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0, accept the new solution; if
e
−
ΔE
k
B
T
temp
>random
[0
,
1](where
k
B
= Boltzmann's constant), accept the new solution; otherwise, go to step3;
Step5: Reduce the system temperature according to the cooling schedule,
T
temp
Step4: If
ΔE
≤
n
=
n
+1
=
T
tem
n
α
n
(where n=0,1,2,...;
α
n
is cooling factor at iteration n;
α
is
ranging between 0.8 and 0.99, here set 0.85);
Step6: If
T
temp
n
<T
temp
min
, the algorithm terminates and outputs the optimized
solution. Otherwise repeat Step 3 to 5 until the stop criterion is met.
5 Numerical Example
The developed model is applied to a particular segment of Shanghai Metro Line.
The experiment segment is a 3200-m-long track route without tunnel, of which
the maximum running time is 232s. The train characteristics and options of SA
algorithm are listed in Table 1.The Track profile and the associated maximum
operating speed are rendered in Table 2.
Tabl e 1.
Parameter Table
Parameters
Values
Train mass
370000 kg
Train
characteristics(AW3)
Train rotating mass
2200 kg
Train length
140.44 m
Energy eciency
0.85
1
.
0079
×
10
−
2
a
Basic resistance
parameter(AW3)
b
0
1
.
0293
×
10
−
4
c
Initial temperature 100
Annealing function Boltzman
Temperature update Metropolis rule
Cooling schedule
Simulated annealing
options
Exponential
Trials per temperature
100
1
×
10
−
3
Minimum temperature
6 Results Analysis
Optimized result is shown in Fig. 3. The solid blue line is track speed limit; the
dashed green line is traction operation curve in time-ecient mode; the dotted
red line is optimized energy-ecient operation curve. Obviously, in the time-
ecient operation mode, the actual running curve of a train is very close to the
speed limit curve, but there exists three coasting sectors in the optimized tractive
curve, starting with X1, X2, X3 respectively. When determining the location of
X2, even if the starting position is set to be the fourth speed limit section, the
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