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3 Optimization Model
3.1 Objective Function
The optimization objective of tractive curve is to minimize energy consumption
while keeping the running time under a given maximum value. Therefore, the
objective function is formulated as follow:
N
e
t
.
minE
=
(10)
t
=1
3.2 Assumption
To develop the model,some following assumptions are made:
1. The maximum tractive effort will be applied to move a train from standstill
until it reaches the speed limit,and the maximum deceleration rate will be applied
for braking to reach the next station.
2. In the time-ecient operation mode, the actual running curve of a train
is tended to be as close as possible to the speed limit curve. If the speed limit
of next section decreases than that of current section, train will have to brake
to guarantee safety. In order to reduce braking energy as much as possible,
here assume that train is only allowed to brake under specific conditions. These
specific conditions are described in section C.
3.3 Decision Variables
The selection of decision variables mainly depends on the distribution of speed
limit sections. Firstly, the speed limit sections between two stations are prepro-
cessed to satisfy following rules [8,9]:
As shown in Fig. 2, for any three adjacent speed limit sections XZ1, XZ2
and XZ3, a train in the entrance of XZ2 executing emergency brake with min[
V1,V2] should slow down to V3 before the train arrive at XZ3. Therefore, for
any two adjacent speed limit sections, if the speed limit of second one is higher
than the first one like XZ1 and XZ2, a train can move forward into the second
section with the maximum allowed speed of the first section. Then there is no
need to set a decision variable in the first section. But if the second one is lower
like XZ2 and XZ3, there must exist a operating mode change point x1 in XZ2
to reduce speed to guarantee safely moving into XZ3. Then x1 is the decision
variable in this section. When a train starts coasting from x1 to the entrance
of next section, if the speed still exceeds the maximum allowed speed of next
section, x1 will move back 3 steps automatically and repeat calculating again
until the speed limit requirements are met; else if x1 move back to the entrance
of the prior section(XZ1) and speed limit requirements are still not met, changes
of speed limit are considered too frequently and brake is allowed in the inter-
station. Accordingly range of x1 is changed from [S0,S3] to [S0,S2], where s2 is
position of start braking. Then repeat optimization of x1.
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