Advanced Mathematics and Numerical Modeling of IoT

Speed limit (km/h)A B C D E F

Distance (km)

xi−1xi

A=휋r^2

Figure 2: Diagram of different running status during one journey.

differential equations and concluded that a maximum eco-
nomic train running strategy should contain four statuses,
maximum traction, cursing, coasting, and maximum brak-
ing. For analyzing station-to-station travel time and distance
profile, it is essential to comprehend the description of
the motion statuses and their mathematical expressions. In
maximum traction, power is used to overcome gravity (if
climbing) and the dynamic resistance so as to accelerate.
When cruising, power is used to overcome the resistance
to maintain the train at the constant speed; at this time,
the acceleration is zero. When coasting, the running train
only suffers from the force of resistance. Applying coasting
when the train runs between stations as much as possible is
consideredtobethemosteffectiveenergyconsumptionway.
When braking, with regeneration technology fitted, energy
canbeproducedusingthemotorasagenerator.
A train’s journey may have variables coast intervals
(Figure 2) to achieve an optimal solution.Figure 2shows a
train’sstatusandchangingpointduringarunningbetween
two stations. In the figure, the points mean the following:
A: traction; B: cursing start point; C: coasting start point; D:
coasting; E: cursing; F: braking.
Now,theaimistofindanoptimalcontrolstrategyfor
minimal energy consumption in a round trip between two
stations. This problem can be seen as a double optimization
problem.
Traction energy module can be described as follows.
Make푋thedistancebetweentwostations,andtraveltime
was fixed푇;[0,푇]canbedividedas

0=푡 0 ≤푡 1 ≤푡 2 ⋅⋅⋅≤푡푛≤푡푛+1=푡, (5)

where푡 0 is the initial time and푡푛+1is the final time; in the
time space[푡푘−푡푘+1]traintraveldistanceis[푥푘−푥푘+1]and
in[0,푥]

0=푥 0 ≤푥 1 ≤푥 2 ≤⋅⋅⋅≤푥푛≤푥푛+1=푥. (6)

Total energy consumed by the train can be defined as follows:

min 퐸=∫

푥푓

푥 0

푢푓(푥)푓(V)푑푥

s.t.

{{

{{

{{

{{

{

푑푡

푑푥

=

1

V

V

푑V

푑푥

=

푢푓(푥)푓(푥)−푢푏(푥)푏(V)

푀푔

−푤 0 (V)−푤푗(푥)

}}

}}

}}

}}

}

700

600

500

400

300

200

100

0

0 50 100 150 200 250

Speed (km/h)

( 1 )

( 2 )

( 3 )

( 4 )

Curve 1 : train traction property

Curve 2 : basic resistance

Curve 3 : adhesion-limited braking force (wet)

Curve 4 : adhesion-limited braking force (dry)

Traction effort (kN)

Figure 3: Train traction property and adhesion-limited braking

force.

푡(푥 0 )=0,푡(푥푓)=푇,V(푥 0 )=0,V(푥푓)=0

V≤푉(푥),푢푓∈[0,1],푢푏∈[0,1],

(7)

where퐸is the energy consumption and푇isafixedtimewhen

the train travels between two stations.푡(푥 0 )is start time,푡(푥푓)

is arrival time, andV(푥 0 )andV(푥푓)represent the start speed

and final speed; it was obvious thatV(푥 0 )andV(푥푓)are equal

to 0.푢푓and푢푏were coefficient of traction power and braking.

Then the train control strategy set was푆={푠푖}=

{traction(T),cursing(CR),coasting(C),Braking(B)} =

{T,CR,C,B}.

Finally, the train control matrix was defined as

퐶=[푐 0 ,푐 1 ,푐 2 ,...,푐푖,...,푐푛−1,푐푛], (8)

where푐푖 =[푥푖,푠푖],푥푖is the position, and푠푖is the control

strategy start at the position푥푖.FromFigure 2,wecansee

that푠푖 ∈S.푥 0 =0and푥푛canbeeasilycalculatedbythe

last braking process.

4. Minimize the Energy Consumption with

Parallel Multipopulation Genetic Algorithm

The genetic algorithm (GA) [ 9 , 10 ]isamethodforsolving

both constrained and unconstrained optimization problems

based on natural selection, the process that drives biologi-

cal evolution. The genetic algorithm repeatedly modifies a

population of individual solutions [ 17 ]. At each step, the GA