XXX -X-XXXX -XXXX -XXXXX.00 20XX IEEE Energy Efficient Train Operation using [612581]

XXX -X-XXXX -XXXX -X/XX/$XX.00 ©20XX IEEE Energy Efficient Train Operation using
Simulated Annealing Algorithm and
SIMULINK model

Daniel Cristian CISMARU
University of Craiova
Faculty of ElectricalE ngineering
Craiova, ROMANIA
[anonimizat]
Abstract —This paper presents an energy -efficient train
operation based on the mathematical model of the train
motion. Starting from the mathematical model it is built the
structural diagram and the SIMULINK model associated of
the train motion. By using the Simulated Annealing Algorith m
it is calculated the minimum value of the consumed energy for
an electric train route
Keywords — modeling; simulation; Simulated Annealing
Algorithm; SIMULINK; energy efficient train operation
I. INTRODUCTION
To calculate and optimize the energy consumption of an
electric vehicle through the chosen driving strategy it is
necessary to build the motion model of the vehicle [1].
The motor torque transmitted of the motor wheels it is
MR=i·η t·M2, where M2 is the developed useful torque of the
traction motor, i is the transmission ratio and ηt is the
transmission efficiency. At the wheel radius r=D r/2, the
motor torque MR it corresponds it the motor force F0[N] at
wheels [4],[5]
2 0 M iD2 = /2DM = Ft
r rR
(1)
In the slip absence, the peripheral speed v of the motor
wheels (which they are turned with the angular speed Ω0)

2D = vr0 (2)
it is the same with the t ranslation motion speed (on the rail)
of the vehicle. As Ω0=Ω m/i, where Ωm it is the angular speed
of the traction motor rotor, it results that
iD = vm r

2
(3)
The relations (1) and (3) are fundamentals in the traction
calculations. They allow the establishment of the vehicle
characteristics depending on the useful torques quantity M 2
and on the angular speed of the shafts of the all its traction
motor (in the equality case of the diameters of the all motor
whee ls). In the motion, both under its traction motors action and
under the rail resistance influence, it achieves the translation
motion of the all vehicle in the long rail.
Moreover, the train motion it is established only of the
external forces action:
– motor active forces (of traction), with the resultant
Ft

– braking active forces, with the resultant
Ff
– train resistance forces with resultant
R .
The active forces
Ft (of traction) and
Ff (of braking)
they operate not simultaneously (presence of one it is
equivalent to the exclusion of other), while the train
resistance it is presented all the time, even in the active
forces absence (in the coast ing regime).
In these conditions the train motion equations is [4], [5]

  m = m ; RF = dtdvm* * (4)
where F it is Ft in traction regime or – Ff in braking regime,
and ξ is the coefficient of increase the mass of the train th at
take account to the presence and weight of the turning parts
from the train structure ( ξ=1.06…1.2).
For the dynamic aspect approach of the train motion it is
necessary a mathematical model. In this purpose it is
considered a electric vehicle of mass m[t] and coefficient of
increase the mass ξ having the specific train resistance
r[daN/t]. On the train motion the speed v(t) and the distance
x(t) they are ruled at the equations

v = dtdx ; R – F = dtdvm (5)
If the train motion has been m ade under the useful
torques action M2 (identical), developed of those " z" traction
motors of the electric vehicle, then in accordance with the
relations (1) and (3)
2M i
D2z = F ; v
Di2 = t
r rm   
(6)

Moreover, if the mass m of the train it is expressed in [t],
the total train resistance R[N] it is established by

10][]/ [ tmt daNr= R[N] (7)
Energy consumption is calculated by
 dtvF Pdt= E
(8)
The equations ensemble (5), (6), (7), (8) they form the
mathematical model of the train motion. Written together, in
the shape of
 
   

dtvF Pdt= Em(x))r(x)+i (v)r ; R = (Mηi
DF= z ; dtv v; x =
Di = Ω ; (F-R)dtξmv =
c de ps t
rrm
1022 1
2
(9)
allow the structural diagram construction of train motion
(Fig. 1). In the mathematical model (9) was replaced the specific train resistance r with its components: the main
specific train resistance rps, the specific train resistance on
level tangent track ide and the specific train resistance due to
curves rc.
By means of this scheme ("c oupled" at the structural diagram
of electromagnetic part of traction motor) can be simulated
the motion of any electric vehicle as compared with the
concrete control modality of this. According ly they are
obtained the motion diagrams v(t) and x(t), too. T he
modification of vehicle mass, of dependences ide(x) or rc(x),
specific to certain vehicle or route, can be easily operated,
obtaining an exact mathematical model, which it respects all
the motion conditions.
In the motor wheels diameters inequalities case, the
scheme suffers a minor change, the total force F resulting
like sum of partial forces developed by each motor partly .
II. SIMULINK MODEL
The SIMULINK model corresponding to the train motion
can be easily implemented, having with a view the
topologic al comparison with the associate structural diagram
(Fig. 2).

Fig. 1. Structural diagram of the train motion

Fig. 2. SIMULINK model of the train motion “um.mdl”

An immediately example of the utilization of the
SIMULINK model it is the motion diagrams drawing, that
illustrate the dynamic aspect of the electric vehicle.
The motion diagrams of the electric train are drawn on
the traction and braking characteristics basis and of the
conditions imposed of route.
The traction and braking characteristics consort any
presentation, however summarily would be, of a hi gh-speed
train. Like example, it has been considered the ETR 500
italian high speed train case, for which, except the traction
and braking characteristics, they have more represented the
train resistance, too, corresponding to the different specific
featur es of the route (through the declivities consideration)
(fig.3).
In the blocks „Ft(v)” and “Ff(v)” (Fig.2) they are
implemented the traction and braking characteristics of the
ETR500 train. The model is based on the train motion model,
at which the main in put variable it is supplied of the block
„F”, which it models the train motion regimes (Fig.4):
– starting regime (acceleration),
– motion at constant speed regime (cruising),
– coasting regime and
– braking regime.
Also, in the "Braking" interpolation b lock ( Fig.2) is implemented the braking diagram of train, that is useful for
calculating of the braking distance .
By means of this SIMULINK model (Fig.2) they have
been simulated the speed diagrams (Fig.5), all
corresponding to a distance of 100km. They have been
obtained the time of 1500 s (case a), of 1512 s (case b), of
1632 s (case c) and of 1982 (case d).
III. SIMULATED ANNEALING ALGORITHM
The SIMULINK model allows the calculation of energy
consumption for different motion regimes and different route
configurations.
Also, can be obtained the motion diagrams for a distance
of 100 km, a maximum speed of 210 km/h and different
values of the starting position braking (Fig.6) [6]. For each of
these cases, is calculated the value of the consumed energy
and can be determined its minimum value . Thus, u sing the
same model , can be obtain the surface corresponding to the
dependence E(T, V) (Fig.7 ) [6], where T is the travel time
and V is the maximum speed corresponding to the different
motion diagrams .
By me ans of this surface, to an imposed value of motion
time timp = 2500 s (for example by schedule reasons ), can be
obtained the dependence shown in Fig.8 [6], that has a
minimum for the maximum speed v = 160km/h.
Similarly, with the Linear Search Algorithm (Fig.9) that
it uses the SIMULINK model, can be finded the minimum
value of the consumed energy for an imposed value domain
of the inte rstation time (time – timp < err) but using a high
number of steps and a long time simulation.

Fig. 3. Traction and braking characteristics and 4 values of train resistance
of ETR 500 italian high speed train resistance, m=600 t
ide=0; b) i de=5; c) i de=10; d) i de=20

Fig. 5. Speed diagram for different values of train resistance (m=600 t)
a) ide=0; b) ide=5; c) ide=10; d) ide=20

Fig. 6. Motion diagrams for maximum speed 58.3 m/s ( 210km/h )

Fig. 4. Four regimes of the train motion

A more modern and faster solution is the use of the
Simulated Annealing Algorithm with SIMULINK model
(Fig.10) and that has advantages of simplicity and efficiency,
and it is also less affected by initial conditions [8].
To prove that, we can compare the results obtained using the two algorithms (Table I). It is noticed that in a two -fold
less time, similar r esults are obtained. This time difference
increases significantly when the search domain expands or
when the interstation time is larger.
TABLE I. COMPARISON BETWEEN ALGORITHMS
Algorithm Number
of steps Simula tion
time (s) Energy
(kWh) Maximum
speed (km/h)
Linear
Search 2100 292 772.54 160
Simulated
Annealing 1001 147 773.19 162
IV. CONCLUSIONS
The presented SIMULINK model take account of the
particularity of the vehicle and route and it is relatively easy
of implemented and of used.
Related to other energy optimization techniques found in
literature [1],[10] the presented Simulated Annealing
Algorithm with SIMULINK model avoid utilization of
complex mathematical methods or time expensive
algorithms.
1000150020002500300035004000 10020030040050010001500200025003000
Speed (km/h)
Time (s) Energy (kWh)
Fig. 7. Energy as a function of time and speed

150 160 170 180 190 200 210 220 230 240 25070080090010001100120013001400
Speed (km/h)Energy (kWh)

Fig. 8. Energy as a function of speed

initialize vmin, vmax {minimum and maximum value for constant speed v2}
initialize xmin, xmax {minimum and maximum value for coasti ng start position x2}
initialize timp {imposed interstation time}
initialize err { interstation time error}
initialize E { energy consumption for v min and x min}
for v ct←v min to v max
for x 2←x min to x max
E'(v ct,x2) {calculate with the SIMULINK model the energy consumption}
if ((E'<E )and( time -timp<err )) then
E←E' { find minimum of E}
end if
end for
end for
output E {best solution}

Fig. 9. Linear Search Algorithm

This Simulated Annealing Algorithm with SIMULINK
model can be useful in the establishment of an energy
efficient train control methods, based on the best utilization
of the installed load. The respective methods are
implemented then in the computer control system on the
electric vehicles, contri buting at the circulation safety
increase, at the decrease of consumptions and allowing even
a possibly ATC (Automatic Train Control).
REFERENCES
[1] P.G. Howlett and P.J. Pudney. Energy -efficient Train Control.
Springer –Verlag London Ltd., 1995.
[2] A. S teimel, “Electric Traction. Motion Power and Energy Supply
Basics and Practical Experience”, Oldenbourg Industrieverlag,
München, 2008
[3] D.C. Cismaru, D.A. Nicola, Gh. Manolea, “Locomotive Electrice.
Rame și Trenuri Electrice”, Ed. Sitech, Craiova, 2009.
[4] D.C. Cismaru, D.A. Nicola, Gh. Manolea, M.A. Drighiciu, C.A.
Bulucea, “Mathematical Models of High -Speed Trains Motion”,
WSEAS Transactions on Circuits and S ystems, Issue 2, Volume 7,
February 2008
[5] D.C. Cismaru, D.A. Nicola, Gh. Manolea, M.A. Drighiciu,
“Simulation of Electric Vehicles Movement”, 7th WSEAS/IASME
International Conference on Electric Power Systems, High Voltages,
Electric Machines (POWER'07), Ve nice, Italy, November 21 -23,
2007. [6] D.C. Cismaru, M.A. Drighiciu, D.A. Nicola, "SIMULINK Model for
Study of Energy Efficient Train Control", Proceedings 12th
International Conference on Applied and Theoretical Electricity,
ICATE 2014 Craiova, Romania
[7] K. Kim, S.I. Chien, "Optimal Train Operation for Minimum Energy
Consumption Considering Track Alignment, Speed Limit, and
Schedule Adherence", JOURNAL OF TRANSPORTATION
ENGINEERING ASCE / SEPTEMBER 2011 137(9): 665 -674
[8] T. Xie, S. Wang, X. Zhao, Q. Zhang, "Op timization of Train Energy –
Efficient Operation Using Simulated Annealing Algorithm", K. Li et
al. (Eds.): ICSEE 2013, CCIS 355, pp. 351 –359, Springer -Verlag
Berlin Heidelberg 2013
[9] X. Li, L. Li, Z. Gao, T. Tang, S. Su, “Train Energy -Efficient
Operation with Stochastic Resistance Coefficient” , International
Journal of Innovative Computing, Information and Contro l, Volume
9, Number 8, August 2013.
[10] X. Vu, “Analysis of necessary conditions for the optimal control of a
train”, Thesis of Doctor of Mathematics, University of South
Australia, July 2006.
[11] M. Larranaga, “Optimization Techniques Applied to Railway
Systems”, Master Thesis, Universidad del Pais Vasco, September
2012.
[12] L.Felixova, “Mathematical Methods of Optimal Control Theory and
their Applications”, Ma ster Thesis, Brno University Of Technology,
Brno 2011.

initialize T {initial temperature}
initialize L {number of trials per temperature}
initialize Tmin {minimum temperature}
initialize timp {imposed interstation time}
initialize err { interstation tim e error}
i←1 {initialize number of iteration}
generate the initial solution s
while T>Tmin
for k←1 to L
generate new solution s'
ΔE← E(s') -E(s) { calculate with the SIMULINK model the energy consumption for
solutions s and s' and the difference ΔE }
if ((ΔE<0)and(time -timp<err)) then
s← s' {accept s' if ΔE<0 and interstation time t is near t imp}
else
if ((exp( -ΔE/T)>random[0,1]) and(time -timp<err)) then

s← s' {accept s' considering probability and interstation time t is near t imp}
end if
end if
end for
i←i+1
T←T*αi {decrease T, where α is cooling factor with ranging between 0,8 and 0,99}
end while
output s {best solution}

Fig. 10. Simulated Annealing Algorithm
𝑒−∆𝐸
𝑇