Study on a Feeding Circuit Model Formulated to
Use Multipurpose Solvers for Multi-Train
Simulators
Ryosuke Murata
Masafumi Miyatake
Graduate School of Electrical and Electronics Engineering
Sophia University
Tokyo, Japan
Email: m-ryosuke@eagle.sophia.ac.jp
Sophia University
Department of Engineering and Applied Sciences
Tokyo, Japan
Email: miyatake@sophia.ac.jp
Takashi Akiba
Power and Industrial Systems R&D Center, Toshiba
Corporation
Tokyo, Japan
Abstract— Reduction of energy consumption is one of the
recent key topics in railway sector. There are many investigations
such as SiC devices, permanent magnet motors, onboard and
stationary energy storage and reversible substations to absorb
regenerative braking, energy efficient driving and traffic
management. Multi-train simulators play an important role in
evaluating the effect of these countermeasures. In the simulator,
power feeding circuit must be modeled to calculate the energy
consumption with high accuracy. There are some literatures of
modeling and solving feeding circuits, however most of them
linearized some nonlinear but important characteristics of trains
and substations. We have also proposed a methodology of precise
modeling and solving power feeding circuits of railway systems
implemented in the multi-train simulator. The methodology is
flexible and characterized by the special node numbering and
introduction of ‘tie nodes’ which is solvable by multipurpose
solvers. In this paper, we analyze the solvability of the
methodology and introduce the virtual ‘tie nodes’ to decrease
errors of circuit equation if the distance of two nodes is very near.
The criterion whether the virtual tie node should be used or not is
finally analyzed.
Keywords—multi-train traffic simulation, energy efficiency, DC
electrification, feeding circuit, solver.
I. INTRODUCTION
In the recent decades, demand growth in public transport
systems has increased rapidly. Reduction of energy
consumption is one of the recent key topics in railway sector.
Most urban mass transit systems require DC traction power
supply to energize their rail vehicles[1]. There are many
investigations such as SiC devices, Permanent Magnet
Synchronous Motors (PMSMs), onboard and stationary energy
storage and reversible substations to absorb regenerative
braking, energy efficient driving and traffic management.
Multi-train simulators play an important role in evaluating
the effect of these countermeasures. In the simulator, power
feeding circuit must be modeled to calculate the energy
978-1-5090-0814-8/16/$31.00 ©2016 IEEE
Masahiro Tajima
Railway & Automotive System Division, Toshiba
Corporation
Kanagawa, Japan
consumption with high accuracy. There are many literatures of
modeling and solving feeding circuits[1]-[10], however, most
of them linearized some nonlinear but important
characteristics of trains and substations. Only a few literatures
deal with enough realistic nonlinear characteristics of trains
and substations.
We have also proposed a methodology of precise modeling
and solving power feeding circuits of railway systems
implemented in the multi-train simulator[11]. Our
methodology is suitable for rapid implementation because it is
flexible and characterized by the special node numbering and
introduction of ‘tie nodes’ which is solvable by multipurpose
solvers.
One of the features of the feeding circuits is that the
topology is often reconfigured as a train moves. Therefore, our
simulator checks the topology and derives the conductance
between adjacent nodes calculated by the multiplication of the
conductivity and the node distance. However, when a train
exists very near from a substation in a snapshot, the
conductance between the adjacent nodes is quite large. The
unbalanced node conductance often influences on the error of
our calculation and finally difficulty in convergence.
This paper describes the method to settle this problem. We
analyze the solvability of the methodology and introduce the
virtual ‘tie nodes’ to decrease errors of circuit equation if the
distance of adjacent two nodes is very near. We also
investigate the criterion whether the virtual tie node should be
used or not.
II. MODELING METHOD
A. Feeding model
Regenerative braking of rail vehicles is commonly used in
these days for higher energy efficiency by transferring
regenerative energy of a braking train to an accelerating train.
Most electrified lines in urban areas use DC parallel feeding
system. Since there are some substations as well as trains in the
feeding circuit, the size of the circuit to be solved becomes so
large and complicated. The circuit calculation of the network
with these elements becomes more important besides energy
calculation of a single train. This makes the circuit analysis
difficult and complicated.
Our methodology uses the concept of ‘tie nodes’ in order to
represent the nodes are connected with wires whose resistance
is neglected. It acts as a constraint in changing voltages of
these nodes simultaneously. The tie nodes are flexible because
they can be set at any places in the circuit topology of the
snapshot. The places of some tie nodes are represented by the
‘tie matrix’. The feeding model can be described as a feedingnetwork in Fig.1. It makes easier modeling and calculation to
express a feeding circuit with substation, train and tie nodes in
a feeding-network. By the tie node representation, nodes
connecting information is divided into the information of nodes
in different voltages and the information of nodes in the same
voltage. It enables to get an expression of Kirchhoff’s current
law in three steps. The first step is to make node current
equations disabling connections between the tie nodes. The
second step is to select the equations of the tie node from all
equations. The last step is to merge the selected equations
together in order to consider current flow among tie nodes
connected each other.
Construction of feeding circuit model is basically the same
in case of large change of a feeder topology. Without the
concept of the tie nodes, it is needed to modify the circuit
topology thoroughly every time in change of a feeding system.
Since only the feeder conductance between adjacent nodes is
considered taking the topological feature of the power feeder
into account, the values of the conductance is given by a vector,
not a matrix. If an element of the conductance vector is 0, it
means that the adjacent nodes are physically separated. For
example, in Fig.1, a conductance between nodes 7 and 8 is 0. A
connection between train and feeding-line is considered as a
point, regardless of train length. Resistance per unit length of
feeders, trolley wires and rails were merged to the resistance in
feeders. Substations and trains are considered as be grounded,
namely 0 [V]. Inductance and capacitance of feeding circuit is
neglected because transient response time is much faster than
the time step of this simulator.
Fig.1. Feeding circuit network model
B. Substation Model
A substation current model is represented by a function
( ) where i and V mean current and voltage, respectively.
With a typical characteristic of a diode substation like Fig. 2
represented with a no-load substation voltage
, a rated
voltage and current and , a substation current is given by
Eq. (1) where max( , ) is a function to choose bigger one of a
and b. This mathematical expression has enough flexibility to
introduce some advanced technologies; e.g. the expanded ( )
up to negative current is used for regenerative inverters or a
PWM substation, and V-I characteristics of each State of
Charge (SOC) controlled by the charge-discharge controller are
used for an energy storage system.
Fig.2. Voltage- Current characteristic of a substation
=
(
0
−
≥ 0)
(otherwise)
(1).
C. Train Model
A train current model is represented by a function ( , )
where i, V and n mean current, voltage and notch that
represents the acceleration/deceleration command by a driver
or Automatic Train Operation (ATO), respectively. In this
paper, induction motors driven by inverters are assumed to be
equipped on trains. There are two types of motor
characteristics of a train; motoring and regenerative braking
characteristics. Each of them is divided into three regions;
constant torque region, constant power region, and high-speed
region, depending on velocity. A motoring characteristic is
and
are given with Eq. (2). In
depicted in Fig. 3, where
the motoring constant torque region, load-compensating
device adjusts tractive effort in proportion to the train weight
dependent on the number of passengers in order to keep the
same acceleration. In the motoring constant power region,
tractive effort is changed in inverse proportion to the voltage.
Therefore a train current is represented by a function
( , , , ). A regenerative brake characteristic is similar to
and used in the
a motoring characteristic; the value of
braking can be different from that used in the accelerating.
Some types of motor drive systems divided into two regions
except for constant braking power region can be modeled as
=
. When voltage of a regenerating train exceeds
voltage limit, train current is squeezed (regenerative squeezing
control, Fig. 4) for protection of onboard apparatus. In order to
get constant deceleration of a train corresponding to a selected
notch, a mechanical brake is added to a regenerative brake. If
a regenerative brake is reduced by the squeezing control, a
mechanical brake should be increased. When train weight W,
voltage V, velocity v and a notch n are given, current i,
regenerative braking current when all braking power is
, regenerative braking current before
provided with it
and current after squeezing are given with Eq.
squeezing
(3)-(8) where min( , ) is a function to choose smaller one
from a and b.
Fig.3. Speed-traction characteristic of a train
Fig.4. Squeezing of regenerative current characteristic
=
=
(2)
Motoring
=
⎧
⎪
⎨
⎪
⎩
∙
, Constant Torque Region
∙
, Constant Power Region
∙
, High Speed Region
Regenerative Braking
=
∙
=
⎧
⎪
⎨
⎪
⎩
= max
= max( ,
Coasting
=0
(4)
∙
, Constant Torque Region
∙
, Constant Power Region
∙
(5)
, High Speed Region
, min max
)
(3)
,
,0
(6)
(7)
(8)
subject to
: rated voltage of train
: rated weight of train driving/braking
: notch
: velocity at end of constant torque region
: velocity at end of constant power region
: rated tractive effort at velocity 0 [km/h]
: brake tractive effort of weight W (
< 0)
η: motor and main circuit efficiency
: start voltage of squeezing regenerative current
: end voltage of squeezing regenerative current
: maximum regenerative current at under start voltage
of squeezing of regenerative current (F < 0)
Because auxiliary current is P/V where P is auxiliary
power, P/V is added to the main circuit current i calculated
already to get total current of a train. After calculating
current i, traction F is calculated according to weight and
voltage (Eq. (9)). Braking traction is obtained by F which
is decided by a selected notch and weight W (Eq. (10)).
=
∙ ∙
=
(9)
(10)
III. FEEDING-CALCULATION METHOD
Fig. 5 shows a flow of feeding-calculation. A flow mainly
consists of three steps; pre-process, optimization process
(feeding-network calculation using an optimization function)
and post-process. In the pre-process, node numbers are
allocated to each of trains, substations and tie positions (1)
based on inputs, and a network model of the feeding circuit is
generated with nodes and edges (2) as a conductance vector
and a tie matrix. It is permitted to allocate consecutive numbers
to disconnected nodes such as a down line and up line. In the
optimization process, node voltages (converged results) are
obtained by feeding-network calculation. First, currents of
feeders, trains and substations with non-linear characteristics as
stated in the last chapter are calculated (3, 4) using initial
values of each node voltage. Essentially, the converged result
at the previous snapshot is used as initial values. Then, sum of
square of current equation’s error is calculated (5) according to
Kirchhoff’s circuit laws. Sum of square of voltage errors
between nodes in ties (positions in same voltage) is also
calculated (6). An objective function (8) is composed of a
weighted sum (7) of sums (5) and (6) (Eq. (11)). A nonlinear-
minimization program is solved by converged calculation until
a value of the objective function becomes small enough to be
judged it has converged, number of iteration is over a limit, or
decrement of the objective function is too small. Because it is
difficult to obtain converged results (voltages) analytically,
approximate solution is applied to solve the program using an
optimization function of a multipurpose solver. This is one of
the proposals to formulate the model with applying the
multipurpose solver for easy implementation. In the postprocess, currents of trains and substations are calculated using
the converged values of node voltages. The non-linear
characteristics and states of each train and substation are
considered in current calculation of trains and substations. The
characteristics and states should be expressed by voltagecurrent formulas and conditional branches.
minimize
=Σ {
+
+ ∙Σ {
}
−
−
}
(11)
subject to
: sum of squared errors of circuit equations
: point of contract
α: weight of the voltage error
: combination of tie connected nodes
, : voltages of tie nodes that should be identical
Fig.5. Flow of feeding-calculation
Parameter
V
v
F
V
I
TABLE 1. Train characteristic parameters
Value
Parameter
1500[V]
W
50[km/h]
V
156[kN]
F
1750[V]
V
-1600[A]
Η
IV. SIMULATION & RESULT
We consider the case with the very short node distance. For
example, if a train is running very near to a substation, the
conductance between them is divergent. Therefore, the
calculation is difficult to find the solution with small errors of
the circuit equations. The ‘virtual tie node’ is used as the way
to settle this problem. It is possible to make two nodes with
small distance the same voltages by using the virtual tie nodes.
Since less error is expected in the calculation, the effect of the
virtual tie nodes is evaluated by comparing the errors between
with and without it against various distances between the
nodes. If the virtual tie nodes are used, these nodes are
connected regardless of the distance between them.
The problem to minimize the sum of squared errors of circuit
equations in (11) was implemented on MATLAB by using the
nonlinear optimization solver ‘fmincon’ of the Optimization
Toolbox.
A. Relation of distance and error
We assume the feeding circuit model of a snapshot of a
commuter line as illustrated in Fig. 6 in order to consider the
relevant condition of applying the virtual tie node. There are
three substations, 24 trains on the double-track layout. The
trains are located with the same distance of 3,000 [m] to
eliminate the influence of uneven distribution as the initial
condition. The location of the substations is in the middle of
the adjacent trains, namely the distance of a train and a
substation is 1,500 [m].
Value
200[ton]
75[km/h]
-194[kN]
1800[V]
0.9
We consider the case with the very short node distance. For
example, the distance between a train and a substation is
virtually changed between 20[m] and 100[m]. We consider
two cases of node pairs: #3 and #4, and #7 and #8. In this case,
the conductance is divergent. Therefore, the calculation is
difficult to find the solution with small errors of the circuit
equations. The ‘virtual tie node’ is used as the way to settle
this problem. It is possible to make two nodes with small
distance the same voltages by using the virtual tie nodes #10
and #11. Fig.7 shows relation between the distance and sum of
squared error of the circuit equation in each case of 3-4 and 78. In the case with the virtual tie nodes, since the two nodes
are virtually connected regardless of the distance, only one
value is indicated in each graph in Fig.7. Generally, if the
error of using the virtual tie nodes is lower than that without it,
this technique contributes to reduce errors. The experimental
result so far shows that the error is increasing when the train is
approaching to a substation without setting the tie nodes. On
the other hand, it is found that setting the virtual tie nodes can
prevent error increasing by setting tie node. However,
increased tendency of the error depending on the distance is
different in the states of the feeding circuit as drawn in Fig.7.
For example, in case of node #3-4 in Fig. 6, node #3-4 should
be tied when the distance was less than 60 [m] between nodes.
But in case of node #7-8, node #7-8 should be tied when the
distance was less than 99 [m] between nodes.
If some minor exceptions with extremely large error can be
neglected, we can find the rough criterion that the virtual tie
nodes should be used if the distance between adjacent train
and substation is less than 50 [m].
Legend: Red arrow: Accelerating train, Black arrow: Coasting train, Blue arrow: Decelerating train.
Fig.6. Assumed double-track model with three substation and 24 trains.
Fig.7. Relation of distance during node, error and using tie nodes in case of node #3-4 and #7-8
B. Typical railway model
Taking the previous results into account, the idea is applied
to a realistic railway line with multiple snapshots in order to
demonstrate the idea of the virtual tie node. A typical Japanese
double-track urban railway line with 10 stations between A
and J and 3 substations was assumed as Fig.8 in order to
ascertain computational error to decrease by simulation.
Timetables for both directions composed of mixed local and
rapid trains are drawn in Figs 9 and 10. The rapid trains
connect A and J stations without stopping intermediate
stations. There are passing loops in E and G for the rapid
trains to pass the local trains. The speed profiles of local and
rapid trains are plotted in Fig. 11.
Simulation was performed for 30 minutes between 8:00 and
8:30, the morning peak time, with one-second increments
(1800 snapshots). The number of nodes of each snapshot is 17
and 21 that depend on the train positions. The computation
time for each snapshot is about 0.3 [s] in this case study.
Compared the criteria of distance, the sum of squared errors
of every circuit equations were calculated to evaluate the
computational error. The result is shown in the TABLE 2. The
number of snapshots with error value less than 100 is the
largest when the virtual tie is set where the distance between
adjacent nodes is less than 100 [m]. Therefore virtual tie node
is effective if it is set the link between a train and a substation
less than 100 [m]. On the other hand, there are still some
snapshots with large errors even if the virtual tie nodes are
properly given. The error must be reduced if the convergence
conditions of the optimization solver are adjusted. However, it
should be noted that the error and computation time has a
trade-off relationship.
12000
Distance[m]
10000
8000
6000
4000
2000
0
8:00:00
Fig.8. Arrangement of stations and substations
8:15:00
Time [h:mm:ss]
8:30:00
Horizontal line: position of three substations
Fig.9. Train timetable for downward direction from A to J
70
12000
60
Speed[km/h]
Distance[m]
10000
8000
6000
4000
50
Local
Train
Rapid
Train
40
30
20
10
2000
0
0
8:00:00
8:15:00
0
8:30:00
2500
Time [h:mm:ss]
5000
7500
Distance[m]
10000
Fig.11. Local and rapid train’s speed profile for one direction
Fig.10. Train timetable for upward direction from J to A
TABLE 2. Computing error of each criteria of distance to make virtual tie node
with virtual Tie where the distance between adjacent nodes are
without
the value shown below
number of snapshots with
virtual
error
values E
shown
on the
right
≤ 100㻌
Tie
30m
50m
75m
100m
125m
150m
300m
930
1099
1134
1134
1134
1130
1125
1084
100 <
≤ 1000㻌
200
190
152
153
158
159
155
168
1000 <
≤ 10000㻌
173
124
125
123
121
120
128
149
497
387
325
389
529
390
733
387
933
391
1093
392
1214
399
1637
283
412
532
654
748
807
1004
42
117
201
279
345
407
633
241
295
331
375
403
400
371
10000 < 㻌
using virtual Tie
(a) error values less than
without virtual Tie
(b) error values more than
without virtual Tie
(a)-(b)
Fig.12. One of the snapshots of railway circuit (Time 8:28:20)
Node
Number
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
Sum of Squared
Errors E
TABLE 3. Voltage and current solution and error value of Fig12
Virtual Tie condition
Virtual Tie condition
Without virtual Tie
50m or less
150m or less
V[kV]
Cur[kA]
V[kV]
Cur[kA]
V[kV]
Cur[kA]
1.5930
0
1.5931
0
1.5931
0
1.5934
-0.1992
1.5931
-0.2064
1.5931
-0.2074
1.5926
0
1.5927
0
1.5931
0
1.5848
0
1.5883
0
1.5885
0
1.5772
-0.6841
-0.4881
-0.4871
1.5837
1.5838
1.5772
0
0
0
1.5837
1.5837
1.5759
0
1.5804
0
1.5805
0
1.5723
-0.8302
-0.9066
-0.9064
1.5698
1.5698
1.5723
0
0
0
1.5698
1.5698
1.5725
0
1.5703
0
1.5703
0
1.5885
0
1.5924
0
1.5923
0
1.5892
0
1.593
0
1.5928
0
1.5897
0
1.5932
0
1.5931
0
1.5904
0
1.5898
0
1.5898
0
1.5915
0
1.5860
0
1.5860
0
1.5923
0
1.5837
0
1.5837
0
1.5449
1.6004
1.5501
1.6004
1.5501
1.6004
1.5609
0
1.5697
0
1.5698
0
2753.2
Fig. 12 shows a snapshot in the simulation. Since most
trains don’t consume/regenerate energy in this situation, we
choose this snapshot for easy consideration of the error.
Three cases of different virtual tie application are
considered. The solved node voltages and currents are
tabulated in TABLE 3. The significant difference of the
results is the node voltages between 2 and 3. Although the
node must have unneglectable voltage difference, the virtual
tie is applied in the last case shown in the right row of
TABLE 3. This increases the error E. Of course, no virtual
tie leads to huge error.
V. CONCLUSION
In this paper we introduce the already proposed circuit
model and newly proposed concept of the ‘virtual tie nodes’
to reduce the calculation error in some specific circuit
topology. The criterion of using the virtual tie node is
derived.
We simulated various snapshots with more realistic traffic
situation to verify the efficacy of our proposal. We
implemented the new circuit model to the multi-train
simulator with more realistic train timetable and speed
profiles. As a result of the simulation, we showed that
virtual tie node introduced by this paper is effective.
Our future work is to balance the computation error and
time. We need to compare different optimization solvers,
0.12421
0.21297
adjust convergence conditions, introduce some heuristic rule
for the initial conditions, etc.
ACKNOWLEDGMENT
This work was supported by the Adaptable and Seamless
Technology Transfer Program through Target-driven R&D
(A-STEP) of Japan Science and Technology Agency (JST),
Japan. Products names (mentioned herein) may be
trademarks of their respective companies.
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