U.S. patent application number 13/803857 was filed with the patent office on 2014-09-18 for system and method for optimizing energy consumption in railway systems.
This patent application is currently assigned to MITSUBISHI ELECTRIC RESEARCH LABORATORIES, INC.. The applicant listed for this patent is MITSUBISHI ELECTRIC RESEARCH LABORATORIES, INC.. Invention is credited to Arvind U. Raghunathan, Satoru Takahashi, Kenji Ueda, Toshihiro Wada.
Application Number | 20140277858 13/803857 |
Document ID | / |
Family ID | 51455306 |
Filed Date | 2014-09-18 |
United States Patent
Application |
20140277858 |
Kind Code |
A1 |
Raghunathan; Arvind U. ; et
al. |
September 18, 2014 |
SYSTEM AND METHOD FOR OPTIMIZING ENERGY CONSUMPTION IN RAILWAY
SYSTEMS
Abstract
A method optimizes energy consumption in a railway system
including a set of trains and a set of substations connected to a
grid. The method optimizes control parameters controlling at least
part of the energy consumption of the railway system to produce
optimized control parameters minimizing a total power provided by
the grid to satisfy a power demand of the railway system. The
optimizing is subject to constraints on operations of the railway
system, which include as complementarity constraint. Next, the
method generates a command to control the energy consumption of the
railway system based on the optimized control parameters.
Inventors: |
Raghunathan; Arvind U.;
(Brookline, MA) ; Wada; Toshihiro; (Tokyo, JP)
; Ueda; Kenji; (Tokyo, JP) ; Takahashi;
Satoru; (Tokyo, JP) |
|
Applicant: |
Name |
City |
State |
Country |
Type |
MITSUBISHI ELECTRIC RESEARCH LABORATORIES, INC. |
Cambridge |
MA |
US |
|
|
Assignee: |
MITSUBISHI ELECTRIC RESEARCH
LABORATORIES, INC.
Cambridge
MA
|
Family ID: |
51455306 |
Appl. No.: |
13/803857 |
Filed: |
March 14, 2013 |
Current U.S.
Class: |
701/19 |
Current CPC
Class: |
Y02T 10/7283 20130101;
B60L 55/00 20190201; B61L 3/006 20130101; B60L 9/04 20130101; B60L
9/00 20130101; Y02T 10/72 20130101; B61L 3/008 20130101; B61L
27/0027 20130101; B60L 15/2045 20130101; G06Q 50/06 20130101; G06F
17/11 20130101; B60L 2200/26 20130101 |
Class at
Publication: |
701/19 |
International
Class: |
B61L 99/00 20060101
B61L099/00 |
Claims
1. A method fir optimizing energy consumption in a railway system
including a set of trains and a set of substations connected to a
grid for providing power to the set of trains, comprising:
optimizing control parameters controlling at least part of the
energy consumption of the railway system to produce optimized
control parameters, wherein the optimized control parameters
minimize a total power provided by the grid to satisfy a power
demand of the railway system wherein the optimizing is subject to
constraints on operations of the railway system, the constraints
include a complementarity constraint; and generating a command to
control the energy consumption of the railway system based on the
optimized control parameters, wherein steps of the method are
performed by a processor.
2. The method of claim 1, wherein the set of trains includes a set
of power consuming trains and a set of power regenerating trains,
and wherein the control parameters include values of voltages and
currents for each substation and for each train.
3. The method of claim 2, further comprising: determining a state
of the railway system at a point of time, wherein the state
includes locations of the trains, and the power demand of the power
consuming trains; and optimizing the control parameters according
to the state.
4. The method of claim 3, further comprising: determining the state
based on run curves of the trains.
5. The method of claim 4, further comprising: modifying the state
based on measurements of the state at the point of time.
6. The method of claim 1, wherein the complementarity constraint
includes a product of a current at a substation and an excessive
voltage at the substation.
7. The method of claim 1, wherein the optimizing comprises:
optimizing the control parameters using iterative relaxation of the
complementarity constraint.
8. The method of claim 1, wherein the complementarity constraint is
relaxed according to a relaxation parameter, and wherein, for each
iteration, the relaxation parameter is reduced monotonically.
9. The method of claim 7, wherein the optimizing further comprises:
optimizing the control parameters using an, interior point
method.
10. The method of claim 7, wherein the optimizing further
comprises: optimizing the control parameters iteratively using
adaptive modification of a relaxation parameter, wherein, for each
iteration, the relaxation parameter is modified only if the
complementarity constraint for a current value of the relaxation
parameter is violated.
11. The method of claim 7, wherein the complementarity constraint
for a substation includes a product of a current at a substation
and an excessive voltage at the substation, and wherein the
optimizing further comprises: reformulating the complementarity
constraint as an inequality, such that the product of the current
and the excessive voltage is less than a relaxation parameter; and
determining the control parameters iteratively based on the
inequality, wherein a value of the relaxation parameter is
adaptively reduced for subsequent iterations.
12. The method of claim 11, wherein the determining for a current
iteration comprises: determining the control parameters based on
values of the control parameters determined during a previous
iteration; testing the inequality using values of the control
parameters corresponding to that of the current and the excessive
voltage of the substation; and modifying the value of the
relaxation parameter, if the inequality is violated.
13. The method of claim 12, wherein the optimizing uses an interior
point method, wherein the relaxation parameter is proportional to a
barrier parameter of the interior point method, and wherein the
barrier parameter is monotonically reduced for each iteration.
14. The method of claim 7, wherein the complementarity constraint
for a substation includes a product of a current at the substation
and an excessive voltage at the substation, and wherein the
optimizing further comprises: reformulating the complementarity
constraint as an inequality, such that the product of the current
and the excessive voltage is less than a relaxation parameter; and
determining the control parameters iteratively using an interior
point method based on a barrier problem.
15. A system for optimizing energy consumption of a railway system
including a set of trains and a set of substations connected to a
grid for providing power to the railway system, comprising: a
processor for optimizing control parameters controlling at least
part of the energy consumption of the railway system, such that a
power demand of the railway system is satisfied and a total power
provided by the grid is minimized, wherein the optimizing is
subject to constraints on operations of the trains and constraints
on operations of the substations, wherein the constraints on the
operations of the substations include a complementarity constraint,
wherein the processor optimizes the control parameters based on a
relaxation of the complementarity constraint.
16. The system of claim 15, wherein the complementarity constraint
includes a product of a current at a substation and an excessive
voltage at the substation.
17. The system of claim 15, wherein the constraint includes V i
.ltoreq. V max F - r i F I i + .alpha. i V i .gtoreq. V min F I i ,
.alpha. i .gtoreq. 0 , I i .alpha. i = 0. } .A-inverted. i
.di-elect cons. N F , ##EQU00012## wherein V.sub.i is a voltage and
I.sub.i is a current at a substation from the set of substation
N.sup.F, V.sub.min.sup.F, V.sub.max.sup.F are minimal, maximal
operational voltages, and r.sub.i.sup.F is an internal resistance
at the substation, wherein a product of the current I.sub.i and an
excessive voltage at the substation .alpha..sub.i is zero.
Description
FIELD OF THE INVENTION
[0001] This invention relates generally to railway systems, and
more particularly to optimizing energy consumption in railway
systems.
BACKGROUND OF THE INVENTION
[0002] It is desired to reduce energy consumption in railway
systems. Several methods are known that reduce energy during an
operation of the railway systems. However, the conventional methods
generally use static models, or only consider energy efficiency of
the individual trains. For example, various run curve optimization
methods can reduce energy consumption of the train. Regenerative
braking can provide additional energy for the train. Although the
conventional methods can increase energy efficiency, those methods
do not consider global optimization of the operation of the railway
system, and do not minimize total energy consumption.
[0003] For example, U.S. 20050000386 describes a railway car drive
system for accelerating and deaccelerating a train by operating a
driving motor, and to improve the power efficiency of the drive
system, and recovering the generative power to reduce the load
borne by the braking system, and to improve the safely and
reliability of the railway car drive system.
[0004] U.S. 20060005738 describes power generation capability
through a traction motor linked to a driving wheel of a train. A
controller selectively operates the traction motor in a motoring
mode, a coasting mode, or a dynamic braking mode. In the dynamic
braking mode, electrical energy is transmitted to an electrical
energy storage system. The controller receives control commands
from an external control source indicating the operating mode for a
particular period of time.
[0005] Similarly, U.S. Pat. No. 7,940,016 relates to generative
braking methods for a locomotive. Four methods for recovering
energy from generative braking and for transferring the energy to
energy storage system are disclosed.
[0006] Accordingly, it is desired to provide a system and a method
for optimizing energy consumption so that the total power provided
by the grid to the railway system is minimized, while the energy
demand of the railway system is satisfied.
SUMMARY OF THE INVENTION
[0007] The embodiments of the invention are based on a realization
that additionally or alternatively to improving the energy
efficiency of the trains individually, the total energy consumption
of the railway system can also be optimized. For example, power,
generated by regenerative braking, decreases with an increase of
the voltage at the braking train. This goes counter to the approach
of operating the trains at high voltages. Therefore, the control
parameters for the voltages at the power regenerating trains can be
optimized to reduce the total energy consumption. Also, if the
railway system does not consume all the regenerative power,
potentially damaging power surges can occur. Thus, the global
optimization can also improve safety in the railway system.
[0008] However, the optimization of control parameters controlling
at least part of the energy consumption of the railway system has
to be performed subject to constraints on various components of the
railway system. The constraints of the railway system can include
discontinuities, which makes the optimization problem ill-posed.
Accordingly, some embodiments of the invention are based on a
general realization that for optimization of the control parameters
of the railway system, the discontinuities of the constraints have
to be reformulated as complementarity constraints. Such
reformulation allows using various non-linear optimization solvers
provided the complementarity constraints are appropriately
handled.
[0009] Accordingly, various embodiments optimize the control
parameters of the railway system subject to the complementarity
constraints. Some embodiments use non-linear optimization methods
to determine the control parameters. However, due to the limitation
on a feasibility region defined by the complementarity constraints,
some non-linear optimization methods solve complementarity
constraint problem with unacceptably high error rate. Therefore,
some embodiments of the invention optimize the control parameters
based on iterative relaxation of the complementarity constraint.
The iterative relaxation of the complementarity constraint can
approximate ill-posed problem as a set of well-posed problems,
which increase an accuracy of the solution and reduce computational
time.
[0010] For example, one embodiment optimizes the control parameters
using an interior point method. This embodiment is based on a
realization that relaxation of the complementary constraints
results in inequality constraints, which is a difficult
optimization problem. However, the interior point method can
efficiently address the inequality constraints and thus can benefit
the iterative relaxation method.
[0011] Alternative embodiments are based on a realization that with
a reduction of the relaxation parameter, the well-posed problems
are progressively transformed into ill-posed problems. Therefore,
it is advantageous to reduce the relaxation parameter only when
necessary. Thus, one embodiment of the invention optimizes the
control parameters iteratively using adaptive modification of a
relaxation parameter. In this embodiment, for each iteration, the
relaxation parameter is modified only if the complementarity
constraint for a current value of the relaxation, parameter is
violated.
[0012] In one embodiment, the complementarity constraint for a
substation of the railway system includes a product of a current
and an excessive voltage at the substation. The optimization
according to this embodiment reformulates the complementarity
constraint as an inequality, such that the product of the current
and the excessive voltage is less than a relaxation parameter.
Next, this embodiment determines the control parameters iteratively
based on the inequality, wherein a value of the relaxation
parameter is adaptively reduced for subsequent iterations. In one
variation of this embodiment, the current iteration includes
determining the control parameters based on values of the control
parameters determined during a previous iteration. The inequality
is tested using values of the control parameters corresponding to
that of the current and the excessive voltage of the substation,
and the value of the relaxation parameter is modified if the
inequality is violated.
[0013] Some embodiments combine adaptive relaxation with an
interior point method to further optimize the solution. For
example, one embodiment determines the relaxation parameter to be
proportional to the harrier parameter. This proportionality
simplifies determining of the modified value of relaxation
parameter in subsequent iterations. This is because the barrier
parameters are monotonically decreasing, and when the relaxed
complementarity constraint, is violated, the relaxation parameter
is modified to match appropriately the reduction in barrier
parameter.
[0014] Accordingly, one embodiment discloses a method for
optimizing an energy consumption of a railway system including a
set of trains and a set of substations connected to a grid for
providing power to the set of trains. The method includes
optimizing control parameters controlling at least part of the
energy consumption of the railway system to produce optimized
control parameters minimizing a total power provided by the grid to
satisfy as power demand of the railway system, wherein the
optimizing is subject to constraints on operations of the railway
system, the constraints include a complementarity constraint; and
generating a command to control the energy consumption of the
railway system based on the optimized control parameters. Steps of
the method are performed by a processor.
[0015] Another embodiment discloses a system for optimizing an
energy consumption of a railway system including a set of trains
and a set of substations connected to a grid for providing power to
the railway system. The system includes a processor for optimizing
control parameters controlling at least part of the energy
consumption of the railway system, such that a power demand of the
railway system is satisfied and a total power provided by the grid
is minimized, wherein the optimizing is subject to constraints on
operations of the trains and constraints on operations of the
substations, wherein the constraints on the operations of the
substations include a complementarity constraint, wherein the
processor optimizes the control parameters based on a relaxation of
the complementarity constraint.
[0016] Following is a summary of variables, terms and notations
used it the detailed description below.
TABLE-US-00001 N set of nodes in the network N.sup.F subset of
nodes in the network corresponding to substations N.sup.A subset of
nodes in the network corresponding to power consuming trains
N.sup.R subset of nodes in the network corresponding to
regenerative trains r.sub.i.sup.L resistance on the electrical line
joining node i and i + 1 r.sub.i.sup.F internal resistance
associated with substation i V.sub.min.sup.F minimal voltage at
substation V.sub.max.sup.F maximal voltage at substation
V.sub.min.sup.R minimal voltage at regenerative trains
V.sub.max.sup.R maximal voltage at regenerative trains
P.sub.max.sup.R maximal power that can be produced by regenerative
trains V.sub.i voltage at node i I.sub.i current supplied to
network from node i I.sub.i.sup.L current flowing between nodes i
and i + 1 P.sub.i.sup.A specified power consumed by accelerating or
coasting trains P.sub.i.sup.R power produced by the regenerative
trains V set of voltages at the nodes I set of currents supplied to
the network from the nodes I.sup.L set of currents in the DC
network P.sup.R set of power supplied to the DC network by
regenerative trains.
BRIEF DESCRIPTION OF THE DRAWING
[0017] FIG. 1 is a schematic of a railway system according to some
embodiments of the invention;
[0018] FIG. 2 is graphical representation the railway system;
[0019] FIG. 3 is a block diagram of a method for optimizing an
energy consumption of the railway system, such as systems shown in
FIGS. 1 and 2, according to one embodiment of the invention;
[0020] FIG. 4A is a plot, of feasible values of voltages for given
currents at substations of the railway system according to one
embodiment of the invention;
[0021] FIG. 4B is a plot of relationship of voltage and current for
power consuming trains in the railway system according to one
embodiment of the invention;
[0022] FIG. 4C is a plot of relationship of voltage and current for
regenerative trains in the railway system according to one
embodiment of the invention;
[0023] FIG. 5 is an example of the optimization subject to
complementarity constraints using the relationships of various
measurements of voltages, currents, resistances at nodes in the
railway system according to one embodiment of the invention.
[0024] FIG. 6 is a schematic of a method for optimizing energy
consumption of the railway system according to some embodiments of
the invention;
[0025] FIG. 7 is an illustration of relaxation of the
complementarity constraints by multiple parameters according to one
embodiment of the invention;
[0026] FIG. 8 is a pseudocode of the method for complementarity
inequality reformulation according to one embodiment of the
invention.
[0027] FIG. 9 is a schematic of another method according another
embodiment of the invention;
[0028] FIG. 10 is a block diagram of an optimization method
according to one embodiment of the invention; and
[0029] FIG. 11 shows a pseudocode for solving the barrier problem
with adaptive relaxation according to one embodiment of the
invention.
DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS
[0030] Railway System
[0031] The embodiments of the invention optimize an energy
consumption of a railway system that includes a set of trains and a
set of substations. The substations are connected to a grid and
provide power to the set of trains. Optimizing energy consumption
includes minimizing the amount of power supplied to the railway
system from the grid. The embodiments can reflect, in real-time,
dynamically varying, states of the railway system, in terms of
locations and identifications of substation and trains, and
measured electrical characteristics such voltages, currents, and
resistances in power lines of DC network connecting the substations
and trains.
[0032] FIG. 1 show a railway system 100 according to some
embodiments of the invention. The railway system 100 includes a set
of trains, such as a train 110 and a set of substations 111. The
train 110 can be an electrical train, wherein the electric power is
converted and supplied to feeders or ground coils (CC) breakers.
The train can use regenerative braking to generate energy. The
energy can be stored in the power supply device 5, or distributed
to other trains in the railway system.
[0033] A control system 101 of the train 110 can include one or
combination of a measurement module 1, a run curve generation
apparatus 3, a power supply device 5, and a control computer 7. The
control system determines the state of the train. The state can
include locations and the power demands of the trains. The state
can also include positions and velocities of the trains and next
actions of the trains. The state can be determined by one or
combination of the control computer 7 and the measurement module 1.
For example, the state can be defined by run curve of the train
determined by the run curve generating apparatus 3. The run curve
can be updated based on the measurements determined by the
measurement module 1. For example, the measurements module can
includes a GPS unit for determining the location of the train. The
measurement module can also include an energy meter far measuring a
power demand or a power excess of the train.
[0034] The implementation of the control system can be performed
within the circuits of the train, in a remote control center 120,
and/or can be distributed between the vehicle and the control
center 120. The communications between the vehicle and the control
center can be achieved using wireless transceivers 11-13. Various
components and modules of the control system and control center can
be implemented using a processor.
[0035] In various embodiments, the control center 120 is also
connected to the substations 111 and to the grid 113. The control
center optimizes control parameters controlling at least part of
the energy consumption of the railway system to produce optimized
control parameters minimizing a total power 117 provided by the
grid 113 to satisfy a power demand of the railway system. The
control center generates a command 115 and/or 125 to control the
energy consumption of the railway system based on the optimized
control parameters. For example, the control parameters can include
values of voltages and currents at each substation and each
train.
[0036] FIG. 2 shows an example of the railway system 100
represented as a graph 200. The graph includes nodes 21, 23 and 25
representing a set of substations receiving power P.sub.1, P.sub.3
and P.sub.5 from an electric grid, a node 22 representing a set of
power consuming trains P.sub.2, and a node 24 representing a set of
power generating trains P.sub.4. The edges joining the nodes
represent power lines 210 in the DC network connecting the
substations and the trains. Resistance 220 of the power lines is
typically known. Additional substations ill and trains can be
incorporated into the graph 200 to represent practical railway
systems 100.
[0037] Optimization Method
[0038] The embodiments of the invention are based on a realization
that additionally or alternatively to improving the energy
efficiency of the trains individually, the total energy consumption
of the railway system can also be optimized. For example, power,
generated by the regenerative braking, decreases with an increase
of the voltage at the braking train. This goes counter to the
approach of operating the trains at high voltages. Therefore, the
control parameters for the voltages at the power regenerating
trains can be optimized to reduce the total energy consumption.
Also, if the railway system does not consume regenerative power,
potentially damaging power surges can occur. Thus, the global
optimization can also improve safety in the railway systems.
[0039] However, the optimization of control parameters controlling
at least part of the energy consumption of the railway system has
to be performed subject to constraints on various components of the
railway system. The constraints of the railway system can include
discontinuities, which makes the optimization problem ill-posed.
Accordingly, some embodiments of the invention are based on a
general realization that for optimization of the control parameters
of the railway system, the discontinuities of the constraints have
to be reformulated as complementarity constraints. Such
reformulation enables using various non-linear optimization
solvers.
[0040] FIG. 3 shows a block diagram of a method for optimizing an
energy consumption of a railway system including a set of trains
and a set of substations 111 connected to a grid for providing
power to the set of trains. The set of trains can include a set of
power consuming trains and a set of power regenerating trains.
[0041] The inputs to the method can include one or combination of a
description of the substations, power consuming trains and power
generating trains. The description can include one or combination
of an identification, location and power consumption and power
generation related information, in general the state of the
railway. The output of the method can include optimal power related
quantities at the substations and trains, particular the amount of
power regenerated by deaccelerating trains and fed back to the
substations.
[0042] The method optimizes 310 control parameters 315 controlling
at least part of the energy consumption of the railway system to
produce optimized control parameters 315 minimizing a total power
317 provided by the grid to satisfy a power demand of the railway
system. For example, the control parameters can include values of
voltages and currents for each substation and for each train. Next,
a command to control the energy consumption of the railway system
based on the optimized control parameters is generated 320. The
steps of the method can be performed by a processor 300.
[0043] The optimization 310 is subject to constraints 330 on
operations of the railway system. The constraints include, e.g.,
constrains on operation of the trains and constraints on operation
of the substation. In various embodiments, the constraints include
a complementarity constraint 335. The complementarity constraints
335 allow solving the optimization problem subject to discontinuity
constraints typically imposed on the railway systems.
[0044] For example, in one embodiment, the complementarity
constraint includes a product of a current at a substation and an
excessive voltage at the substation. This formulation allows
transforming discontinuities of the constraint of the substation
into the complementarity constraints, which, respectfully,
reformulates optimization subject to discontinuity constraints into
optimization subject to complementarity constraints 335.
[0045] In some embodiments, the method optimizes the control
parameters based on a state 345 of railway system. The state can be
determined 340 at a particular time, and can include location 348
of the trains, the power demand 346 of the power consuming trains,
and/or power excess of the power regenerating trains. For example,
a total power demand of the railway system can be determined based
on states of each train of the railway system, e.g., by subtracting
the power excess of the train from the power demand of the trains
and considering energy loses based on resistance of the power
lines. In one embodiment, the method determines the state based on
run curves 342 of the trains. The usage of the run curves allows
avoiding or minimizing real time measurements in the railway
system, which can be expensive. Additionally, one embodiment
modifies the determined state based on measurements 344 of the
state at the point of time, which can improve the accuracy of the
determination of the state while minimizing the amount of
measurements.
[0046] The method can be performed repeatedly to dynamically
determine a state of operation of the railway system, and optimize
power usage accordingly. For example, the optimization is performed
every five seconds or less. In addition, the processing can be
performed in a distributed manner, e.g., at the substations and/or
at trains.
[0047] The method can take as input, various measurements of
electrical conditions, and dynamically changing configuration of
the railway system as the trains move. The conditions are
constrained as described below. The input can be transformed to
determine a state of the railway system based on the sensed data
using, for example a supervisory control and data acquisition
(SCADA) system.
[0048] Substations
[0049] FIG. 4A shows feasible values 410 of voltages as a function
of currents at substations. As defined, herein, feasible means
capable of being according to, e.g., a feasible plan.
[0050] If no current is consumed, then the entire voltage axis
above V.sub.min.sup.F is feasible. The substations receive power
from an external electric grid and power the trains. A voltage and
current model for substation i is
V i .di-elect cons. { [ V min F , V max F - r i F I i ] if I i >
0 [ V min F , .infin. ) if I i = 0 } .A-inverted. i .di-elect cons.
N F , I i .gtoreq. 0 ( 1 ) ##EQU00001##
where V.sub.i is the voltage, I.sub.i is the current consumed,
V.sub.min.sup.F, V.sub.max.sup.F are minimal, maximal operational
voltages, and r.sub.i.sup.F is the internal resistance 220 at the
substations.
[0051] In one embodiment, a non-negativity requirement on the
current I.sub.i ensures that the substation does not feed power
back to the electric grid if the DC network does not include an
inverter. This restriction reflects the problem instances that are
considered, and is not a limitation for other embodiments. However,
if the DC network cannot consume all the regenerative power,
potentially damaging power surges can occur.
[0052] The voltage at the substation has a discontinuity 420 at
current 430 I.sub.i=0. In other words, there is an upper limit on
the voltage V.sub.i when power is consumed from the substation.
There are no upper limits when power is not consumed from the
substation. Discontinuities are not desirable for well-posed
nonlinear programs (NLP), which assume differentiability of
functions and constraints.
[0053] Therefore, various embodiments reformulate the model of Eqn.
(1) using complementarity constraints,
V i .ltoreq. V max F - r i F I i + .alpha. i V i .gtoreq. V min F I
i , .alpha. i .gtoreq. 0 , I i .alpha. i = 0. } .A-inverted. i
.di-elect cons. N F , ( 2 ) ##EQU00002##
where a product of a current at a substation I.sub.i and an
excessive voltage at the substation .alpha..sub.i is zero. This
constraint is precisely a complementarity constraint, which
requires that at any feasible solution either I.sub.i or
.alpha..sub.i vanishes. The value .alpha..sub.i is a measure of a
maximum voltage violation at a feeding substation.
[0054] Consequently, when I.sub.i>0, the scalar .alpha..sub.i=0
and the upper bound of V.sub.max.sup.R-r.sub.i.sup.F I.sub.i are
imposed on the voltage. When I.sub.i=0, the scalar
.alpha..sub.i.gtoreq.0 can be positive to allow the voltage to
exceed V.sub.max. The above constraints are differentiable. Hence,
the complementarity constraints enable us to model the otherwise
unsmooth substation behavior using smooth constraints.
[0055] Power Consuming Trains
[0056] FIG. 413 shows the relationship 440 of voltage and current
for power consuming trains (i.epsilon.N.sup.A). The power
P.sub.i.sup.A consumed by each of these trains can be expressed
as
V i I i = P i A I i .ltoreq. 0 } .A-inverted. i .di-elect cons. N A
. ( 3 ) ##EQU00003##
[0057] A non-positively bound on the current ensures that only
these trains consume power.
[0058] Regenerative Trains
[0059] FIG. 4C shows the relationship 450 of voltage and current
for the regenerative trains (R). The regenerative trains
(I.epsilon.N.sup.R) can, supply power generated from braking to the
DC network. The amount of generated power is
P i R .ltoreq. { P max R if V i .ltoreq. V min R P max R V max R -
V i V max R - V min R if V min R .ltoreq. V i .ltoreq. V max R P i
R .gtoreq. 0 P i R = V i I i I i .gtoreq. 0 } .A-inverted. i
.di-elect cons. N R , ( 4 ) ##EQU00004##
where P.sub.max.sup.R is the maximal power that is available from
the regenerative train, and V.sub.min.sup.R,V.sub.max.sup.R are
bounds on the voltages. The amount of power that can be recovered
is greatest when the voltage at the train is less than
V.sub.min.sup.R. At higher voltages, the amount of power available
for recovery decreases linearly with increasing voltage
V.sub.i.gtoreq.V.sub.min. For voltages higher than V.sub.max.sup.R,
power is not available from the regenerative trains.
[0060] FIG. 5 shows an example of the optimization 510 subject to
complementarity constraints using the relationships 520 of various
measurements of voltages, currents, resistances at nodes in the
railway system 100. Notably, the optimization ensures that the
voltages are within minimal and maximal limits.
[0061] Complementarity Constraints
[0062] FIG. 6 shows a schematic of a method for optimizing energy
consumption of the railway system according to some embodiments of
the invention. The optimization of the energy consumption is
formulated 610 subject to complementarity constraints. The example
of such formulation is optimization 510 of FIG. 5.
[0063] The plot 620 shows a feasible region of points representing
the solution satisfying the complementarity constraints. In the
plot 620, feasible points are on one of the axes. For example, the
feasible points include a feasible point 622, i.e., I.sub.i=0, or a
feasible point 624, i.e., .alpha..sub.i=0.
[0064] Some embodiments use non-linear optimization methods to
determine the control parameters. However, due to the limitation on
the feasibility region defined by the complementarity constraints,
some non-linear optimization methods solve complementarity
constraint problem with unacceptably high error rate. For example,
interior point methods, which are routinely used for solving
nonlinear programs, require (i) strictly feasible interior, and
(ii) gradients of active constraints that are linearly independent.
However, the complementarity constraint problem is degenerated and
not well suited for solving by interior point algorithms.
[0065] Based on this realization, some embodiments of the invention
optimize the control parameters based on iterative relaxation of
the complementarity constraint. The iterative relaxation of the
complementarity constraint can approximate ill-posed problem as a
set of well-posed problems, which increase accuracy of the solution
and reduce computational time.
[0066] Accordingly, some embodiments relax 630 the feasibility
region with relaxation parameter, e.g., .mu.>0 635. The
relaxation parameter 633 defines the curve 635 such that feasible
points of the solution lay between the axis 620 and the curve 635.
There exists a strictly feasible interior point 639, and the
problem is well-posed.
[0067] The optimization using relaxation of the complementarity
constraint is solved iteratively. In one embodiment, the relaxation
parameter is modified monotonically 640 forming, e.g., the curves
635-637. For example, the interior point method iteratively reduces
645 value of .mu. to zero. As .mu. approaches zero, the method
recovers the complementarity constraints. This yields strictly a
feasible interior for some complementarity constraints even as .mu.
approaches zero, with a better performance. Specifically,
approximating ill-posed problem as a set of well-posed problems
increases accuracy of the solution and reduces computational time.
Below are the various method used by some embodiments of the
invention for solving a program with complementarity
constraint.
[0068] Mathematical Program with Complementarity Constraints
(MPCC)
[0069] In one embodiment, the optimization problem is formulated
as
min i .di-elect cons. N F V i I i Total feeding power s . t . V i -
V i + 1 = r i L I i L .A-inverted. i = 1 , , N - 1 Voltage drop I 1
= I 1 L I i L + I i + 1 = I i + 1 L .A-inverted. i = 1 , , N - 2 I
N - 1 L + I N = 0 } Current balance Constraints in ( 2 )
Constraints in ( 3 ) Constraints in ( 4 ) Feeding station Power
drawing trains Decelerating train . ( 5 ) ##EQU00005##
[0070] in the above formulation (5), the optimization is formulated
as a minimization of the sum of the power (.SIGMA.V.sub.iI.sub.i)
supplied by the electric grid. The constraints include the voltage
drop due to the resistance in the lines connecting the nodes, the
need to balance current flowing on these lines, and the constraints
as in Eqns. (2-4). The formulation above assumes the DC network is
represented by a line graph for ease of illustration. This is not a
restriction of the approach and it can accommodate more general
graph representations.
[0071] Because the embodiment uses the complementarity formulation
of Eqn. (2) for the substations, the above problem is an instance
of MPCC. The MPCC is a class of .degree. nonlinear programs (NLPs)
that includes complementarity constraints.
[0072] For any MPCC, a Linear Independence Constraint Qualification
(LICQ) fails to hold at any feasible point. LICQ is typically
assumed at a solution point of a NLP, and this ensures uniqueness
of multipliers. The lack of this property implies that the
multiplier set is not unique at a minimizer of the MPCC.
Furthermore, the failure of LICQ at any feasible point implies
difficulty in step calculation as the Newton system is singular.
Furthermore, there is no strict interior for the feasible region.
This implies the failure of a weaker Mangasarian Fromovitz
Constraint Qualification (MFCQ). The failure of MFCQ renders the
multiplier set unbounded at the solution.
[0073] Interior point methods 360 can solve large-scale inequality
constrained NLPs. Interior methods require a strictly feasible
interior for the constraint set of the NLP. The lack of strict
interior also makes it difficult to apply interior point methods to
MPCC. However, the complementarity constraints can be reformulated
to allow a strictly feasible interior to which interior point
methods can be applied.
[0074] Interior Point Method
[0075] Some embodiments of the invention are based on a realization
that approximation reformulates complementarity constraints as
inequality constraints. Optimization under inequality constraints
is a difficult optimization problem. However, the interior point
method was designed for inequality constraints, and thus,
advantageously used in the iterative relaxation method.
[0076] For example, some embodiments use at least two
reformulations of the MPCC, which enable interior point methods to
be applied. For the purposes of brevity, we express the MPCC in
Eqn. (5) as,
min f ( V , I ) s . t . h ( V , I , I L , .alpha. ) = 0 g ( V , I )
.gtoreq. 0 I i , .alpha. i .gtoreq. 0 , I i .alpha. i = 0
.A-inverted. i .di-elect cons. N F , ( 6 ) ##EQU00006##
Where V=(V.sub.1, . . . , V.sub.|N|) and I=(I.sub.1, . . . ,
I.sub.|N|) are sets of voltages and currents supplied to the DC
network by the electric grid and the generating trains and drawn by
the consuming trains at the N nodes, I.sup.L=(I.sub.1.sup.L, . . .
, I.sub.|N|-1.sup.L) is the set of en between the nodes, and
.alpha.=(.alpha..sub.i).sub.i.epsilon.N.sub.A is the set of
variables denoting an amount by which the substation voltages
exceed an operational maximal at the substations. In (6), the
reformulation f(.epsilon.V.sub.iI.sub.i) includes the equality and
inequality constraints.
[0077] The function
h:R.sup.3|N|-1+|N.sup.A.sup.|.fwdarw.R.sup.m.sup.e denotes the set
of real equality constraints in Eqn. (5), with the exception of the
complementarity constraints, the function
g:R.sup.2|N|.fwdarw.R.sup.m.sup.i represents the inequality
constraints in Eqn. (5) with the exception of bounds on
I.sub.i,.alpha..sub.i for and i.epsilon.N.sup.F, and f represents
the objective function in Eqn. (5).
[0078] The interior point method applied according to Eqn. (6)
solves the following equality constrained problem. The problem is
obtained by posing the inequality constraints in the objective
function with a barrier term
min f ( V , I ) - .mu. i = 1 m i ln ( s i ) - .mu. i .di-elect
cons. N A ( ln ( I i ) + ln ( .alpha. i ) ) s . t . h ( V , I , I L
, .alpha. ) = 0 g ( V , I ) + s = 0 I i .alpha. i = 0 .A-inverted.
i .di-elect cons. N F , ( 7 ) ##EQU00007##
where .mu.>0 is the barrier parameter, and S.sub.i:i.epsilon.{1,
. . . , m.sub.i} are slack variables for the inequality constraints
g. In an optimization problem, a slack variable is a variable that
is added to an inequality constraint to transform it to an
equality. Barrier methods are an alternative for constrained
optimization. Barrier methods use the barrier parameter to force
the iterates to remain interior to the feasible domain, and biases
the iterates to remain away from the boundary of the feasible
region.
[0079] Conceptually, the interior point method starts from a high
barrier value of .mu.>0, and solves a sequence of problems of
Eqn. (0.7) for decreasing values of .mu.. An initial iterate, such
that s>0, is
I.sub.i.alpha..sub.i>0.A-inverted.i.epsilon.N.sup.A. The barrier
parameter iterates in interior of the nonnegative orthant
(hyperoctant) as the objective approaches infinity on the boundary.
However, as .mu. decreases, the iterates are allowed to approach
the boundary of the orthant, thus recovering a solution that lies
at the bounds. In the limit, under certain assumptions, the
sequence of solutions for different .mu. approaches the solution of
Eqn. (6). However, in the case of MPCCs, there are no
.alpha..sub.i,I.sub.i>0 that are feasible in Eqn. (7).
[0080] Penalty Reformulation
[0081] In this embodiment, the complementarity constraints in Eqn.
(6) are penalized in the objective function as
f .pi. ( V , I ) := f ( V , I ) + .pi. i .di-elect cons. N A I i
.alpha. i , ( 10 ) ##EQU00008##
where .pi. is a penalty parameter.
[0082] With the above formulation, the optimization problem always
possesses a strict interior. This is in contrast with a prior art
approach where the problem loses the strict interior in the limit.
Thus, the penalty formulation removes some of the deficiencies of
the inequality formulation. However, it is possible that at a
solution of the penalized formulation some of I.sub.i,.alpha..sub.i
can violate the complementarity constraint 335. The barrier problem
for this relaxation is
min f .pi. ( V , I ) - .mu. i = 1 m i ln ( s i ) - .mu. i .di-elect
cons. N A ( ln ( I i ) + ln ( .alpha. i ) ) s . t . h ( V , I , I L
, .alpha. ) = 0 g ( V , I ) + s = 0 , ( 11 ) ##EQU00009##
[0083] Inequality Reformulation
[0084] To implement complementarity relaxation, some embodiments
formulate the complementarity constraint as inequalities. Some
variations of those embodiments use the barrier parameter to modify
the constraint evaluation for the complementarity constraints.
[0085] FIG. 7 shows plots 710 and 720 illustrating relaxation 730
of the complementarity constraints by multiple parameters, e.g.,
the barrier parameter .mu., and the relaxation parameter .delta..
Both parameters can be tightened dependently. In this embodiment,
the constraints are strictly feasible interior for some
complementarity constraints even as .mu. approaches zero, which can
increase a performance of the optimization method.
[0086] For example, in one embodiment, the complementarity
constraint is reformulated as
I.sub.i.alpha..sub.i.ltoreq..delta..mu., (8)
where .delta.>0. The complementarity constraint is increased by
an amount proportional to the barrier parameter. Hence, for all
.mu.>0, there exists a strict interior for the barrier problem.
As the barrier parameter .mu. approaches zero, we converge to a
solution of the MPCC in Eqn. (6). The barrier problem for this
relaxation is
min f ( V , I ) - .mu. i = 1 m i ln ( s i ) - .mu. i .di-elect
cons. N A ( ln ( I i ) + ln ( .alpha. i ) + ln ( s i c ) ) s . t .
h ( V , I , I L , .alpha. ) = 0 g ( V , I ) + s = 0 I i .alpha. i +
s i c = .delta..mu. .A-inverted. i .di-elect cons. N F . ( 9 )
##EQU00010##
[0087] FIG. 8 shows the psi of the method for the above
complementarity inequality reformulation. Steps 1-3 of the method
initialize the variables, set the iteration parameters, and select
constraints. Steps 5-7 solves iteratively the above barrier problem
(9).
[0088] Adaptive Relaxation
[0089] FIG. 9 shows a schematic of another method according another
embodiment of the invention. This embodiment is based on a
realization that with reduction of the relaxation parameter, the
well-posed problems are progressively transformed into ill-posed
problems. Therefore, it is advantageous to reduce the relaxation
parameter only when necessary. Thus, one embodiment of the
invention optimizes the control parameters iteratively using
adaptive modification 910 of a relaxation parameter. In this
embodiment, for each iteration, the relaxation parameter is
modified 720 only if the complementarity constraint of the solution
tot a current value of the relaxation parameter is violated.
[0090] For example, in one iteration of the method for optimizing
the control parameters iteratively using adaptive modification of a
relaxation parameter, a current value of the relaxation parameter
define the feasible region under the curve 635. If the solution 930
of the method for this iteration is feasible, i.e., under the curve
635, the next iteration does not update the relaxation parameter,
skip the relaxation parameter that defines the curve 920, and uses
the relaxation parameter of the curve 635 instead. Thus the next
iteration optimizes the solution 930 using the relaxation parameter
of the curve 635, which is more well-posed problem than problem for
relaxation parameter of the curve 920. If the solution of the next
iteration violates the relaxed complementarity constraints, the
relaxation parameter is modified, e.g., to correspond to the curve
637.
[0091] Inequality Reformulation with Adaptive Relaxation
[0092] FIG. 10 shows a block diagram of a method according to one
embodiment of the invention. In one variation of this embodiment,
the complementarity constraint for a substation of the railway
system includes a product of a current at the substation and an
excessive voltage at the substation. The optimization according, to
this embodiment reformulates the complementarity constraint as an
inequality, such that the product of the current and the excessive
voltage is less than a relaxation parameter 1060. Next, this
embodiment determines 1010 the control parameters 1050 iteratively
based on testing 1020 the inequality, wherein a value of the
relaxation parameter is adaptively reduced for subsequent
iterations. In one variation of this embodiment, the current
iteration includes determining the control parameters based on
values of the control parameters determined during a previous
iteration. The inequality is tested 1020 using values of the
control parameters corresponding to that of the current and the
excessive voltage of the substation, and the value of the
relaxation parameter is modified 1030 if the inequality is violated
1040.
[0093] Some embodiments combine adaptive relaxation with an
interior point method to further optimize the solution. For
example, one embodiment determines the relaxation parameter to be
proportional to the barrier parameter 1070. This proportionality
simplifies determining of the modified value of relaxation
parameter in subsequent iterations. This is because the barrier
parameters are monotonically decreasing, and when the relaxed
complementarity constraint is violated, the relaxation parameter is
modified to match appropriately the reduction in barrier
parameter.
[0094] For example, the embodiment relaxes each complementarity
constraint as
I.sub.i.alpha..sub.i.ltoreq..eta..sub.i, (12)
where .eta..sub.i>0. The relaxation is adaptively tightened. The
barrier problem for this relaxation is
min f ( V , I ) - .mu. i = 1 m i ln ( s i ) - .mu. i .di-elect
cons. N A ( ln ( I i ) + ln ( .alpha. i ) + ln ( s i c ) ) s . t .
h ( V , I , I L , .alpha. ) = 0 g ( V , I ) + s = 0 I i .alpha. i +
s i c = .eta. i .A-inverted. i .di-elect cons. N F . ( 13 )
##EQU00011##
[0095] FIG. 11 shows the pseudocode for solving the barrier problem
(13) with adaptive relaxation 1100.
[0096] The above-described embodiments of the present invention can
be implemented in any of numerous ways. For example, the
embodiments may be implemented using hardware, software or a
combination thereof. When implemented in software, the software
code can be executed on any suitable processor or collection of
processors, whether provided in a single computer or distributed
among multiple computers. Such processors may be implemented as
integrated circuits, with one or more processors in an integrated
circuit component. Though, a processor may be implemented using
circuitry in any suitable format.
[0097] Further, it should be appreciated that a computer may be
embodied in any of a number of forms such as a rack-mounted
computer, a desktop computer, a laptop computer, minicomputer, or a
tablet computer. Also, a computer may have one or more input and
output devices. These devices can be used, among other things, to
present a user interface. Examples of output devices that can be
used to provide a user interface include printers or display
screens for visual presentation of output and speakers or other
sound generating devices for audible presentation of output.
Examples of input devices that can be used for a user interface
include keyboards, and pointing devices, such as mice, touch pads,
and digitizing tablets. As another example, a computer may receive
input information through speech recognition or in other audible
format.
[0098] Such computers may be interconnected by one or more networks
in any suitable form, including as a local area network or a wide
area network, such as an enterprise network or the Internet. Such
networks may be based on any suitable technology and may operate
according to any suitable protocol and may include wireless
networks, wired networks or fiber optic networks.
[0099] Also, the various methods or processes outlined herein may
be coded as software that is executable on one or more processors
that employ any one of a variety of operating systems or platforms.
Additionally, such software may be written using any of a number of
suitable programming languages and/or programming or scripting
tools, and also may be compiled as executable machine language code
or intermediate code that is executed on a framework or virtual
machine. In this respect, the invention may be embodied as a
computer readable storage medium or multiple computer readable
media, e.g., a computer memory, compact discs (CD), optical discs,
digital video disks (DVD), magnetic tapes, and flash memories.
Alternatively or additionally, the invention may be embodied as a
computer readable medium other than a computer-readable storage
medium, such as a propagating signal.
[0100] The terms "program" or "software" are used herein in a
generic sense to refer to any type of computer code or set of
computer-executable instructions that can be employed to program a
computer or other processor to implement various aspects of the
present invention as discussed above.
[0101] Computer-executable instructions may be in many forms, such
as program modules, executed by one or more computers or other
devices. Generally, program modules include routines, programs,
objects, components, data structures that perform particular tasks
or implement particular abstract data types. Typically the
functionality of the program modules may be combined or distributed
as desired in various embodiments.
[0102] Also, the embodiments of the invention may be embodied as a
method, of which an example has been provided. The acts performed
as part of the method may be ordered in any suitable way.
Accordingly, embodiments may be constructed in which acts are
performed in an order different than illustrated, which may include
performing some acts simultaneously, even though shown as
sequential acts in illustrative embodiments.
[0103] Use of ordinal terms such as "first," "second," in the
claims to modify a claim element does not by itself connote any
priority, precedence, or order alone claim element over another or
the temporal order in which acts of a method are performed, but are
used merely as labels to distinguish one claim element having a
certain name from another element having a same name (but for use
of the ordinal term) to distinguish the claim elements.
[0104] Although the invention has been described with reference to
certain preferred embodiments, it is to be understood that various
other adaptations and modifications can be made within the spirit
and scope of the invention. Therefore, it is the object of the
append claims to cover all such variations and modifications as
come within the true spirit and scope of the invention.
* * * * *