caenary in orer o ensure a coninuous an uninerrupe supply of energy o he racion vehicle []. Moeling of he escribe sysem in erms of ynamic properies is a complex ask ha is affece, among oher hings, by: vehicle spee an parameers of he OCS sysem (from Over Conac Sysem). Oher influenial facors also inclue inerfering facors relae o he environmen of he OCS sysem, such as []: geomerical characerisics of he rack an heir changes ha occur uring he operaion, amospheric coniions like air emperaure an is humiiy, icing of racion neworks, win spee an is irecion changes, an ohers. As can be seen, because of he mulipliciy of various aspecs, he opimizaion of he parameers of he panograph caenary sysem requires comprehensive research an analysis. Simulaions of he OCS sysem are presene iner alia in publicaions: [ 5].. Moel of he panograph an caenary Figure shows he physical moel of ineracion in he OCS. The panograph in his moel has four egrees of freeom an consiss of a lower frame, an upper frame an a panhea. The moel of he panograph consiss of four effecive moving masses, which represen he componens lise above. They are inerconnece by spring an amping forces [3]. The effecive masses, springs an ampers are mache o he acual behavior of he panograph in ynamic coniions. Fig.. The panograph moel (illusraion on he lef sie from [4])
43 The ynamic properies of he panograph are eermine by parameers, such as a concenrae mass (reuce o he conac poin wih he caenary), saic force of he panograph on he conac wire, aeroynamic force, elecroynamic force an oher facors. The escribe moel of ineracion in he OCS also comprises he moel of he caenary sysem, aking ino accoun he insananeous value of he reuce mass as a funcion given by he relaion (3a). The auhors formulae he assumpions for he caenary moel in he following way [3]: he sysem only performs verical oscillaions, he caenary sysem is symmeric, he caenary span lengh is fixe an eermine, he caenary is mae of a homogeneous maerial, he maerials use for he consrucion of he caenary are isoropic, he equivalen mechanical parameers of he caenary change perioically an mee he coniions of Dirichle, he caenary affecs masses m 3 an m 4 in he same way, a any ime, he loss of conac beween masses m 3 an m 4 an he caenary elemen m c (x()) is no possible, an equivalen cener of graviy was assume for he elemens wih masses m 3, m 4, m c (x()), whose isplacemen an velociy coorinaes were escribe by he coorinaes of he sae x c () an x () c. The following parameers were assume in he escripion of he iscree caenary moel: m c (x()) funcion of he reuce mass of he caenary, D c (x()) funcion of viscous amping of he caenary, k c (x()) funcion of he siffness of he caenary, r c (x()) funcion of ry amping of he caenary, Q c (x()) he sum of he insananeous value of he weigh acing on he panhea of he panograph an he weigh of he masses m 3, m 4, L span lengh, x() insananeous isance measure along he caenary from he neares pole o he panograph, v() insananeous velociy of he ranslaional moion of he panograph, measure in he longiuinal irecion of he caenary, a acceleraion of he ranslaional moion of he panograph, measure in he longiuinal irecion of he caenary, iniial velociy of he ranslaional moion of he panograph, measure in he V 0 x 0 longiuinal irecion of he caenary, iniial isance measure along he caenary from he neares pole o he panograph. The moel akes ino accoun, iner alia, he impac of he aeroynamic force on he panograph, which is usually skippe in less complex moels. No informaion abou he values of he coefficiens of air resisance K 3, K 3, K for he panograph moel cause ha hey were assume as consans. However, he assume values may no correspon o he acual values of hese coefficiens.
44 A he same ime, he moel presene in Fig. oes no ake ino accoun he impac of vibraion of he racion vehicle on he vibraion of he panograph base frame nor he possibiliy of loss of conac of masses m 3 an m 4 wih he caenary elemen m c (x()). In fac, he vibraions of a locomoive boy are irecly ransferre on he panograph base frame, as well as he menione loss of conac is possible. In he escripion of he sysem vibraions corresponing o masses m an m he isplacemen coorinaes were assume relaive o heir equilibrium posiions. The mahemaical moel for he moel of he panograph-caenary ineracion assume in he aricle is presene by (). The coefficiens an variables in his equaion are explaine above an in Fig.. xc () ( mc ( x() )+ m3 + m4) xc () xc () kc x() xc() Dc x ()) r h ( + ) c g κ k3 k3 xc () y () = ( ) ( ( ) ( D + D xc() y () ) ( r + r ) xc() y () 3 3 3 3 gh κ F F Q x + + ( ) 3() 3 () c () y xc () y ( ) xc () y () m = k k x y D D ( 3 + 3 ) ( c () () )+( 3+ 3 ) r r h k + ( + ) 3 3 g κ y() y () y() y() D r y () y () gh κ F + () ( ) () y y() y() m = k y y D ( () () )+ + r y y gh κ k y D y () () () () r gh y () κ + F The mahemaical moel of he ime erivaive of he prouc of he mass of he caenary m c (x()) an he verical velociy x () c. for he moel assume in he aricle is given by equaion (a) or (b). Equaion (b) inclues he impac of he prouc of he ime erivaive of he mass an he verical velociy x () c on he behavior of he OCS sysem. In he presen paper, he moel of he OCS ineracion, which uses equaion (a) in he sysem of equaions (), is calle he moel wihou he mass ime erivaive erm, while he moel of he OCS ineracion, which uses equaion (b) in he sysem of equaions (), is referre o as he moel aking ino accoun he ime erivaive of he mass. xc() xc() mc ( x() ) m x() = ( ) c (a)
45 Where: x () m x() mc ( x( x) ) x () Disance as a funcion of ime: x () x () c c c = ( ) + c ( ) 3 πi m x m m L x c( ())= co + ci cos ( ) i= 3 πi D x D D L x c( ())= co + ci cos ( ) i= 3 πi k x k k L x c( ())= co + ci cos ( ) i= x () = 05. a + v + x v () = a + v 0 0 0 m x( x) xc () (b) Forces acing on he caenary: Qc( x() )= ( mc( x ())+ m3 + m4 ) g (5) F () = K v () 3 3 F () = K v () 3 3 F () = K v () 3. Simulaion resuls From among many exising moels of panographs, he ype WBL 85 3kV was selece as he simulaion moel. Is parameer values were aken from [3]. This paper presens he analysis of movemen for he chosen panograph ype ineracing wih a caenary in which he erivaive of he prouc of caenary mass an verical velociy x () c. is represene by equaion (a) or (b). Simulaions were carrie ou uner he following assumpions [3]: he sysem performs only verical oscillaions, he amping in kinemaic pairs of he panograph is a ry amping, as well as viscous amping, elemens connecing he reuce masses of he panograph arms m, m have linear characerisics wih consan coefficiens k, D, r, elemens connecing he lower mass m wih he panograph s base frame have linear characerisics wih consan coefficiens k, D, r, elasic elemens connecing he reuce masses of panoheas m 3, m 4 wih he upper panograph arm m have a nonlinear characerisics, he value of he saic force F () being he impac of he panograph lifing sysem on he caenary is he same a any spee an oes no epen on he posiion of he boy wih mass m, (3a) (3b) (3c) (4) (6)
46 he velociy v() of he moving panograph is consan, he oscillaions of he sysem for he elemens m an m ake place aroun he saic equilibrium posiions, which resul from he acion of he force of graviy coming from hese masses, he simulaions were performe wih iniial coniions assume as zero. The following parameers of he panograph were assume in calculaions of he moel of he OCS [3]: Mass [kg]: Siffness [N/m]: Viscous amping [kg/s]: m = 0.5; k = 3 500; D = 60; m = 8.73; k = 3 500; D = 0; m 3 = 7.93; D 3 = 0; m 4 = 7.93; D 3 = 0; Dry amping [N]: Drag coefficien [kg/m]: r =,5; K = 0.36 0.7/3; r =,5; K 3 = 0.36 0.7; r 3 = ; K 3 = 0.36 0.7; r 3 = ; The value of he saic force F () was assume as [3]: F = 0 N; Figure shows he nonlinear characerisics of he spring F k3j = f(y () x c ()) (where j =, ) [3], corresponing o he magniues k 3, k 3 (Fig. ) a specifie inervals of he isplacemen ifference y () x c (). Fig.. The nonlinear characerisics F k3j = f(y () x c ()) (where j =, ) of a spring [3] referring o he values k 3, k 3 (Fig. ) in specific inervals of ifferences of isplacemens y () x c ()
47 The caenary parameers assume as in [5]. They have he following values: Mass [kg]: Siffness [N/m]: Viscous amping [kg/s]: m c0 = 95; k c0 = 7000; D c0 = 40; m c = 00; k c = 3360; D c = 40; m c = 0; k c = 650; D c = 50; m c3 = 5; k c3 = 60; D c3 = ; Dry amping [N]: Span lengh [m]: r c = 0; L = 65.5; I was assume ha he panograph move wih uniform moion along he caenary: a = 0 m/s ; x 0 = 0 m; V 0 = 0 m/s. The value of acceleraion ue o graviy was aken as g = 9.8 m/s. Figure 3 shows he funcion of he reuce mass epening on ime for velociy V 0 = 0, 0 an 40 m/s an simulaion ime equal o 6.55 s. This ime refers o raveling, respecively, once, wice an four imes he assume isance L a given spees V 0. The line enoe by correspons o he spee V 0 = 0 m/s, he soli line correspons o he spee V 0 = 0 m/s, an o he line enoe by correspons o he spee V 0 = 40 m/s. The funcions in Fig. 3 are rawn for he coniion of he iniial panograph posiion x 0 = 0 m. Fig. 3. The funcion of he reuce mass versus ime for velociy V 0 = 0, 0 an 40 m/s of he panograph moving wih respec o he caenary
48 The nex par of his paper presens he resuls corresponing o he spee V 0 = 0 m/s (7 km/h) for he simulaion ime from 0 s o 3 6.55 s. The simulaion ime was increase ue o he occurrence of an aperioic componen. The simulaion was performe for he coniion of he iniial panograph posiion x 0 = 0 m. Figure 4 shows he imeline of he verical isplacemen x c () calculae from he moel of he sysem wihou he ime erivaive of he mass componen. This isplacemen is calle x c () in he graph. Figure 4 also shows he imeline of he verical isplacemen x c () calculae from he moel, aking ino accoun he ime erivaive of he mass. This isplacemen is calle xcpoch() in he figure. Fig. 4. Time courses of verical isplacemen x c () calculae for he iscusse moels of he OCS for movemen of he panograph along he caenary wih spee V 0 = 0 m/s. The soli line enoes he moel aking ino accoun he ime erivaive of mass; he ashe line is for he moel wihou he ime erivaive of he mass Figure 5 epics he ime course of he insananeous ifference in he verical isplacemen x c () calculae for he compare moels of he OCS for movemen of he panograph along he caenary wih spee V 0 = 0 m/s. The ifference was calculae as he ifference beween he isplacemen x c () from he moel, aking ino accoun he ime erivaive of mass an he isplacemen x c () from he moel wihou he ime erivaive of mass componen. Figure 6 shows ime courses of verical velociy x () c calculae for he iscusse moels of he OCS for movemen of he panograph along he caenary wih spee V 0 = 0 m/s. Figure 7 gives he ime course of he insananeous ifference in he verical velociy x () c calculae for he compare moels of he OCS for movemen of he panograph along he caenary wih spee V 0 = 0 m/s. The ifference was calculae as he ifference beween he velociy x () c calculae from he moel, aking ino accoun he ime erivaive of mass an he isplacemen x () c from he moel wihou he ime erivaive of mass componen.
49 Fig. 5. Time course of he insananeous ifference in he verical isplacemen x c () calculae for he compare moels of he OCS for movemen of he panograph along he caenary wih spee V 0 = 0 m/s Fig. 6. Time courses of verical velociy x () c. calculae for he iscusse moels of he OCS for movemen of he panograph along he caenary wih spee V 0 = 0 m/s Vcpoch() velociy calculae for he moel incluing he ime erivaive of mass; V c () is for he moel wihou he ime erivaive of he mass componen Fig. 7. Time course of he insananeous ifference in he verical velociy x () c calculae for he compare moels of he OCS for movemen of he panograph along he caenary wih spee V 0 = 0 m/s
50 4. Conclusions The presene suy akes ino accoun he effec of aeroynamic forces acing on he panograph. Moeling of he panograph was simplifie by omiing he impac of vehicle vibraions on he verical movemens of he panograph. In he moel ineracion OCS, i was assume ha he loss of conac beween masses m 3 an m 4 an he caenary elemen m c (x()) is no possible (Fig. ). Simulaions were performe in Malab using meho oe3. The simulaion resuls inicae ha, for he chosen zero iniial coniions, he calculae isplacemens x c () an velociy x () c afer cerain ime reach a seay sae for he consiere moels of he OCS for he movemen of he panograph along he caenary wih spee V 0 = 0 m/s. Base on he obaine simulaion resuls, i can be conclue ha he ifference beween he values of he insananeous verical isplacemens x c () calculae for he iscusse moels of he OCS, for movemen of he panograph along he caenary wih spee V 0 = 0 m/s, wih given iniial coniions, for he ime inerval wihou he aperioic componen of he verical isplacemen x c (), can reach values larger han 0.00 m. The simulaion resuls also inicae ha he ifference beween he values of he insananeous verical velociies x () c calculae for he iscusse moels of he OCS, for movemen of he panograph along he caenary wih spee V 0 = 0 m/s, wih given iniial coniions, for he ime inerval wihou he aperioic componen of he verical velociy x () c., can reach values exceeing 0.0 m/s in absolue value. I shoul be noe ha he simulaion resuls have no been verifie in real coniions an require furher work an analysis. The iniial coniions were aken as zero, which causes he ime courses presene here o have an aperioic characer. Such a sae exposes he mos significan impac of he prouc of he ime erivaive of he mass an velociy mc ( x() ) x () xc () x () on he OCS ineracion. The auhors sugges appropriae choice of iniial coniions for he analysis of he OCS ineracion. References [] Kobielski A., Wyznaczenie wpływu paramerów mechanicznych ukłau obierak prąu sieć rakcyjna na jego własności ynamiczne meoami symulacji kompuerowej, Rozprawa Dokorska, Poliechnika Krakowska, Wyział Mechaniczny, Kraków 98. [] Pisano A., Elio U., Conac force esimaion an regulaion in acive panographs: an algebraic observabiliy approach, 46h IEEE Conference on Decision an Conrol, New Orleans, LA, USA, Dec. 4, 007.
[3] Pisano A., Elio U., Conac force esimaion an regulaion in acive panographs: an algebraic observabiliy approach, Asian Journal of Conrol, Vol. 00, No. 0, Publishe online in Wiley InerScience, DOI: 0.00/asjc.0000Monh 008, 9. [4] Pisano A., Elio U., Conac force regulaion in wire-acuae panograhs via variable srucure conrol, 46h IEEE Conference on Decision an Conrol, New Orleans, LA, USA, Dec. 4, 007. [5] Pisano A., Oupu Feeback Regulaion of he Conack-Force In High-Spee Train Panographs, Journal of Dynamic Sysems, Measuremen, an Conrol, Vol. 6/87, March 004. [6] Kumaniecka A., Meoy analizy rgań ukłaów yskreno-ciągłych i ich zasosowanie o problemów ransporu szynowego, Wyawnicwo PK, Kraków 007. [7] Roman Z., Symulacja współpracy obieraka prąu z siecią rakcyjną, TTS Technika Transporu Szynowego, 00, 7 8. [8] Pisano A, Usai E., Conac force regulaion in wire-acuae panographs via variable srucure conrol an frequency-omain echniques, Inernaional Journal of Conrol, vol. 00, no. 00, January 006, 0. [9] Kobielski A., O pewnym aspekcie moelowania ukłau rgającego obierak prąu sieć rakcyjna, in XIV Warszay Naukowe PTSK, Krynica Zrój, 6 9 września 007. 5