| N G H I E N C L f U K H O A H O C
^NGHIEN CUU PHUONG PHAP TINH TOAN SUC KEO CUA TAU DIEN C H A Y T R E N D U O N G R A Y
REASEARCH METHODOLOGY OF CALCULATION TACTION OF ELECTRIC TRAINS R U N N I N G O N R A I L S
NCS. Trjnh Lirong Mien. E-mail: trinhmienfa gmail.com
Dai hoc tong hop giao thong duxmg sat Matxcova (MIIT)
Phan bien 1. PGS.TS. LE HUNG LAN - Trirong Dai hoc Giao thong Van tai
Phan bien 2. PGS.TS. PHAM ANH T l AN - Vien Cong nghe vu try - Vien Khoa hoc va Cong nghe Viet Nam
Tom tat
Bai bao nay dua ra phuong trinh toan mo ta chuyen dpng tau dien va cac gidi han vat ly cua no. Dua vao do. tac gia nghien cuu hai phuong phap tinh toan sue keo: giai tich va xap xi so. Phan tich ket qua nghien ci'ai cho thay. phuong phap so Ole dupe su dung kha hieu qua.
Abstract
In this paper, mathematical equation of motion and physical constrains of electric train is proposed to calculate traction of trains. Two methods of calculation are studied, such as analysis method and numerical method to approximate the solutions. The results demonstrate that Euleris numerical method is usued quite efficiently.
Thiet lap nhiem vu
Cling vdi su tang tnrong kinh te tai Viet Nam. dong ngudi do ve cac khu do thi tai cac thanh pho Ion ngay cang nhieu. Keo theo do la cac loai hinh tham gia giao thong ngay cane gia tang, ma kha nang phan luong giao thong de khdng xay ra un tac la bai toan rat khd.
Cling da co nhieu giai phap duoc dua ra nham giam toi da cac loai hinh phuong tien di chuyen tren mat dat. Va viec xay dung tau dien chay tren dudng ray la mot lira chon tot nhat de van tai hanh khach chinh yeu trong thanh pho. Van tai hanh khach bang tau dien chay tren ray cho phep chuyen chd mot khdi lupng hanh khach rat Ion. an toan va van minh. No se gop phan phan bo lai mat dp ngudi tham gia giao thong.
ciing nhu thay doi thdi quen su dung cac dich vu van tai hanh khach cong cong (YTHKCC) cua ngudi dan do thi.
Dac thu cua YTHKCC bang tau dien la chuyen chd hanh khach tir nha ga nay den nha ga khac (khoang each giiia hai nha ga duoc gpi la khu gian) tren tuyen duong ray doi doc lap theo ehieu xac dinh. Mot qua trinh chuyen cho hanh khach bang tau dien tir nha ga A (vj tri SI) den nha ga B (vi tri S2) dupe gpi la mot chu trinh chay tau. hinh 1.
Hinh I: Chu trinh chay tau dien tren khu gian (LK-luc keo.
OT-qudn tinh. HM-ham)
Chu trinh chuyen dong cua tau la ACB. gdm hai giai doan: chay vdi sue keo Ion nhat AC. sau do ham vdi lire ham Ion nhat CB (chu trinh nay tuong ung vdi thdi gian chay nho nhat). Nhung trong thuc te. van toe chay tau tren khu gian ludn bi gidi han boi Vm a x. Khi do. chu trinh van hanh tau tau dien se la ADGB. bao gdm ba giai doan: tang tdc vdi luc keo Idn nhat cho den khi dat van toe Ym a x (quy dao AD): duy tri luc keo de tau chuyen ddng vdi van tdc Vm a x (quy dao DG); ham GB vdi lire ham Idn nhat. Ydi mong mudn tiet kiem nang lupng. tan dung ddng nang tich luy trong thiet bj chay tau. trudc khi thuc hien ham.
chiing ta ngimg cap nang lupng dien. luc nay tau se chay theo quan tinh (quy dao EH), sau do tien hanh ham cue dai (quy dao HB). Khi chay theo quan tinh.
chuyen ddng cua tau dupe dam bao an toan thong qua he thong giam sat toe dp. Nhu vay. chiing ta thu dupe chu trinh chay tau trong trudng hop nay la: ADEHB.
Chu trinh chay tau tiet kiem nang lupng nay bao gdm bdn giai doan.
Van de dat ra a day la tim tpa dp cac diem (vi tri quang dudng trong hanh trinh chay tau) chuyen tu che dp lai tau nay sang che dp lai tau khac (tpa dp diem D.
E. H. G). Dieu nay dan chiing ta den viec can phai tinh toan sure keo cua doan tau. tiic la can phai biet dupe phuong uinh chuyen ddng cua doan tau chuven ddng tren khdng gian. Viec tinh toan sue keo doan tau dua vao md hinh hoa qua trinh chuv en ddng tau dien la phuong phap hieu qua nhat. la lira chpn duy nhat de phan tich. tong hop va md phong qua trinh dieu khien chuyen ddng cua tau dien. Bang phuong phap nav.
ngudi ta xac dinh dupe quy dao chay tau (tinh toan quy dao van tdc chuyen ddng theo quang dudng). thdi
Tu dong hoa ngay nav 7. 2011
gian chay tau (de xay dung ke hoach chay tau) va nang luong tieu hao trong qua trinh chuyen dpng,.. .Cung tir day, cho phep chung ta giai quyet bai toan toi uu nang lupng chay tau tren khu gian. bai toan toi uu nang lupng chay tau tren tuyen, bai toan
w , Ifi-
dieu dp toi uu trong toan mang ludi duong sat thanh pho.
T h a n h l a p p h i r c m g t r i n h c h u y e n d o n g Nghien cuu doan tau sue keo dien gdm mot toa dpng co (toa sinh luc keo/luc ham dien) va n toa keo theo (toa khdng sinh ra luc keo) nhu hinh 2.
— • V
m i Wo,
Oi x0 xn/o Qn
Xn
Hinh 2: So do tinh toan sice keo dien Nghien cuu chuyen dpng ciia doan tau o dang lien ket
"thanh cung" giua cac toa, tren he true toa dp trung vdi hudng chuyen dpng: x0-toa dp trong tarn cua doan tau; xn-toa dp trong tarn cua toa n; Xn/0-toa dp tuong doi giua trong tarn ciia toa n so voi trong tarn doan tau.
Ta co phuong trinh vi phan the hien moi lien he ve gia toe cua doan tau nhu sau:
d2xn d2xn d 2x + - n/0
~ a 0 + 3 n/0 (1 dt dr dt
Ta nhan thay, su dch chuyen cua diem trong tarn doan tau (voi gia toe a()) se lam cho doan tau chuyen dpng, do la dich chuyen co ich. Ngupc lai, dich chuyen tuong doi ciia diem trong tarn toa n (voi gia toe an/o) khong lam cho doan tau lan banh (tren he toa dp co dinh gdc O), ma chi lam tang nguy co mat an toan cho doan tau. Bo qua djch chuyen gay hai nay, ap dung dinh luat 2 Newton, ta co phuong trinh chuyen dpng ciia doan tau:
dx
— = v dt
dv ( 2 )
(1 + y)m — = F ( v ) - W ( v , x ) - B , ( v ) dt
trong do: Q=XQi; M=P+Q [kg]; y- he so quan tinh doan tau; dx/dt=v-van toe chuyen dpng cua doan tau [m/s]; F(v)- luc keo doan tau [N]; Bt(v)=B(v)+R(v)- tdng luc ham doan tau; B(v)-luc ham doan tau do phanh co khi sinh ra [N]; R(v)-lirc ham doan tau sinh ra boi bp phan ham dien; W(v,x)=W0(v)+Wd(x)- tong lire can tac dpng len doan tau; W0(v)-lyc can chuyen dpng chi'nh [N]; Wd(x) luc can chuyen dpng bo xung.
gay ra bdi goc nghieng, dp cong cua duong ray, sue can ciia ham [N];
Khi nghien cuu chuyen dpng cua tau tau dien. ngudi ta thudng bien ddi phuong trinh (2) o dang luc tac dpng rieng va nghien cuu mdi lien he giua van tdc vdi quang dudng tau di dupe s = x trong he don vj do luong cua nganh dudng sat: s[m], v[kmh], t[s], fm(v)=eFm(v)/M [N/kN]-luc keo chuyen ddng rieng cua doan tau, Fm(v)-dac tinh keo Idn nhat;
bm(v)=eBm(v)/M [N/kN]-luc ham co khi rieng cua doan tau, Bm(v)-dac tinh ham co khi 1cm nhat:
rm(v)=sRm(v) M [N kN]-luc ham rieng bang eo cau
doan tau 1 toa ddng co va n toa keo theo
phanh dien cua doan tau, Rm(v)-dac tinh ham bang phanh dien Idn nhat; w(v,s)=w0(v)+wd(s)- tong lire can rieng len doan tau; w0(v)=sW0(v)/M [N/kN]-luc can co ban cua doan tau; wd(s)=eWd(s)/m [N/kN]-lyc can bd xung ciia doan tau; £=£,/( 1 +y)-he sd bien ddi don v j do luong. Trong trudng hop nay phuong trinh (2) chuyen ddi thanh:
' dt_ _X6_
ds v dv (3)
ds
fm (V) " Urrm (V) ~ U' bh>n, (V) ~ W( V> S)
trong do: ur = F/Fm(v)- tac ddng dieu khien lire keo doan tau; ur = R/Rm(v)- tac ddng dieu khien luc ham phanh dien doan tau; ub = B/Bm(v)- tac ddng dieu khien lire ham phanh co khi doan tau;
Ta cd cac dieu kien ve tac ddng dieu khien nhu sau:
0< uf <1; 0< ur <1; 0< ub <1; uf.ur = 0.
Tu phuong trinh (3) de dang nhan thay, he thong dieu khien tau dien la phi tuyen "ddng", khdng tu tri (non- automon). Vi vay, viec nghien cuu va phan tich no se rat phtic tap. Khi ur = 0 ta cd he phuong trinh chuyen ddng doan tau chi sir dung ham bp phan ham co khi.
Tinh toan sire keo tau dien
Viec tinh toan siic keo tau dien (luc keo/luc ham/luc can ...) la qua trinh giai phuong trinh vi phan chuyen ddng doan tau, tire la xac dinh mdi quan he giua van tdc vdi quang dudng tau chay (bieu do chay tau theo khoang each) va xac dinh thdi gian chay tau.
Phan tich phuong trinh vi phan chuyen ddng doan tau va cac nghien cuu trong [1-6], nhan thay cd hai nhdm phuong phap chi'nh de giai, do la: phuong phap giai tich va phuong phap giai sd. De ap dung dupe phuong phap giai tich ddi hoi phai cd mot sd gia thiet nhat djnh: luc keo/ham phai bieu dien duoc dang phuong trinh giai tich, hinh dang dudng ray (profm-dd nghieng/ddc/cong ...) khdng qua phuc tap. Trong khi do phuong phap so cho phep tinh toan vdi bat ky dac tinh luc keo/ham, dang profin va ngay ca khi cd gidi han van tdc chay tau.
3.1. Phuang phap giai ti'ch
Xet tau chuyen ddng tren quang dudng cd dp ddc khdng ddi: wd(s)=hang sd. ddng thdi. bieu dien duoc
Automation Today / 7, 2011
quan he luc keo. luc ham. luc can doan tau dudi dang bieu thuc giai tich lien he voi van toe. tue la:
ulf J v ) - i i rr j Y ) - i t J i J v ) - M v . s ) = f ( r ) (4) Tir (3), ta co dt=dv f(v). sau khi tich phan hai ve ta thu duoc van toe chay tau la ham cua thdi gian:
v = <p(t) (5) Phuong trinh (5) cho phep chung ta tinh dupe thdi
gian chuyen ddng cua tau tren mot doan dudng cho trudc (dp nghieng khdng ddi) vdi su thay ddi van tdc xac dinh va cho trudc.
Tiep tue bien ddi va ket hpp vdi (5). de y rang v = ds dt. t = q> (s), ds=vdv f(v). ta cd:
v = (p(s) (6) Bieu thuc (6) chinh la quan he giua van tdc chay tau
va quang duong tau da di qua. Tiep rue chia nhd quang dudng tau chay ra thanh tirng doan cd dp doc khdng ddi, chiing ta cd the tinh duoc lire keo cho doan tau chuyen dpng den cudi ga. Tuy nhien phuong phap nay chi ap dung cho cac doan tau cd dac tinh keo.
ham don gian va protin don gian. chua tinh den van tdc gidi han chay tau.
3.2. Phuong phap so
Thuc te chuyen ddng tau ludn bj gidi han ve van tdc khi tau vao ga. rdi ga. hay d nhung doan xudng ddc nguy hiem.... Chiing ta cd the bieu dien dieu kien nay nhu sau:
v<vm(s) (7)
Ciing vdi dieu kien (7). viec giai phuong trinh chuyen ddng tau cang tro nen khd khan. Cac nghien cuu [1-4]
chi ra rang, phuong phap hieu qua nhat de giai phuong trinh (3) la phuong phap xap xi sd nhd su trp giup cua cac thiet bi ky thuat. Phuong phap sd thudng dupe ap dung de giai (3) la phuong phap Ole [5].
Ngoai ra de nang cao dp chinh xac ngudi ta cd the ap dung phuong phap Runge-Kutta, Adams. Milne, ...[5]
hay phuong phap khai trien chudi Taylor [6].
Vdi phuong phap sd Ole. ta cd the tich phan phuong trinh chuyen ddng tau theo thdi gian. theo quang dudng, hoac theo van tdc.
;+l
°i ~ G , = f
= v, +ar(tM - t :) (9) sM = si + ( v M+ vi ) . ( tM- t i ) / 7 , 2
3.2.2. Giai phirong trinh theo biroc tich phan quang dudng As
Neu giai phirong trinh chuyen ddng theo budc tich phan la quang dudng As. khi ay:
t = 3,6 [ds v = 3.6 l i m V As, I v , ; J A/,.->0 ^ ' '
f ( 1 0 >
v = ja.ds v = l i m at .As, v(
Neu budc tich phan quang dudng du nhd thi at coi nhu la hang so. khi do chiing ta cd cac bieu thuc tinh toan sau:
a, * a, = f
vi+] = v , + 2.ar(sM - s s) / ( vM + Vf) hay
tM =ti + 7 , 2 . ( jf + J - 5f) / ( vf + 1 + vf) 3.2.3. Giai phuong trinh theo buoc ti'ch phan van toe Av
Neu chiing ta ehpn budc tich phan de giai (3) la van tdc Av, thi ta cd:
t = ids a = l i m V A v la ;
jv.dv (3.6.a) = lim ^v,..Av, /(3,6.a,) n : Khi budc tich phan van tdc du nho thi at coi la hang sd. Gpi fi la lire tac ddng don vi trung binh len doan tau o budc thay ddi van tdc thir i: Av,, luc nay ta cd:
3.2.1. Giai phuong trinh theo buoc tich phan thoi gian At
Cluing ta gpi a [(kmh) s] la gia tdc chuyen ddng cua tau, khi do:
v = \adt = l i m V a At, :
s = \vdt = lim V vAt (8)
Neu At, du nho. va gia su tinh toan dupe lire don vj tac ddng vao doan tau tai thdi diem t, la:
/; = u rfm( vi) - urrm( v . ) - ubbm( vi) - w ( vnsi) Khi do. ta cd cac bieu thdc tinh toan dudi day:
«, = f
5,., =j,.+vf.(ri+1 -t,)/3,6
(13)
Bang phuong phap Ole. viec tinh toan luc keo kha don gian. thdi gian tinh toan nhanh. sai sd phep tinh phu thupc chu yeu vao dp Idn cua budc tich phan dupe lira chpn va thdi gian tinh toan dat ra.
Ngoai ra. ngudi ta con ap dung phuong phap Runge- Kutta bac bdn de xap xi nghiem cua phuong trinh (3) theo cdn« thuc dudi day:
vM = v +(kl+2k2+2k3+k4)/6 ,14)
•>,_, =t, +7.2.A5. (v:_, + v , )
Tu dong hoa ngay nay 7, 2011
trong do:
/ ( v , ,s,) = u, f j v , ) - urm (v,) - ubbm (v,) - ut V;,s,);
= A sif ( vi, si) / vi;
/ ( v , . + 0 , 5 ^ , s , + 0 , 5 A v , ) . k2 = As
k3 = As:
kA =Asi
v, + 0,5*,
/ ( v , . + 0 , 5 £2, ^ + 0 , 5 As,) v, + 0 , 5 £2
/ ( ^ +k3,sk +As,)
luc can Wo
80 100 0 20 40 60
van toe v(km/h]
///«/? 3. Dac tinh lire keo va dac Qua trinh tinh toan sue keo doan tau se giup ta xay dung duoc quy dao chuyen dong doan tau tren khu gian. Ngay nay, nhd su trp giup cua cac thiet bi tinh toan toe dp cao, ngudi ta thudng sir dung phuong phap giai sd true tiep de giai phuong trinh vi phan chuyen ddng doan tau. Phuong phap giai sd nay ket hpp vdi cac nguyen ly cue dai, nguyen ly quy hoach ddng, ... cho phep nghien cuu nang cao hieu suat van hanh tau, nhu tdi uu lai tau theo chi tieu tiet kiem nang luong dien; dieu dp tdi uu cac tau tren mot tuyen va tren cac tuyen, ...
Hinh 4 the hien quy dao chay tau tren khu gian cd chieu dai 875m; khu gian nay cd ba muc gidi han van tdc: 45km/h khi rdi khoi nha ga vdi do dai 200m. muc gidi han tiep theo la 7()km/h vdi chieu dai 550m va muc gidi han cudi ciing khi vao nha ga tiep theo la 45km/h vdi do dai 125m.
Phuong phap Runge-Kutta cung cd the tien hanh theo budc tich phan thdi gian hoac van tdc gidng nhu phuong phap Ole. Vdi cac thu nghiem cung du lieu dau vao va sai sd cho trudc, thdi gian tinh toan ra ket qua theo phuong phap Runge-Kutta gap 5-10 lan so vdi phuong phap Ole, phuong phap nay cung ddi hdi dung lupng nhd bp nhd Idn [7].
4. Ket luan
Be minh hoa cho nghien cuu tren, chiing ta tinh toan quy dao chuyen dpng cho doan tau 6 toa, mdi toa duoc truyen ddng bdi 4 ddng co dien xoay chieu cd dac tinh dupe nhu hinh 3.
R(v) I 15
tr fl 10 s
1 | £ 5 : 32.5[T1 Tare/toa rong 7t : 45.9fT) Raled/dlnh muc
•9 / 57.7[T1 Max/lrong lai toi da O — •
0 10 20 30 40 50 60 70 80 90 100 van toe v(km/h]
tinh lire ham eita I dong co dien
muc gidi han cudi cung khi vao nha ga tiep theo la 35km/h vdi do dai 103m.
Thoigian 87 846[s]
Quang duong s(m|
Hinh 4. Quy dao chay tau tinh theo phirong phap Ole Theo ket qua tinh toan tren hinh 4, hanh trinh chay tau tren khu gian la ehudi ke tiep cac che dp lai tau xac dinh trong rung khu vuc bi gidi han van tdc. do la:
LK-C, LK-QT-HS,QT-HS-HM. Trong do: LK-dieu khien luc keo, C-chay on dinh tdc dp Lie keo. QT- chay theo quan tinh, HS-chay a che dp ham tai sinh.
HM- che do ham dung bang co cau co khi.
Khi ung dung phuong phap Runge-Kutta de tinh toan sue keo, xay dung quy dao chay tau. ta ciing thu dupe ket qua nhu tren hinh 5. Trong ket qua nay. khu gian khao sat cd chieu dai 1103m. cd ba muc gidi han van tdc: 65km h khi rdi khoi nha ga vdi dp dai 750m. muc gidi han tiep theo la 50km h vdi chieu dai 250m va
Quang duong s[m]
Hinh 5. Tinh toan quy dao chay tau theo Runge-Kutta Dua vao quy dao tinh toan tren hinh 5, ta cd the dua ra trinh tu de lai tau tuong ting vdi ba muc gidi han van tdc tren, do la: LK-QT-HS, QT-HS, QT-HS-HM.
Qua nghien cuu va phan tich sd lieu thu dupe, tac gia nhan thay ket qua tinh toan theo phuong phap Runge- Kutta thudng cham hon phuong phap Ole khoang 5- 10 lan vdi ciing du lieu dau vao va dp chinh xac tinh toan cho trudc. Tuy nhien phuong phap Runge-Kutta cho thay dp hdi tu nghiem bai toan cao hon. va ket qua cd dp chinh xac tot hon vdi ciing sd budc tinh theo phuong phap Ole.
Tai lieu tham khao
[1] Brenan, K.E.; Differential-Algebraic Equations Issues in the Direct Transcription of Path Constrained Optimal Control Problems, Aerospace Report, No. ATR-94 (X489)-l. December 1993. American Railway Engineering and Maintenance-of-Way Association (AREMA), 1999.
[2] Andrews, H. I, Railway Traction The Principles of Mechanical and Electrical Railway Traction. Elsevier. New York, 1986.
[3] BapaHOB JI.A., EpocfteeB E.B.. AcTpaxaH B.H.. MSKCHMOB B.M. H
ap. CucmcMH aemoMammecKoeo u meneMexaHunecKozo \npanlemm
•jneKmponodsuoic-HbiM cocmaeoM, TpaHcuopT, 1984.
[4] JI.A. EapanoB. d.M, ro/iOBHiep. E.B. EpoijieeB. B.M. MaKCHMOB,
MiiKponpoifcccopubic cucmeMbi aemoeedemiH i.TeKmponodeujKHoaa cocmaea; TpaHC-nopT. 1990.
[5] ZteMHUOBiiw B.n.. HuaietmueMemodu aiiaiuia -\1 Ha\Ka. 1967.
[6] Ralston A. A first course in numberical analwic McGraw-Hill, New York. 1965
[7] BapaHOB JI.A., EajiaKHHa E.n.. BopoobeBa Jl.H. AieopumMbi 0m noeddoe Mempononimeua. Mnp TpaHcriopTa. 2007 i . .Vj2. c. 104-11 3.
Automation Today / 7, 2011
