XXX -X-XXXX -XXXX -XXXXX.00 20XX IEEE Energy Efficient Train Operation using [612579]

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 Electrical Engineering
Craiova, ROMANIA
[anonimizat]
Abstract —This paper presents a n energy -efficient
algorithm starting from the mathematical model of the train
motion. Using the mathematical model it is created the
SIMULINK model of the train motion. By using the Simulated
Annealing Algorithm it is determined the minimum value of
the energy consumed by the electric train for a specified
route .
Keywords — modeling; simulation; Simulated Annealing
Algorithm; SIMULINK; energy efficient train operation
I. INTRODUCTION
In order to calculate the minimum value of the energy
consumed by the el ectric vehicle it is mandatory to use the
mathematical model of the motion that can be written in the
following form [1], [3],[4], [6] :

  
dtvF Pdt= E dtv v; x =
Di2 = Ω (F-R)dtξm1v = m (x))r (x)+i (v)r R = ( Mηi
D2F= z
rmc d p2 t
r
(1) where M2 is the traction motor torque , i is the transmission
ratio, ηt is the transmission efficiency , Dr is the wheel
diameter , x is the vehicle position , v is the vehicle speed , Ωm
is the angular speed of the traction motor rotor , F is Ft (motor
active force ) in traction regime or – Ff (braking active force )
in braking regime , m is the mass of the vehicle , r is the
specific train resistance , R is the total train resistance and ξ is
the increase coefficient of the train mass [3], [4], [6] .
The motion model (1) contains the components of the
specific train resistance r : the specific main resistance rp, the
curves specific resistance rc and the gradient specific
resistance id .
With the mathematical model (1) can be s tudy by
simulation the vehicle motion and to draw the motion
diagrams v(t) and x(t). The accuracy of the model take into
account of the all vehicle motion and route conditions by
using vehicle mass and dependences id(x) and rc(x) .
II. SIMULINK MODEL
The SIMULINK model correspondi ng to the train motion
was built based on the mathematical model of the vehicle
motion (1) (Fig. 1) [6].
This SIMULINK model (Fig.1) can be used to draw the
motion diagrams and to simulate the dynamic behavior of the
train.
In order to draw the motion diagrams of the vehicle must
be known the traction characteristic, the braking

Fig. 1. SIMULINK model of the train motio n

characteristic and the route features .
Thus, in the SIMULINK model (F ig.1), the function
block Ft(v) contains the traction characteristic and the
function block Ff(v) contains the braking characteristic of
the italian ETR500 train (fig.2) . Also, the purpose of the
function block F (Fig.1) is to select the type of force (active,
braking or resistance) in order to simulates the train motion
regimes (Fig. 3):
– starting regime (acceleration),
– cruising regime,
– coasting regime or
– braking regime.
The interpolation block Braking (Fig.1 ) simulates the
braking diagram of ETR500 , necessary to estimate the
braking starting position .
III. SIMULATED ANNEALING ALGORITHM
The SIMULINK model (Fig.1) can be used to calculate
the value of the energy consumed by the train in all motion
regimes and for all route configurations.
For an imposed interstation distanc e, a different cruising
speed and different braking starting positions it results
different motion diagrams (Fig.4 ) and different value s of consumed energy [6]. The purpose is to find the minimum
value of consumed energy .
If, by schedule reasons it is imp osed an interstation time,
then with the SIMULINK model (Fig.1) it has found the
minimum value of consumed energy for the cruising speed
v = 160km/h (Fig.5) [6] .
Similarly, with th e Linear Search Algorithm (Fig.6 ) that
it uses the SIMULINK model, can be f ound the minimum
value of the consumed energy for an imposed value domain
of the interstation time ( time – timp < err) but using a high
number of steps and a long time simulation.
A more modern and faster solution is the use of the

Fig. 2. ETR 500 traction characteristic, braking characteristic and resistance characteristics
a) id=0; b) i d=5; c) i d=10; d) i d=20

Fig. 3. Four regimes of the train motion
150 160 170 180 190 200 210 220 230 240 25070080090010001100120013001400
Speed (km/h)Energy (kWh)

Fig. 5. Consumed e nergy as a function of cruising speed

Fig. 4. Motion diagrams for cruising speed 58,3 m/s

Simulated Annealing Al gorithm with SIMULINK model
(Fig.7). The Simulated Annealing Algorithm is simpl e,
efficien t and 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 results 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 Simulation
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 pr opos ed SIMULINK model can be used to draw the
motion diagrams , to simulate the dynamic behavior of the
train and to calculate the value of the e nergy consumed by
the train in all motion regimes and for all route
configurations .
Also, the SIMULINK model can be integrated in to a
Simulated Annealing Algorithm to find the minimum value
of consumed energy , avoid ing the utilization of complex
mathematica l methods or time expensive algorithms energy
used in other optimization techniques [1], [10].
This Simulated Annealing Algorithm with SIMULINK
model can be also used by the rail operators to implement
energy efficient train strategies .
initialize T {initial temperature}
initialize L {number of tria ls per temperature}
initialize Tmin {minimum temperature}
initialize timp {imposed interstation time}
initialize err { interstation time 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. 7 . Simulated Annealing Algorithm initialize vmin, vmax {minimum and maximum value for constant speed v2}
initialize xmin, xmax {minimum and maximum value for coasting 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. 6 . Linear Search Algorithm
𝑒−∆𝐸
𝑇

REFERENCES
[1] P.G. Howlett and P.J. Pudney. Energy -efficient Train Control.
Springer –Verlag London Ltd., 1995.
[2] A. Steimel, “Electric Traction. Motion Power and Energy Supply
Basics and Practical Experience”, Oldenbourg Industrieverlag,
München, 2008
[3] D.C. Cismaru, D.A. N icola, 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 Systems, 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/IASM E
International Conference on Electric Power Systems, High Voltages,
Electric Machines (POWER'07), Venice, 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 12 th
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 TRAN SPORTATION
ENGINEERING ASCE / SEPTEMBER 2011 137(9): 665 -674
[8] T. Xie, S. Wang, X. Zhao, Q. Zhang, "Optimization of Train Energy –
Efficient Operation Using Simulated Annealing Algorithm", K. Li et
al. (Eds.): ICSEE 2013, CCIS 355, pp. 351 –359, Springer -Verla g
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, “Ana lysis 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, Se ptember
2012.
[12] L.Felixova, “Mathematical Methods of Optimal Control Theory and
their Applications”, Master Thesis, Brno University Of Technology,
Brno 2011.