Mathematical Models of High-Speed Trains Movement

Transcription

1 Mathematical Model of High-Speed Tain Movement CISMARU CRISTIAN DANIEL NICOLA DORU ADRIAN MANOLEA GHEORGHE DRIGHICIU MIRCEA ADRIAN BULUCEA CORNELIA AIDA Faculty of Electomechanical Engineeing Univeity of Caiova Decebal, 107 ROMANIA Abtact: - In the pape it i peented the mathematical model of the ueful movement of the high-peed tain. On the mathematical model bai it i built the tuctual diagam aociated of the ueful movement. The obtained tuctual cheme can be eaily implemented in SIMULINK. An immediately example of the utilization of the ueful movement model it contitute the dawing of the movement diagam, which it illutate the dynamic apect of the ueful movement. Like application, by mean of the SIMULINK model they ae taced, on the bai of the taction and baking chaacteitic and of the impoed condition of the oute, the movement diagam of the high-peed tain. Key-Wod: - modeling, imulation, electic vehicle, high-peed tain, ueful movement 1 Intoduction In any electic vehicle, the developed electomagnetic toque of the taction electic moto ha been tanmitted of the moto wheel. Thee, though tuning, etablih the tanlation movement of the vehicle on the ail. Becaue the otation movement of the moto wheel it i "attached", though tanmiion, of the taction moto oto movement, it eult that the moto wheel movement equation can be deduced fom the moto equation with which thee ae coupled. Thu, the moto toque tanmitted of the moto wheel it i M R =i t M, whee M i the developed ueful toque of the taction moto. At the unning adiu =D /, the moto toque M R it coepond it the moto foce F o [N] at wheel [1], [], [3], [4] M R Fo = = i M D D t (1) / In the lip abence, the peipheal peed v of the moto wheel (which they ae tuned with the angula peed 0 ) v = 0 () it i the ame with the tanlation movement peed (on the ail) of the vehicle. A 0 = m /i, whee m it i the angula peed of the taction moto oto, it eult that D v = m (3) i The elation (1) and (3) ae fundamental in the taction calculation. They pemit the etablihment of the vehicle chaacteitic depending on the ueful toque quantity M and on the angula peed of the haft of the all it taction moto (in the equality cae of the diamete of the all moto wheel). In the unning, both unde it taction moto action and unde the ail eitance influence, it achieve the ueful tanlation movement of the all vehicle in the long ail. Fig.1 Extenal foce which they etablih the movement Moeove, the ueful movement of the tain it i etablihed only of the extenal foce action. Thee can be (fig.1): - moto active foce (of taction), with the eultant F t (of contollable magnitude, which they opeate on the ene and diection of the ueful movement); - baking active foce, with the eultant F f (of contollable magnitude, which they opeate on the 67

2 ueful movement diection, but on the contay ene of the peed vecto v ) and - tain eitance foce with eultant R. The active foce F t (of taction) and F f (of baking) they opeate not imultaneouly in none unning egime (peence of one it i equivalent to the excluion of othe), while the tain eitance R it i peented all the time, even in the active foce abence (in the coating egime without cuent). Modelling of taction moto In the powe cheme of moto electic vehicle, the taction induction moto it i the final element fom the equipment chain by conveion of enegy. Afte all, it achieve the electomechanical conveion of enegy making thu poibly the movement. Like complex electomechanical ytem, the induction moto will have decompoed functionally between an electomagnetic pat and a mechanical pat (fig.). Both the M electomagnetic toque and the m mechanical peed of oto, they intevene like intenal vaiable, by inteaction, between thoe two functional Fig. Functional pat of taction induction moto pat. In the moto vehicle cae, the mechanical pat of taction induction moto it i coupled (though the tanmiion intemediation) with the motive axle and can be modelled in the hape of the ueful movement o/and the elatic mechanical tanmiion [1]. Fo can be connected, the model mut be achieved in accodance to the ame pinciple, indiffeently that they decibe the phenomenon by electic natue o mechanical natue. In the choice cae of fixed efeence ytem, connected at tato, the electomagnetic pat of induction moto it will have decibed by the equation [5], [6], [7], [8], [9]: d! = u - R i dt d! = j p m! - R i (4) dt! Lu!! - Lu -! i = L ; i = L ' L ' L 3 * M = p Im{ i! } whee u = tato voltage vecto i = tato cuent vecto i = oto cuent vecto = tato flux vecto = oto flux vecto L u = magnetizing inductance L = tato inductance L = oto inductance p= numbe of pole pai R = tato eitance R = oto eitance and Lu ' = 1- = leakage total coefficient of moto. L L On the equation bai (4) they ae epeented the tuctual diagam and mak block of the electomagnetic pat of induction moto (fig.3). Fig.3 Stuctual diagam and mak block fo electomagnetic pat of induction moto 68

3 The tuctual diagam of electomagnetic pat can be coupled both with the tuctual diagam of machine convete though the intemediation of the input vaiable u and output vaiable i and with the tuctual diagam of mechanical pat though the intemediation of input quantitie m and output quantitie M. 3 Modelling of Ueful Movement The tain eitance foce, between with the taction chaacteitic of the high-peed tain, they pemit the ueful movement tudy of the vehicle. In thee condition the ueful movement equation i [1], [], [3], [9] * dv * m = F - R; m = m. (5) dt whee F it i F t in taction egime o - F f in baking egime, and D i the coefficient of inceae the ma of the tain that take account to the peence and weight of the tuning pat fom the tain tuctue (D= ). Thu, though the coefficient of inceae the ma agency it can make abtaction of the tuning pat peence, eplacing the eal ma "m" of the tain with a "fictitiou ma" m*=m. found in tanlation movement with the peed "v", the ame of the conideed tain. Fom the phyical viewpoint, thi i equivalently with fictitiou eplace of the mechanical ytem of igid olid pat though a mateial point with inetial ma m*=m.. Fo the dynamic apect appoach of the ueful movement it i neceay a mathematical model. In thi pupoe it i conideed a electic vehicle of ma m[t] and coefficient of inceae the ma D having the tain eitance [dan/t]. On the ueful movement duation, the peed v(t) and the ditance x(t) they ae uled at the equation dv dx m. = F - R ; = v (6) dt dt If the movement ha been made unde the ueful toque action M (identical), developed of thoe "z" taction moto of the electic vehicle, then in accodance with the elation (1) and (3) i m = v ; F = z i t M (7) Moeove, if the ma m of the tain it i expeed in [t], the total tain eitance R[N] it i etablihed with R [N] = [dan/t] m[t] 10 (8) The equation enemble (6), (7), (8) they fom the mathematical model of the ueful movement. Witten togethe, in the hape of 1 i v = (F-R)dt ; m = v; x = v dt ; m. (9) F= z i M ; R = ( p(v) ± ide(x)+c(x)) m 10 t they pemit the tuctual diagam contuction of ueful movement (fig.). Fo the mak block they have been conideed like input quantity the M toque and like output quantity the m peed, time vaiable quantitie on the ueful movement duation. By mean of thi cheme ("coupled" at the tuctual diagam of electomagnetic pat of taction moto fig.3) can be imulated the ueful movement of any electic vehicle a compaed with the concete modality by leadehip (o by contol) of thi. Accodingly they ae obtained the unning diagam v(t) and x(t), too. The modification of vehicle ma, of dependence i de (x) o c (x), pecific to cetain vehicle o oute, can be eaily opeated, obtaining an exact mathematical model, which it epect all the unning condition. In the moto wheel diamete inequalitie cae, the cheme uffe a mino change, the total foce F eulting like um of patial foce developed by each moto patly. UM Fig.4 Stuctual diagam and mak block of ueful movement 69

4 4 Simulink Model of Ueful Movement The SIMULINK model coeponding to the ueful movement of the electic vehicle can be eaily implemented, having with a view the topological compaion with the aociate tuctual diagam. In the taction electic moto conideation cae, too, at the electomagnetic pat model MAT it i coupled the block UM (fig.5). The two block, fo the electomagnetical pat and fo the ueful movement pat, ha been achieved on the coeponding tuctual diagam bai (fig. 6). Within the famewok of the ueful movement (fig.4), the tanmiion ha been uppoed ideal. In imulation in which they have been deied to be taken account the othe mechanical phenomenou (like example the tick-lip [10]), too, can be intepeed the affeent elatic mechanical tanmiion model [1], [], [3]. UM Fig.5 Model of taction induction moto at conideation ueful movement of tain a) mak block MAT b) mak block UM Fig.6 SIMULINK model a) fo electomagnetic pat of taction induction moto; b) fo ueful movement 70

5 Fig.7 Taction and baking chaacteitic and 4 value of tain eitance of ETR 500 italian high peed tain eitance, m=600 t i de =0; b) i de =5; c) i de =10; d) i de =0 5 Simulation An immediately example of the ueful movement model it contitute it the unning diagam dawing, which they illutate the dynamic apect of the dynamic apect of the electic vehicle. The unning diagam of the electic tain they ae dawn on the taction and baking chaacteitic bai and of the condition impoed of oute. The taction and baking chaacteitic conot any peentation, howeve ummaily would be, of a high-peed tain. Like example, it ha been conideed the ETR 500 italian high peed tain cae, fo which, except the taction and baking chaacteitic, they have moe epeented the tain eitance, too, coeponding to the diffeent pecific featue of the oute (though the declivitie conideation) (fig.7). Fo the unning diagam dawing coeponding to a oute of the ETR500 it i ued a SIMULINK model baed on the ueful movement model (fig.8). In the block F t (v) and F f (v) they ae implemented the taction and baking chaacteitic (fig.7). The model i baed on the ueful movement model, at which the main input vaiable it i upplied of the block F, which it model the ueful movement phae: - tating phae, - unning at contant peed phae, - coating phae and - baking phae. Fig.8 SIMULINK model ued fo unning diagam dawing 71

6 Fig.9 Running diagam of high peed tain By mean of thi SIMULINK model (fig.8) they have been dawn the unning diagam coeponding to a tet oute of the high-peed tain, on it oute ha been eached the peed of 44,44 m/ (160 km/h) and epectively of 83,33 m/ (300 km/h). They have been dawn the vaiation of the foce, of the tain eitance, of the acceleation, of the peed and of the ditance (fig.9). In the peviou cae ha been taken account only the main tain eitance coeponding to a oute with condition fom cae a, fig.7. The next imulation efeed to the ame highpeed tain, with the ame pefomance (taction and baking chaacteitic), fo which it i deied the peed vaiation obtainment on a ditance of 100 km. They ae conideed anothe value of the tain eitance etablihed of the diffeent declivitie value exitence, too (fig.7). The maximum eached peed they coepond the conjunction between the taction foce and adequate tain eitance, excepting the cae a) (fig.7) whee the maximum peed ha been limited at 83,33 m/ (300 km/h). Fig.10 SIMULINK model ued etablihment of baking ditance 7

7 Fig.11 Speed diagam fo diffeent value of tain eitance (m=600 t) a) i de =0; b) i de =5; c) i de =10; d) i de =0 Fo the epecting of 100 km ditance mut achieved an obtainment algoithm of the point fom which the tain tat to bake. Fo thi it i ued a SIMULINK model (fig.8), what it pemit the detemination of the baking ditance when it i known: the baking beginning peed, the baking foce (vaiable depending on peed), the ma m of the tain, the tain eitance (vaiable depending on peed and ditance). It i thu poibly the knowledge of the point fom which it begin the baking and the topping at fixed point. Fo thoe, fo cae (fig.7), by mean thoe two SIMULINK model (fig.9 and fig.10), they have been imulated the peed diagam (fig.11), all coeponding to a ditance of 100km. They have been obtained the time of 1500 (cae a), of 151 (cae b), of 163 (cae c) and of 198 (cae d). 5 Concluion A compaed to othe model of the ueful movement of the electic vehicle, which they ae epecially fo the taffic contol, [13], [14], [15], [16], [17] the mathematical model peented in thi pape ha a high geneality degee. The tuctual diagam of ueful movement (fig.4) can be eay coupled at the tuctual diagam of the electomagnetic pat of the taction moto, being thu poibly a fomed contuction aociated of ailway vehicle. Hee it i illutated the taction induction moto cae but, only though the modification of the electomagnetic pat, can be tudied the movement of the high peed tain with D.C. taction moto and even ynchonou taction moto. Alo, the model can be eaily adapted of the concete condition of the electic vehicle o the oute. The tuctual diagam (fig.4), elatively eay of implemented and of ued, it ha a geat impotance in taction, can be ueful in the etablihment of a contol method, on a pecial oute, baed on the bet utilization of the intalled load. The epective method ae implemented then in the compute contol ytem on the electic vehicle, contibuting at the ciculation afety inceae, at the conumption deceae and pemitting even a poibly ATC (Automatic Tain Contol). The model pemit, though the pecific featue intoduction of the oute fom any county within the famewok of the ueful movement model, the teting and the compaion of the pefomance of diffeent poible vaiant of the electic vehicle. The topological imilitude between the tuctual diagam and the SIMULINK model it pemit an eay implementation in SIMULINK, both fo the peented tuctual diagam and fo the imulation of a concete ituation. In concluion, the peented model i ueful in: - contuction of the unning diagam; - the taffic contol; - the implementation of a new contol method of the tain moving; - the implementation of a diagnotic ytem; - contuction of a main electic cicuit model of the vehicle [18]; - the tudy of a anti-lip contol method; - the oute influence about the electic component woking of the tain. - the achievement of the calculu model of the conumed enegy of tain etc.; - optimization of the tain movement. Thee element how the advantage of the popoed method and open a new diection fo futue eeach. 73

8 Refeence [1] D.A. Nicola, D.C. Cimau, Electic Taction, Ed. Sitech, Caiova, 006. [] R. Kalle, J.M. Allenbach, Taction électique, PPUR, Lauanne, [3] F. Peticaoli, Sitemi elettici pe i tanpoti. Tazione elettica. Ed. Maon, Milano-Paigi- Bacellona, [4] D.C. Cimau, D.A.Nicola, Gh. Manolea, M.A. ighiciu, Simulation of Electic Vehicle Movement, WSEAS POWER'07, Venice, Italy, Novembe 1-3, 007 [5] D.A. Nicola, D.C. Cimau, Stuctual Diagam fo Induction Machine, Annall of Univeity of Caiova, No.0/1996, pg (1997). [6] D.A. Nicola, Electic tanfome. Induction machine, Ed. Sitech, Caiova [7] D.A. Nicola, D.C. Cimau, Modelling and Simulation of the atuated Taction Induction Moto, Poc. ICATE 000, Caiova, (000). [8] A. Câmpeanu, Intoduction in AC electic machine dynamic, EA, BucueOti, [9] I. Boldea, Electic tanfome and machine, EDP, BucueOti [10] I. SebeOan, Railway vehicle dynamic, Editua TehnicP, BucueOti, 1996 [11] H. Bühle, Réglaje de ythme d'électonique de puiance, Vol.I, PPUR, Lauanne [1] Chee-Mun Ong, Dynamic Simulation of Electic Machiney Uing Matlab/Simulink", Pentice Hall Inc., New Jeey [13] A.Zimmemann, G. Hommel, A tain contol ytem cae tudy in model-baed eal time ytem deign, Paallel and Ditibuted Poceing Sympoium, 003. [14] U.S. Depatment Of Tanpotation, Fedeal Raiload Adminitation, Development of a Geneal Tain Movement Simulato fo Safety Evaluation, Reeach Reult, RR07-01 Febuay 007 [15] S. Guangwei, R. Meiinge, S. Gang, Modeling and Simulation of Shanghai MAGLEV Tain Tanapid with Random Tack Iegulaitie, Sondeduck Schifteneihe de Geog-Simon- Ohm-Fachhochchule Nünbeg N. 39, Juli 007 [16] L. KePing, G. ZiYou,, An impoved equation model fo the tain movement, Simulation Modelling Pactice and Theoy Volume 15, Iue 9, Octobe 007, Page [17] M. MandiT, Z. PopoviT, I. UglešiT,, Calculation of powe upply fo electic ailway ditibution ytem, IYCE 007, Budapet, Hungay [18] D.C. Cimau, D.A.Nicola, Gh. Manolea, M.A. ighiciu, Modelling and Simulation of Locomotive with Taction Induction Moto and Thee Level Convete, WSEAS ISTASC 07 Vouliagmeni, Athen, Geece, Augut 4-6,