From: AAAI Technical Report WS-02-06. Compilation copyright © 2002, AAAI (www.aaai.org). All rights reserved.
En-Route Sector Metering using a Game-theoretic Approach
Goutam Satapathy*,
Vikram Manikonda*, John Robinson #, #Todd Farley
*IntelligentAutomation
Inc.
7519StandishPlace, Suite 200
Rockville, Maryland20879
goutam@i-a-i.cora,
vikram@i-a-i.com
Abstract
Currently, Traffic Management
Coordinators in the
Air Route Traffic Control Centers establish flow
constraints at various sector meter fixes, based on
their best estimates of the predicted traffic demand
into their sector. Sector flow rates are not coordinated
between neighboring sectors or centers. Incorrect
sector meteringrates can lead often lead to a cascade
effect resulting in delays in the schedulesof aircraft
several sectors away. In this paper we develop an
approach for dynamicsector metering based on game
theory. The approach is based on a Bayesian game
with communication,wherein the sectors determine
mutually beneficial metering rates (based on
collaboration and exchange of metering rates and
negotiated ScheduledTimesof Arrival) chosen so as
to optimize delay, controller workloadand capacity.
Weformalize the model for a simplified scenario
consisting of two sectors belonging to two different
centers, attempting to set the flow rates at their
boundaries. The simplified modelsfor the two-player
(sector) game capture the coupling of dynamics
between the two sectors and possible interactions
between incoming and outgoing traffic flows. The
inboundaircraft from other sectors and outboundflow
rate restriction to other sectors are generatedfrom a
stochastic time series model. To demonstrate
feasibility we implementour approachon a simplified
version of agent-based decision support to captures
inter center/sector communication
and decentralized
decision-making
#NASAAmesResearch Center
Moffett Field, CA94035-1000
jerobinson@~rc.nasa.gov
ffarle¥@~rc.nasa.gov
Currently, Traffic ManagementCoordinators (TMCs)
the Air Route Traffic Conlrol Centers (ARTCC)
establish
flow constraints at various sector meter fixes, based on
their best estimates of the predicted traffic demand,or
sectors affected by adverse weather conditions. Arrival
rates into a sector are usually determinedfor sectors close
to airports with high demand,and these rates typically flow
down(radially outward) to en route sectors. Typically
controllers add a safety buffer to this rate to accommodate
for any unforeseencircumstances.Sector flow rates are not
coordinated betweenneighboring sectors or centers based
on the estimated demandon their sectors. Furthermore,
metering times at sectors are assigned based on miles-intrail restrictions (distance-based spacing). While this
approachworksreasonably well for low traffic densities,
for higher densities (coupled with the dynamics and
uncertaintyof the traffic) this approachfails. Inappropriate
sector meteringrates near a busyairport can often lead to a
cascade effect resulting in delays in the schedules of
various aircraft several sectors away.In addition, it has
beenobservedthat fixed miles-in-trail restrictions basedon
quantization to maintain the metering rates can be
inefficient. Controller workloadcan be improvedby giving
the controller the flexibility to imposevariable in-trail
restrictions betweenaircraft, as long as an averagemetering
rate over a certain duration is maintained,and safety is not
compromised.
Time-basedspacing (or "metering")of arrival traffic flows
has been shownto be more efficient than distance-based
spacing, or "in-trail restrictions," as shownby Sokkappa
in
a theoretical study (Sokakapa.1989). Sokkappa’sfindings
were validated by Swenson, who documentedsignificant
improvements in delay and throughput versus in-trail
spacing using NASA’s Traffic Management Advisor
(TMA)in operational field tests at Dallas-Fort Worth
International Airport (DFW)(Swenson, Hoang, et.
1997)
Introduction
As air traffic congestion and delay have increased in the
post-deregulationera, the air transportation community
has
sought to identify moreefficient methodsof Traffic Flow
Management(TFM) in order to best utilize existing
capacity. AdvancedTFMstrategies have the potential to
improve system throughput and reduce delay without
requiting substantial rework of the National Airspace
System (NAS)infrastructure (e.g., airspace redesign,
Communication, Navigation and Surveillance (CNS)
upgrades, etc.) and without imposingnewaircraft equipage
requirements.
Researchers at NASA
AmesResearch Center are pursuing
a distributed scheduling concept to implementtime-based
meteringin constrained, transition airspace (Farley, Foster,
Hoangand Lee 2001). Instead of relying on a single,
monolithic scheduler to computea workableand efficient
schedule for the entire region, as is the ease for current
time-based metering systems, a distributed scheduling
Copyright
©2000,American
Association
for ArtificialIntelligence
(www.aaai.org).
Allrightsreserved.
66
scheme relies on a loosely integrated network of
schedulers, each governing small airspace regions. A
scheduler might govern an area as small as a single
airspace sector or as large as an enroute Center. The
envisioned time-based metering operation will be more
sensitive to local ATCconstraints and goals, but will still
enable facilities to implementan efficient, time-based
metering approachto air traffic management
on a regional
scale.
acceptance rates and minimumseparation requirements
only. Uncertainty in arrival time estimates was included.
Lowfidelity models for computation of estimated and
scheduled boundary-crossing times (ETAs and STAs,
(Scheduled Time of Arrival) respectively), trajectory
conflicts, and controller advisories werealso developed.A
simplified implementationof the algorithms is adoptedfor
this study, capturing inter-sector communicationand
decentralized decision-making.
A key challenge in this workis to determinehow---andto
what extent--to couple the distributed schedulers in order
to generate a set of schedules which are individually
workableand collectively beneficial. That is, collectively
they producea significant throughputbenefit.
Due to space restriction, we only briefly describe the
domain. Readers mayrefer to (Nolan, 1994) for a more
detailed description of current NAS
operations.
Notation
This paper develops a game-theoretic approach for
coupling distributed scheduling algorithms. The approach
is based on a Bayesian gamewith communication,wherein
distributed schedulers negotiate acceptance rates to
optimize system-wide delay, controller workload, and
throughput. Oneinstance of the scheduler is assumedto be
computingtrajectory projections (estimated boundarycrossing times, or ETAs(Estimated Timeof Arrival))
each defined airspace region. The ETAsare exchanged
between neighboring agents and are compared against
negotiated acceptance rates. Overflow situations are
resolved by negotiating changesin acceptancerates or by
assigning delay to the offendingaircraft. For the purposes
of this initial investigation, a modelwasformalizedbased
on a simplified airspace consisting of two neighboring
sectors (Figure 1). The sectors set acceptance rates for
arrivals across the shared boundaryto their respective
sector. This simplified modelfor a two-player (i.e., twosector) game captured the coupling of dynamicsbetween
the two sectors, and it captured the possible interactions
SiC": Sectori in center Cu
mf : Sector meter fix (inboundor outbound)
ctrl: A control point (A control point is an intersection
or mergingpoint of two airways or a bifurcation
point of an airway).
m,: The indices of the inbound meter fixes with
respect to the sector
mo: The indices of the outbound meter fixes with
respect to the sector
r=y, r,,,, rmo: The sector meterfix flow rate through
meter fix mfor meter fix indexed by m~or mo
At: Thetime unit in the specified flow rate (i.e., 10
minutes if the flow rate is specified as 4/10
minutes).
sarc,: Sector arrival rate - the rate at whichaircraft
enter the sector S/c" . This is equal to the sumof
flow rates into the sector S/c’- E r,,, .
J
sdrc~ : Sector departure rate - the rate at which
}I ulanIImmmm~mulllllmu¯~
aircra~ depart the sectorsc" . Thisis equal to the
sumof flow rates fromthe sector S/C" into other
sectors-
E r,,omo E~u
The capacity of a sector - the maximum
number
of aircraft that can be present in a sector at a
given time. It differs from sector to sector
depending the sector controller’s ability to
manageaircraft
to~ The absolute value of time whena stage game
starts. (The gametheoretic approachesproposed
for the sector metering problem consists of
several stage gamesand is played an infinite
numberof times.)
AT: The duration of the stage game
c:
oj
Figure1: A layoutof the airwa~,sin twosectors for the two
player game-theoreticmodelfor sector metering
betweenincomingand outgoing traffic flows. The inbound
aircraft from other (non-playing)sectors and the outbound
flow rate restrictions imposedby those other sectors were
generated from a stochastic time-series model. The
metering times assigned within each sector to complywith
the negotiated acceptancerates were not based uponfixed,
in-trail spacing restrictions. Rather, they were basedupon
assignment of minimumdelay subject to the negotiated
67
s:
The minimum
aircraft separation distance based
on safety requirements
v,,~: The maximum
cruise speedof an aircraft.
v~n: The minimum
cruise speed of an aircraft.
The minimum
separation time that must be kept
based on FAA’s restriction
on the minimum
separation distance and the assumption of
uniform cruise speed maintained by all aircraft
(Notethat 6 is not sames/v,=).
j, k: The aircraft are indexed in such a waythat the
PSTAof a lowindexedaircraft is earlier than the
PSTA
of a higher indexedaircraft.
!:
A control point in sector’s airways. The control
pointsare indexedas/1, 12, .... li, ... suchthat the
STAs of the incoming aircraft
must be
determined
at li beforel~+1.
du, :The distance betweentwocontrol points.
d,,,m,
:Thedistance
distancebetween
betweenatwo
meter
dmd
meter
fixfixes.
and a control
" :The
aj,m,
point.
:Thefh aircraft enteringat meth inboundmeterfix
ETAo~,mf
: ExpectedTimeof Arrival of an aircraflj
at a meter fix mf, a control point, inboundor
outboundmeter fix
STAoj,~.m
f : ScheduledTimeof Arrival of an aircraftj
at a meter fix mf(STA
is alwaysequal or greater
than ETA)
PSTAoj,,,,m,: Predicted STAof the aircraft aj,m, at
meth inboundmeter fix such that PSTAoj,~,mr<
PSTA%~.,,,m. (Predicted STAsare used only
for optimization)
PSTAo/~,t,: Predicted STAof the aircraft a/,me at lw
control point. (Predicted STAsare used only for
optimization)
PSTAoj,.,,m,: Predicted STAof the aircraft a/,m, at
moth outbound meter fix. (Predicted STAsare
used only for optimization)
PETA%,,.mo
: Predicted ETAof the aircraft a/,me at
th
mo
outbound meter fix, given that
PETAoj~.m,= PSTAoj,,,mt " (Predicted ETAs
are used only for optimization)
B(/, u, at, az): Abeta probabilitydistribution overl_<x_<
u, whereat and a2 are the two shape parameters
68
Formulation of the sector
a game-theoretic
metering problem as
problem
Game-theoretic models are well suited to determining
sector metering rates across sector boundaries as this
involves negotiation betweenthe sectors. It is only by
negotiating the flow rates that sector controllers can reduce
their workloadand increase airspace safety. For example,
one sector can negotiate with another sector to set a flow
rate so that the former sector controller does not get
overloadedwith advisories, while the latter sector does not
obtain any significant increase in its workloadby holding
or delaying aircraft. In this paper we formalize our
approach using a simplified scenario consisting of two
sectors belongingto twodifferent centers, attemptingto set
the flow rates at their boundary.The simplified modelfor
the two-player (sector) game captures the coupling
dynamicsbetweenthe twosectors and possible interactions
between incoming and outgoing traffic flows. Figure 1
showsthe layout of the scenario chosen as a basis of our
formulation.
In Figure 1 S~/" denotes Sector i in center u, rnf denotes
the sector meterfix, ctri denotesa control point. Thesolid
and the dashed lines showairways in the west-east and
east-west directions respectively. The problem in this
context is describedas
The sectors Scl and SI q play a game to set an
equilibriumflow (acceptance)rate across their sector
boundary,that should be maintainedfor duration of
AT seconds into the future. After the AT seconds,
they play the gameagainto set the sector flow rates
for the next AT seconds.
q
sector
change
in order if,
to given
increase
its
TheflowSratecannot
is said
to be rmf
in ’equilibrium
rmL
, the
utility, and given rmf’ the sector "Sc2 cannotchangermf
~ in
order to increase its y utility. Hererm/, denotes the sector
meterfix flow rate throughthe meterfix. Notethat wehave
assumedthat the twosectors lie in twodifferent centers. In
this paper, we model the game as a Bayesian game with
Communication.
Bayesian Game with Communication
In this formulation, the gameconsists of two instances of
TMA,one for each sector controller deciding a rate that is
fixed for a duration of ATsee (the duration of the game).
Eachcontroller will "push" aircraft into the other sector
and schedule aircraft to the sector boundarysuch that the
actual time of arrival at the sector boundaryconformsto
the rate (action). Notethat "conformingto a flow rate"
an importantpart of the action. For example,if the decision
is to push2 aircraft per AT,then the sector can push the
two aircraft immediately,one after the other, and not send
any aircraft for the rest of the interval. Alternatively, the
equilibrium ~ of actions. In the Bayesian game with
communication, a message can be considered as the
decision regarding the flow rate madeby the players. Note
that in this context the action is determining the
corresponding STAsof aircraft to the meter fix that
conform to the prescribed flow rate. Therefore, the
exchangeof messagesdoes not really reveal the actions to
be played by the players z. In fact, a player may
"cheat/defect" by not strictly conformingto the flow rate
that was exchangedin the last message.Figure 3 showsthe
relationship between the duration of negotiation/
communication,
the action and the duration of the game.
sector could send one aircraft at the beginning of the
interval and the other towards the end. The waythe sector
conformsto the flow rate will affect the utility of the other
sector. Figure 2 summarizesthe various componentsof the
game.
This game can be modeled as a strategic game with
incompleteinformation.Recall, in a strategic game,players
take actions simultaneously and independently. In a game
with incompleteinformation (i.e., payoff/ utility due to
their actions are not knownto each other - Bayesian
games) and absence of commonknowledge (i.e., prior
probabilities of the opponents’action that is knownto or
assumedby each player is not a commonknowledge),the
notion of equilibrium is hard to define. In fact, the games
without commonknowledgeplayed by two players can be
described as each player playing distinctly different games
without each other’s knowledgeand hence equilibrium
cannot be defined in such cases. Therefore, we transform
the Bayesian game to be played by the sector
Absolute
start timeor
7
T
AT
~’x
Communication
/ NegotiationPeriodto Set SectorBotmdary
FlowRatefor NextATPeriod
ATcan be I hour, 4 hours, or morning,aftemoonetc. or basedon sector
¢or~oller’sshift Ix’riod
Game:The game consists of negotiating for a flow
rate and executing an action (see action definition)
accordingly over a AT period during which sector
controllers use TMA
to monitorand advise the aircraft
pilots entering his/her sector andhandover the aircraft
leaving his/her sector to the other sectors. At the
beginning of every AT period, the player also
communicates with the neighboring player by
exchanging messages about the cross-boundary flow
rates.
Player: Sector controller and his/her game-theoretic
tool assisting him/herto set the flowrate.
Message:The rate at whicha sector (e.g., ci )will
push/ handoff the aircraft into the neighboringsector
(e.g., S~t2 ) duringthe game,that is the ATperiod.
Action:Theaction of a sector controller (e.g., cl )is
Figure 3: Evolution of Timein the Game
Playing
the Game
The solution of the sector-metering problemas formulated
in the earlier section requires the identification and
development of a methodology/approach to address the
following aspects (inputs, models,optimization algorithms
etc.) of the game:
¯
¯
to maintain the STAsof the aircraft leaving his/her
sector into the neighboring sector (e.g., SC2),
conformingto the flow rate that he/she had exchanged
as a message.
¯
Figure 2 Componentsof the Bayesian Gamewith
Communication
Identification of the utility functionto be optimizedin
the game, and the inputs, variables and decision
parametersof the game
Developmentof traffic flow models needed for data
generation (e.g. ETAsof the flights entering a sector)
and optimization
Development
of an optimization algorithm, tailored to
meet the specific needs of this problem. The players
choose an action (i.e., decides STAsof the flights
leavingits sector) that maximizes
its utility.
Anequilibriumis Pareto efficient if a specified function
of the utilities of the players at the equilibrium is
maximum.
For example, the function could be sum of the
equilibrium utilities or product of equilibrium utilities
IZeuthenstrategy)
Unlike a typical strategic gamediscussed in literature,
here the duration to complete the action is not
instantaneous.Asa result, a player / sector controller can
see part of its opponent’saction before he completeshis
ownaction. In that sense, the gamebeing modeledis not
strictly a strategic gameas is discussedin the literature.
controllers/players every ATseconds into a game with
communication.The motivation is to exchange messages
(or pre-play communication) in order to uncover the
commonknowledge and arrive at a Pareto-efficient
69
inflow rates. Thus, sarff, = rmf° + r,,fc + r,,f, and
In the following subsections we discuss the technical
approachadoptedin this paper to address these aspects of
the game
sar~2=rmf" +r~f f +rmfy.In this paper,rmA, rmA, r~f ,
and rmff are calculated from a time series modelthat
Sector Utility
Wemeasure a sector’s utility as a wei~ted sum of the
following elements (e.g., for sector S~ , componentsof
the utility functionare):
1. Aircraft
delay due to traffic,
i.e.
predicts the inflow rate throughthe inboundmeter fix
for every AT period. These flow rates are not
negotiable, rmf. and r,,,fyon the other hand are
negotiable and are determinedthrough a stage game.
3.
E E PSTAaia~.mo-PETAaj~",’too ]for each
aircraft ay,m, leaving the sector at a meter fix
indexed by mo
2,
E r=- E
rmo)XAT]/cl"
E
l,
2.
4.
+rmA+rraf~
In
the proposed set up
PETAoj,~,,,o
The Predicted
is thetheETA
of an
aircraft
ay.m" :(that
has not yetETA
entered
sector’s
airspace) at the outboundmeter fix mo,. Note that
ETAsare calculated as soon as the aircraft enters the
sector’s airspace. ETAsof the aircraft at the sector
boundary leaving the sectors Scl and Sl c2 are
typically calculated using extensions of the Route
Analyzer (RA) and Trajectory Synthesis (TS) modules
of the CTAS- DynamicPlanner (DP) (Wong2000).
These tools must be used to calculate PETAstoo.
However,in our experimental setup, we calculate the
PETAsof the aircraft leaving the sectors S1c~ and
Sc2 based on a simple calculation that does not
Workloadon the sector controller (Numberof
advisories issued)
Variables, and Decision parameters
The variables and decision parametersof interest to us in
this gameinclude capacity, sector arrival rate, sector
departure rate, ETAsand STAsto meter fixes. A key
design issue is the ability to either measureor accurately
estimate these variables. In somesettings, someof these
parameters can be directly obtained from high fidelity
TMAtools such as Center-TRACON
Automation System
(CTAS)(Erzberger 1994). However,for the purpose of
investigation, we developed somesimple approaches to
determinethese parameters.
=rmfg
rmA , rmfa, rmf , andrmfhare calculated for everygame
(i.e., for every ATperiod) based on a time series
model. These rates are also not negotiable. Based on
the set-up of the game, we mayconsider somerates as
completelyinflexible (hard constraints) and somethat
can changesubject to a penalty.
rra,
(same as Sarsc. ) and E rmo (same as sdrsc. )
moEff"
are the arrival and departure rates of the sector
andc is the capacityof the sector. Xois the number
of aircraft at the beginningof ATperiod, i.e., just
priorto to.
3.
sdrs~12
Ratio of estimated numberof aircraft at a
given
time
to
capacity,
i.e.
( [x°+(
sdr: Thesector departure rate is the rate at whichthe
aircraft
are departing
sector,
sdr is
theasum
of sector
outflow rates.
Thus, asdrff,
= rmA
+ rmf
+ rmfy,
and
consider speed restriction at mergingpoints and the
restrictions due to aircraft modelsand weathermodels
in the calculation of PETAs,as is done currently with
CTAStools.
5.
Capacity (c): This is an upperboundon the number
aircraft that can be presentin a sector at a giventime,
andis limited bythe ability of the sector controller to
managemorethat a certain numberof aircraft. With
improveddecisionsupporttools, it is expectedthat this
numberwill increase.
PSTAojm,mo
: The Predicted STAof an aircraft at a
meterfix or control point is the STAof the aircraft that
has not yet entered the sector’s airspace, but is
predicted to enter the sector airspace through an
inbound meter fix at a certain time in the future.
Hence, PSTAof that aircraft is calculated assuming
the entry-time of several such aircraft. PSTAo/m.m,
is
used to denote the PSTof an aircraft aj,m, at the
inboundmeterfix me. PSTA~j~,,,m,is calculated given
sar: TheSectorarrival rate is the rate at whichaircraft
are entering into a sector, sar is the sumof the sector
the flow rate rm, (which in turn is given by a time
7O
NowSlq calculates rmf~ using the given r,,fx and
series). GivenPSTAaj,~.m,
of several such aircraft for
3.
all inboundmeter fixes, PSTAaj,~:,o at the outbound
meter fLX mo can be computed using extensions of
existing tools such as the DP. Whilethe use of DPwill
provide a more accurate computationof PSTAs,due to
time constraints in this effort we used relatively low
fidelity heuristics to determinethe PSTAs.
.
Wemeasure the sector controller workload by the
numberof advisories issued by the controller and the
complexityof those advisories (i.e., mustthe aircraft
be prescribed a holding pattern or does it need to be
diverted before it is returned to the original airway?).
Weassume that an advisory is issued when an
aircraft’s STAat any control point or exit point is not
equal to its ETA.The complexityof the advisory is
hard to measure,and at this stage of the development
we do not consider complexity while computingthe
workload. Anyadvisory that results in a holding
pattern is treated withan equal weight.
Repeated Strategic Bayesian Game
A Bayesian game with communicationis similar to an
extensive game(Osborne and Rubinstein 1994). In the
case of an extensive game, one player exchanges a
message,the other player respondswith a message,and this
exchangecontinuesfor a finite numberof iterations until a
terminal state is reached. The players then take the action
simultaneously. EveryATseconds (duration of the game),
the gamewith communicationis played in the extensive
form as follows (since the Bayesian gameis played after
everyAT,it is called a repeatedBayesiangame):
1. A sector, say SC’ predicts rmL, calculates
optimal rmf~ and exchanges it with Sc’ . This
involves
a.
Predicting PSTA~j,/x.mLof each aircraft
aj,~L using a set of probability distributions
that conformsto rmL.
b.
Calculating optimal PSTA~/,.,.mfy using
PSTAo,,.:r.mL
SO on
The gamereaches a terminal point when,given the arrival
rate from the other sector, a sector’s calculated rate of
departure to the other sector remains the same as that
computedearlier. Figure 4 showsa tree called an extensive
Figure 4: Extensive gametree of the Bayesiangamewith
communication
tree representing the sequential actions of each sectorplayer. The design of the gameinvolves desiL, ning the
extensive tree in such a way that it leads to faster
convergenceto a terminal point.
In the above description of the Bayesian game with
communicationas an extensive game, the gamestarts with
only one of the sector players initiating a message
exchange(Satapathy 1999). However,in a strategic game
form of the Bayesian game with communication, the
prediction and computation is started by both sectors
simultaneously and independently. Hence, in a strategic
gameform of the Bayesian game, two such extensive game
trees can be constructed that progress simultaneously.The
convergence occurs when both trees lead to a terminal
point simultaneously.Note that it is possible that we may
be unable to design the gamesuch that it converges. In
such a case, we imposea hard constraint on the numberof
times the messagescan be exchangedand study the Paretoefficient equilibriumof the strategic gameplayedafter the
exchange of messages. The design task requires
determiningsuitable constraints on the numberof times the
messagescan be exchangedso that it results in equilibrium
closer to the Pareto-efficient equilibrium.In this paper, we
adopt a strategic form of the Bayesian game discussed
above. The primary tasks associated with this gametheoretic formulationof determiningequilibrium flow rate
is to developalgorithmsto:
c.
Calculating rmfy using PSTA~j,,,.mf~ and
sendingr,,f, c2
to. S
c2
Given rsf r, S calculates an optimal rmL and
1. Predict a probability distribution over the different
possible sets of arrival times of the aircraft arriving
from the neighboring sector that conformsto a flow
rate.
2.
2.
exchangesit with Sc’ .
71
Calculate the optimal outboundflow rate given all the
constraints discussedin formulation.
PSTAsof the aircraft entering from the non-playing
sectors. The PSTAsof the aircraft entering from nonplaying sectors can be estimated based on the past
observations. Wedescribe each estimation procedure
below:
Developmentof traffic flow models needed for
data generation and optimization
Figure 5 presents the details of input data required for
optimization to computethe flow rate. Details with respect
to the sectorslCi, are shown.Theinputs are:
1. The inbound flow rates at the meter fixes mfa and
mfc, and PSTAsconformingto these flow rates.
2.
The inboundflow rate at the meter fix mfx from the
sector Sc2 and the PSTAsconformingto the flow rate.
3.
The outbound flow rate at the meter fixes mfb and
o~..~.. ~,o
Routine
tufa that the sector controller Slq mustmaintain.
I
I
Based on this input data, the sector sC~ calculates the
Figure5: Theinput data to sector controller’s optimization
routine to calculate optimaloutboundflow rate at the meter
optimal STAsat the meter fix mfy and thus calculates the
optimal sector boundary flow rate from the computed
optimal STAs.These optimal STAsare exchangedwith the
sector sIC2as Negotiable STAs(NSTAs). The optimal
STAssent by opponent sectors are called NSTAbecause
the sector Stq believes that these will be the STAsof the
Estimation of the PSTAsfrom non-nlaving sectors based
on past observations:
The estimation of the PSTAsof aircrat~ from non-playing
sectors involvesthree parts:
aircraft at the meter fix mfy in the ATperiod. Thesector
Sc2 is free to calculate PSTAsbased on these NSTAs
that
conformto the exchangedoptimal flow rate, or generate
PSTAsthat conformto the exchangedoptimal flow rate.
Similarly; the sector Stc2 exchangesthe optimal flow rate
at meter fix mfx and the NSTAsof the aircraft leaving
through mfx. The sector SIc~ calculates PSTAsbased on
1.
Predictionof flow-rate at a meter-fixthat feeds aircraft
from a non-playingsector
2.
Generationof aircraft with its trajectory that will enter
throughthe meter-fix
3.
Estimationof PSTA
of the aircraft that conformsto the
predicted flow rate
The flow rate from non-playing sectors can only be
predicted since no negotiation occurs betweenthe sectorplayer and non-playing sector. Weassume that the flow
rates can be predicted froma time series modelconstructed
using the historical data. Theassumptionthat a time-series
model is an accurate flow rate prediction model is a
reasonable assumption. This assumption can be validated
given the Enhanced Traffic ManagementSystem (ETMS)
and Traffic ManagementUnit (TMU)log data. Note that
the order of the time series is as importantas the accuracy
of the prediction model.Generally,higher order time series
models are more accurate, but the parameters of higher
order modelsare difficult to estimate given the ETMS
and
TMU
log data. For simplicity, we assumea first-order time
series, whichcan be expressedas follows:
these NSTAsthat conformto the exchangedoptimal flow
rate and uses the calculated PSTAsas inputs to the
optimizationroutine as illustrated in Figure 5. Wedescribe
our approachfor the input data collection and generation
and the optimizationroutine in the followingsubsections.
The PSTAsof the aircraft entering from the opponent
(sector) can be estimated based on NSTAsgiven by the
opponent. However,in order for the opponent sector to
send NSTAs,it needs to calculate NSTAsusing the
optimization routine, whichin turn requires PSTAs
fromits
opponentsector and other non-playing sectors. Becauseof
the cyclical dependencyof the data requirement,i.e., the
data required for optimization dependson data output from
the optimization, we resolve the issue by allowing the
sectors to only exchangethe set of aircraft (for the first
optimization iteration) that they believe will enter from
their ownsector to a neighboring sector during the AT
period. This set of aircrat~ can be calculated basedon the
r,, (n) =/l +q~ rm, (n 1)+ s(n- 1)
72
t?
Inl~u~lflowrote(r) m~mum
sq~u-~ondJsu~c~
Figure 6: Estimationof PSTAs
of the inboundaircraft from
the probabilitydistribution functions
wheren is the index of the nth stage game,g and ~ are the
parametersof the first order time series. Theparameter6 is
randomly generated from a Gaussian distribution
constructed from the difference between the last (n-l)
predictions using the sametime series modeland the last
(n-l) actual observations. Notethat the order of the time
series does not affect the convergence of iterative
optimization,but it affects the fidelity of the solution.
rate. Onerequires a probability function(s) to represent the
time whenthe aircraft arrive at the sector meter fix. As
shownin Figure 7, the PSTAof the aircraft a0can lie
anywherein the ATperiod, but the probability of PSTAat
a particular time is given by a probability distribution
function (the red curve). HavingPSTAof o s et ( as s hown
by red dot), the PSTAof I can be s et t o l ie a nywhere
betweena 0 + 5 (where ~i captures the minimum
allowable
separation distance) and (t0+AT) as long as PSTA
al does not violate the inboundflow rate calculated based
on the moving average method. The PSTAsof the other
aircraft are estimatedin a similar fashion.
The estimates of PSTAsof the aircraft are determinedby
using a set of probability distribution functions as
illustrated in Figure 7. The modelestimates PSTAsof all
aircraft entering a given meterfix (e.g., me) such that the
flow rate of less than or equal to r,, is maintained. For
simplicity, rra, is denotedby r and PS~Aaj
~,,.~ is denoted
by PSTA/
If r = 0,
Noaircraft needsto be generated,return.
else
°)
PSTA
o = t o +bo where, bo 13 B(O, AT, a°,a
Forallj > 0,
If PSTAj_~+d~ < t o +AT
Generatej~ aircraft and assign PSTAs
as follows:
PSTAj =
PSTAj_I +g~+b/ where, b/13 B(O,t o +AT-PSTA/_
l-8,a[,aj2)
J
if )~g((PSTAj_
1 + ~; - PSTAj_t),At)
r
l=l
j)
PSTA
k +At+bj where, b/[3 B(O,t o +AT-PSTA
k -At-5,ctlJ,ct
otherwise
where,
g(a,b) = 1 if a ___
= 0 otherwise
and
PSTAj+c~-PSTA
k < At
~ < PSTA/+g/-PSTAk_
Else
Donot generate any moreaircraft for the ATperiod.
Figure7 Themodelto estimate PSTAs
of the aircraft at a particular meterfix, givenits flow rate
The model in Figure 7 estimates
PSTAs only
probabilistically. In order to estimate the PSTAs
accurately
(i.e., the PSTAsto be used in optimization do not vary
significantly from the actual STAduring the execution of
the game), we need to ensure that the variance of the
distributions is as lowas possible. If the variance of the
distribution (which is the error associated with the
Thesecondand third part of the problemis howto estimate
a set of PSTAsat the sector meter fix, given a predicted
flow rate, and how to assign those PSTAsto randomly
generated aircraft (objects). There can be a number
PSTAsof the aircraft arriving at the inboundmeter fixes
(e.g., mfaand mf~ ) that will conformto a specified flow
73
prediction) is low, then b.- in the modelcan be substituted
with the meanof the distn/bution i.e., /.tbj. Thus,the PSTA
of the aircraft at the meterfixes can be ~stimatedgiven the
predicted inboundflow rates. The modelassumesthat the
probability distributions associated with the estimation of
each PSTAdo not change from one stage game to the
other, or the shapeparametersof the distributions (al and
a:) can be calculated from historical data and past
observations(e.g., time series models).
Onlya subset of the set of aircraft that is generatedusing
this modelcan enter the neighboringsector¯ This subset is
calculated as follows:
LetPSTAa~"
~_, be theo estimated
¯
oj ¢,
PSTA
of the a,,,, aircraft. Theaircraft a, ,, will
enter the
neighboring
sec’to~"betweento and(to+AT),’i~" PSTAa~,,
:too
calculated for the sector boundary meter fix mo usmg
PSTAaj~,m, ,. the distance between mo and me and
maximum
cruise velocity of the aircraft ay.m, is less than
(to+AT).
indexed 2 is the gameplayed for the period between(to
2AT)and (to + 3AT)and the gameindex 1 (i.e., previous
game)is the gameplayed for the period between(to + AT)
and (to + 2AT).In such cases, the time series maynot make
muchsense since the aircraft indexed 1 in the game
indexed 1 does not have any relationship with the aircraft
indexed1 in the gameindexed2. This is especially true if
the ATis short (e.g., lhr). Therefore,one maydivide a day
into several ATslots or stages, and represent the x-axis in
Figure 8 as the gamesplayed at a particular ATstage over
several consecutive days. By modelingthe (STA-NSTA)
this way,we can extract the trends/patterns associated with
a particularflight.
/--- 4~ stage game
# of Gamesplayed -I~
Estimationof the PSTAs
giventhe set of aircraft:
Estimation of PSTAsgiven the set of aircraft that enters
between to and (to+AT) is similar to the estimation
PSTAsbased on the past observation. The only difference
is that the aircraft need not be generated. However,the
flowrates still needto be predictedfroma times-serieslike
modeland PSTAsassigned to the aircraft that conformto
that flow rate. This approach is followed when the
neighboring
sector submitsa set of aircraft that are likely to
enter between to and (to+AT). In this model, as in the
previous case, if PSTAj_t + c~ < (t o + AT) then
.PSTA/=(t0+AT) or PSTAj =PSTA/_I +8, whichever
is greater.
# of Gamesplayed -~
Figure 8: Variation of(STA-NSTA)
of two aircraft
indexed 1 and 2 with the each game
Let 0st AU(n) /,/th
denote this correction time series model
where n is the stage game. This time series essentially
predicts or estimates howmuchthe actual STAwill differ
from the NSTA.The PSTAjat the (n+l)th ATperiod game,
given NSTAjof the neighboring sector-player, are now
calculated from a model similar to the model shown in
Figure7.
The estimated
PSTAo
asNSTAi+ 0STA (n)
for
J is the same
¯
.
all atrcraft
exceptwhenthe optimal
flow rate (ropt)j sent by
the neighboring sector-player is violated¯ In this case,
PSTA/is PSTA
k + At + b" (added to delay PSTA/to
later timethat doesnot violate the flowrate). This addition
is justified, because according to a sector-player’s past
observation, the sector-player believes that aircraft will
enter at the estimated PSTAsand will also maintain the
flow rate¯ In other words, a sector believes that the
neighboring sector-player will push the aircraft at its
estimated PSTAs,since exchangedNSTAsonly serve as a
sourceof reference.
Estimation of the PSTAsbased on exchangedNSTAs:.
The exchanged NSTAsand the flow rate are used to
estimate PSTAsfor next optimization cycle¯ Neither the
NSTAsexchanged during communication nor the final
NSTAthat have been converged upon can be assumed to
be the sameas PSTAs.This is because the STAsduring the
execution of the gameneed not necessarily conformto the
exchangedNSTAs.The difference between STAsobserved
during the execution and the exchangedNSTAsis captured
in a correction modelwhichis applied to the estimation of
PSTAs.Weassumethat a time series modelis appropriate
to model the difference between STAsand NSTAsfor the
jth aircraft entering at everynth stage game.Therefore,the
players are required to develop a time series modelof the
variation of the difference betweenNSTA
and actual STA
(STA-NSTA)
with each stage-game played, for each jth
aircraft. Notethat the NSTA
in the time series modelmust
be the NSTA
that the players convergeto before the game
is played. Figure 8 illustrates howthe (STA-NSTA)
of two
aircraft indexed 1 and 2 varies with the gamesthat have
been played.
The error in the estimation is captured by the Gaussian
error present in the time-series model¯ Similar to the
estimation of PSTAs,wherethe error in the estimation is
due to the large variancein the probabilitydistribution, the
error in this estimationis due to the inaccuraciesinherentin
the timeseries.
Optimization algorithm
The following data is available for the maximization
problem:
In Figure 8, the indices of the gamesplotted on the x-axis
represent consecutive games. In other words, the game
74
1. PSTA‘,j,,.,,, of the aircraft aj,,,, at meth inboundmeter
1.
fix suchthat PSTA‘,j,~,,,,< PSTA‘,/.,,,,m,¯
2. Trajectories of each aircraft a/,m, (e.g., we denote
trajectory of an aircraft al 3 as [3, 1~, 15. 2]. Thismeans
the aircraft enters at the’ 3’d inboundmeter fix and
leaves at 2"d meter fix by passing through the control
points/2, Is). Let the trajectory of an aircraft aj,m, be
+dm,mo/ Vm~,
ETA,,/..,,too =PSTAa/~..,mr
f
2.
denotedby a set
that p~ is a control point except Poand /~eo/~,l which
are meterfixes.
3. The rate of outboundflow at the moth outboundmeter
ETA‘,j~,,t,= PSTAj,
m, + din,2, / Vm~x
Create a bucket Bo and add all the aircraft along with
ETAsinto the bucket as STAsof the corresponding
aircraft at specified control points and meterfixesk
3.
Usethe algorithm shownin Figure 9 to create (branch)
a list of bucketsfromBo. Initially, the list of bucket
contains only B~
4.
A bucket is infeasible if it violate the following
fix(r~o).
4. Minimum
aircraft separation distance based on safety
Calculate ETAsof each aircraft at the control points
and meterfixes using the following formula(Note that
control points must be contained in the trajectory of
the aircraft aj,m, ):
¯ ") For each bucketB, fromthe list of buckets
For each control point 1~ (this set of points includesboth control points and outboundmeterfixes)
¯~ Separate the aircraft into two buckets/~, and Bk such that/~k contains ETAsof those aircraft whose
trajectory containsIt and Bkdoesnot aircraft withpoint 1~
¯ "~ Sort
the <aircraft
in /~k bythe ascendingorderof STA‘,j.,t, of the aircraft a/,. suchthat
STA%..t
’ STA‘,/~,..,t,
¯") Starting fromj = 0, if Vm~(STA%,..t, - STA%.,t,< s, then make twocopies of/~k
¯ ") In
one copy
(/~ ), replaceSTAa/.,..,t, by STA%.,t,+ s / Vma
x and in the other (/~ ) replace
STA%.,t
’ by STAoj.i..,t~ -S/Vma
x
¯") AdjustSTA‘,/.j.., of aj+l,, at all control points l that comesafter 1~ to compensate
the delay at
t
l,.
¯") AdjustSTA%..t
of a j.. at all control points l that comesbeforeli to ensureearly arrival at l~.
¯ ") AddBk to ~ and ~, and replace k i n t he l ist o f b uckets with t hese two buckets
¯ ~ Applybucket elimination procedureto eliminate infeasible buckets.
<
¯ ") Repeatuntil the conditionVma
~ (STA%,..,t,- STA‘,/..t, ) s for all aircraft holdsfor all the buckets
Figure 9: Steps of the Branchand BoundAlgorithm
5.
6.
requirement(s).
Maximum
allowable cruise speed (v,,~).
Capacity(C) of the sector.
constraints:
Vm~
(STA‘,/.,t~- PSTAaj:.m.
) din.t,,
Vmax
(STAa/.,t.,
- STA%.,t,
) < dt~t,.,
STA%.,t
’ < PSTAa/e.m,
It is required to determine STA‘,i;"m of
the
aircraft
o
th
aj.o[with trajectory containing mo] at ttie mooutbound
meter fix. The following steps illustrates the branch and
boundalgorithm:
So bucket contains both ETAsand STAsof the aircraft,
but both are equal in Bo
75
,K--a
Eg((STA%.,mo -PSTAok ,),0)/C
an agreed-upon 2.a rm. and 2.a %0" In that case, the
2motif
,,o~ff’
> 1 for all ak, and
Oj°
flow rate
E r,.oiS
E rmo(i)and
E r,.o is
"o~’
"o~111
moe~"
E r~o(i ) . The sectors also need to converge at
aj, as long as PSTAoj...,, < PSTAa,..., (This constraint
evaluates the numberof aircraft present in the sector’s
airspace whenan aircraft enters the sector’s airspace)
/ COIC,,
Thecalculatedflow rate at m
o t rmo ) doesnotviolate r,,~.
2moeS~l
equilibrium
Theflow rate a,tc.
r~ Is calculated by the followingequation:
Vaj.., if(STA%..m
° < (t o + AT)),
5¯
S~l I converges to
equilibrium STA,if STAa~...,mo(i)0STAaj....mo(i-1) for
each aircraft a/,. leaving any of the outboundmeterfixes
betweento and (to + AT). The sameis true for the sector
r~a/c = max(
a,..
PSTAs. The sector
~
c2
S
g(STAa,...,no -, 0))
^(STA,t..~0 >STA,I...u 0) STAaj..,rao
)
For each bucket from the list of remaining buckets,
calculate the utility value. Pick the bucket with
maximum
utility. Note that rerouting is required, if we
have
following
conditions:
>
and
Vmin
(STAa;..,t,
- STAa;..
,m, ) din,t,
Vmin
(STAa;...t
,t~ ) >dt~t,÷~
m- STAa;..
NSTAs
andFlowrate
fromthe opponent
sectot-~aye~
Send~
opt~alST/ks
Rou~
.[ mama. PsrAs
(no,H~yeO
~cto~
In our two sector-player gamesetup, the sector Scl sends
the optimal STAa.°. m
as. NSTAom"’.too Sl c2 (where mo is
o
mfx) and I C2sends o ptimal S TAo masNSTAo. ,, to
S~la (wheremois mf~)as longas optfma°lSTAsis less’ t~an
(to + AT).Thesectors also exchangeoptimalflow rate c°tc
calculated using optimal STAo.
:,r... The optimal flow ~te
calculated using a movingaverage methodis as follows:
the numberof aircraft countedto cross the sector boundary
meter fix mo between¯ STAo.
m and (STAo,,".°. -o At) is - the
.*. o
flow rate whenthe a|rcraft a.. crosses the meter fix mo.
The average of all such flow rate over the number of
aircraft crossing betweento and (to + AT)is the optimal
flow rate rmoat the meterfix mofor the period betweento
and (to + AT).
Figure 10 illustrates howoptimization routine is invoked
iteratively until both sector convergesto and equilibrium
flow rate and STAs.The sector Sl cl converges to a flow
rate~ ~ rmo if at the z ~h iteration,
it finds
moeS~,’
rmo(i)D r~ o(i-1)" Si
milarly, Th
e se ctor
sC2converges to a ~ r,,o if at the t ~h iteration,
it
Figure1 O: Iterative optimization
process
until calculated
flow rate for twoconsecutive
iterations are the same
The numberof iterations required for convergencecan be
assessed from the airspace dynamics(i.e., howfrequently
the flow rates change from one stage gameto another and
the trajectories of the aircraft) andthe predictability of the
PSTAs.Westudy this through simulation described in the
next section.
Implementation
for GameSimulation
To study the robustness, sensitivity, stability (i.e.
convergence) and uniqueness of the solution (i.e.
equilibrium sector boundaryflow rate), we developedand
implementedsoftware for optimization and simulation of
the two-playergame.As a part of the simulator we modeled
some features of decision support tools and sector
controller rules used for issuing holding directives. The
simulation and the optimization were adequately
coupled/interfaced since the optimization routine requires
data (set of aircraft still flying the airspace when
optimization gets started prior to AT)from simulation and
vice versa. In the following sections we discuss the
implementation
aspect of the software in somedetail.
2moE~ll
finds E rmo (i) 0 E rm° (i- I). Note that both sector
2
2moC~
moe~ll
need to convergeat the sameiteration in order to determine
76
User set parameters
Theuser is allowedto set up AT,start time (to), flow rate
unit (At), minimumseparation distance (s), minimum
aircraft cruise velocity (Vmi,), maximum
aircraft cruise
velocity (v,~), capacity (C), and weights in the utility
function. The default values of these parametersare: AT=
lhr (3600seconds), to = 00:02, At = 20 minutes,s = 5miles,
Vm~n= 6miles/min, (v~ = llmiles/min, c = 10. The
parameters such as airway segment information are read
from files (sectorl_config.txt &sector2_config.txt). Each
meter fix has its ownconfiguration file that sets up
parametersrequired for the time series modelfor flow rate
generation, the Gaussianerror parametersassociated with
the time series model, the shape parameters of the Beta
probability distributions used to determinePSTAsof the
incomingaircraft, and the list of trajectories that can be
randomlyassigned to the aircraft entering through that
meter-fix (this is true only for the inboundmeterfix from
the non-playingsectors). Thechoice of these parametersis
discussedat lengthin the followingsection.
Data collection and generation
Tuningparametersfor estimation of the PSTAsbased on
hast observations
Thefirst set of parametersrequiredfor the estimationof the
PSTAsbased on past observation are the values of la, dp,
and the meanand variance of the Gaussian error e of the
time series model. These parameters can be determined
based on ETMSdata. Weused several combinations of IX
(0 - 8), d~ (0.0 - 0.25), Gaussianerror mean(0 - 2)
Gaussianvariance (0 - 0.44) for our experimentaldesign.
In a time series analysis, the meanand variance of the
Gaussianerror is calculated basedon the past observations,
but in our experimental design we used the constant
Ganssian meanand variance. Whenthe value of the ~b,
Gaussian mean and variance is 0.0 then the maximum
inflow rate at the inboundmeter fix for which the time
series is relevantis Ix at the specifiedflowrate unit (At).
there is somenon-zero Gaussian variance associated with
it, then the maximum
inflow rate at the meterfix oscillates
th
around~t. Thevalue of the ~b indicates howstrongly the n
stage flow rate dependson the (n-l) th stage.
The modeldescribed in Figure 7 estimates the PSTAsof
each aircraft given the flow rate fromthe time series model
for the nth stage game. The shape parameters of the
probability distributions are set in such a way that the
distribution is moreskewedtowardsthe left for the first
few aircraft entering the meter fix starting from to and
gradually the skew shifts towards the right as the PSTAs
calculated from the model as described in Figure 7
approach(to + AT). For example,for the meterfix mf~the
shapeparametersare arrangedas: (2, 40), (3, 42), (5,
(6, 36), (8, 34), (10, 33), (12, 32) .... As far
optimizationis concerned,the PSTAjof the jth aircraft is
determinedby replacing bj (refer to the modeldescribedin
Figure 7) by the meanof the distribution bj, which is
defined by the shape parametersas: (cq/(cq+c~2)). When
PSTAjis determined to lie betweento and (to + AT),
aircraft object is generated whichis assigned with that
PSTAjto be the time at whichit is expected to enter the
meterfix. Eachaircraft generatedis assigneda trajectory.
77
The inbound meter fix configuration file lists all the
trajectories that canbe assignedto an aircraft if an aircraft
originates from that meter fix. For example, the aircraft
originating at the meterfix mf~can havetwotrajectories mf~---~trlo---~trla--~mfy---~trlf---~fh
andmf~--octrla---~trla--~
mfy--gctrlf---~fg. Thetrajectories are assignedat random.
The generated aircraft objects are sent to the neighboring
sector, if their PSTAj
at the enteringmeterfix will result in
their exiting into the neighboringsector betweento and (to
AT).Eachsector obtains a list of such aircraft objects that
just contains information about the sequence, but does not
have information regarding the time at whichthe aircraft
would enter the opponent’s sector though the sector
boundarymeterfix. Thesectors predict their flow rate and
timeof entry (i.e. PSTAjat the sector boundarymeterfix).
Tuning parameters for estimation of the PSTAsgiven the
set of aircraft
Theset of aircraft objects sent by a sector is the sumof two
subsets:
1. Thesubset of the generatedaircraft objects for the
period betweento and (to + AT)whosePSTAjat the
sector boundarymeterfix is betweento and (to + AT).
2. Thesubsetof aircraft objects that are alreadyin the
airspace, but whoseETAsat the sector boundarymeter
fix is betweento and(to + AT)
Estimation and assignmentof PSTAjat the sector boundary
meter fix to each of the aircraft objects sent by the
neighboring sector is similar to the previous case. The
values of ~t, d~, and the meanand variance of the Gaussian
error 6 of the time series model reflect the numberof
inboundmeterfixes of the neighboringsector feeding into
and /tmfsector
values of
/.t of
twoexample,
time series
a/-Gf,
particular
boundary
meter
fix.theFor
if
~ are the
modelsfor the meterfix mf~and mf~.then the/~ of the time
series model for mfy is approximately (/G/o +Pr#c)"
Similarly other parametersof the time series modelfor mfy
reflect a similar relation with the parametersof the time
series model for the meter fix mf~ and mf~. The shape
parametersof the probability distributions that determine
the PSTAjof each ff aircraft for the optimization are
howeverset independently of the shape parameters of the
probability distributions that determinesthe PSTAsat mf~
and mf~
Tuning parameters for estimation of the PSTAsbased on
exchanged NSTAs
The NSTAsare exchanged after the first optimization
iteration. The PSTAsare calculated from NSTAsand the
optimal flow rate sent by the opponentsector. Hence,there
is no need to predict the flow rate or generated PSTAs
using a probability distribution. Instead a time series model
is used as a correction model to convert an NSTAto a
PSTA.Theparametersof this time series model(i.e., Ix, ~bl,
~Pz, dp3... etc. dependingthe order of the model)must be
built or trained with data assumingseveral gameshave
been played so that we have (STAj-NSTAj)data for each
fh aircraft. However,in order to play the games,we need
to assume a time series model with its parameters that
approximatelyreflects a realistic situation ~. For this
experimentaldesign, we assumeda first order time series
modelwith la = 20, dpl = 0.5, Ganssianerror mean= 0, and
Gaussianvariance = 10000.Note that the variance reflects
the reliability of the correction model.A varianceof 10000
indicates that the PSTAscalculated using the correction
model might lie between _+300 seconds. For this
experimentaldesign we kept the samela and d~ for eachff
aircraft.
Optimization
The optimization routine is invoked with the flight
informationof all the aircraft that will be flying throughthe
sector’s airspace betweento and (to + AT).For example,the
optimization routine of the sector Scl takes into
considerationthe followingaircrat~ into its initial bucket:
I.
The set of aircraft entering throughmf, betweento and
(to + AT)as calculated by the model- "Estimation
PSTAsbased on past observations".
2.
Theset of aircraft entering throughmf~betweento and
(to + AT)as calculated by the model- "Estimation
PSTAsbased on past observations"
3.
Theset of aircraft entering throughmf~betweento and
(to + AT)as calculated by the model- "Estimation
PSTAsgiven the set of aircraft" (for the fast
optimization routine) or by the model- "Estimationof
PSTAs given the NSTAs" (for subsequent
optimizationiteration).
4.
Theset of aircraft that has enteredthe airspaceprior to
to andare still in the airspaceafter to.
A bucket is branchedoff into two buckets as described in
the optimizationroutine. Figure11 illustrates howa bucket
is branched off into two buckets. Each new bucket of
aircraft is consideredfurther if the newSTAsassigned to
the aircraft indicating STAsat the control points can be
maintained without exceeding the maximumcruise
velocity.
The control point STAsassigned to the aircraft mayneed
to be increased or adjusted depending on the holding
Onewayto estimate the parametersof the time series is to
collect the difference betweenSTAs(not ETAs)at the
sector boundarywhenan aircraft departs an airport and
whenit actually arrives at the sector boundary.This
information can be collected from ETMS
data and sector
controller’slog file.
78
ALiA.~I2A
Z0,2740,
32~0,
37401 AR2.1’~d2Z20,.
................. ]
ALlY~II g40,21~0,2670,3160] ARI
2,’-~19~0
.................... I
,hl,t
2~:1114~
I
~
2070,
2r~
AP,2~I’/30
................... I
I
1250,1~401 ARI.I~I420,16~)
.......... ]
Jl)eL~ycarl b~al~ol-t~tl
I 150,16401 .~RLI~
[1/,0~ 1410,1510,
lg30
I
by reducing vdocit~| ..................|,AIAI~I
...................... I
/
or
Delayneedsto be incr~sed /
dependlng on holding pallern
~
//t,,i r i~’~ ~ with I~[]e di~l-eao4~ 150 ~
But requited time difference 300 based on
This Call becomein feasibh
{in fe&’~iblehi Ibis ¢x~e)
rainseparationtime
Figure11: Illustration of howa bucketis branchedinto
two buckets and whatvalues are changed(In this case
STAsat the control points are changed)
pattern to be enforced. Someother rules applied to check
the validity of the STAsare illustrated in Figure 12. For
example,in mergingairways, the aircraft to be held depend
on the location of the other aircraft causingit hold.
If thifis is delayedthenthis mustalso be delayed
/
°’°’°°’°’°’;I’O,o...0oO..,.o.o.,..
°
:
#"
.........
"
."
/
If tJds is delayed
so that it hasto be held
[ close to the control point
~thenti~s must also be delayed
~.......:o.,......,~....
o
X
This mustbe hel
,<
4~
Figure12: Safety criteria applied to adjust the STAs
Software and Simulation of the Game
Weimplementedthe optimization and simulation modules
using OpenCybeleTM agent infrastructure in order to
implementthe interface easily (Refer www.openeybele.org
for more details on OpenCybele).OpenCybeleallows easy
decompositionof problem into agents and activities and
allows the programmers
to write decision support tools that
run concurrently with the simulation threads. In the
implementation,the optimization for the two sectors takes
place concurrentlyin different executionthreads.
The simulator is used to simulate the behaviorof a sector
controller issuing advisories to aircraft and introducing
randomnessto the time of entry of aircraft from nonplaying sectors. The mainfeatures of the simulator are as
follows:
1. The rates at whichaircraft are injected into the sectors
during the simulationare different fromthe rates used in
the optimization algorithm. This implies that the values
of ~t, dp, and Gaussianerror parametersof the time series
modelused for simulation are slightly different from the
ones used in optimization. This was done to show the
robustness of our algorithm. Weconjecture that a large
difference in the aircraft-feed rate will demonstratethe
need to develop accurate time series modelor mandate
exchange of NSTA
similar to the exchange of NSTA
by
the twosector players in this experimentalsetup.
2. The probability distributions shape parameters used to
calculate PSTAsis slightly different in simulation to
demonstratethe robustnessof the algorithm.
3.The simulator contains features to issue control
advisories such as (1) reduce speed or (2) hold
aircraft. Thesimulatordoes not issue tactical advisories
such as howto hold an aircraft due to limited time and
resource. Since this part of the simulation can be
eventually be replacedby other high fidelity TMA
tools.
Analysis of Simulation Results
Weran several simulation scenarios, by changing the
maximum
inflow rate at meter fixes mf~, mf~, mf,, and mj~
to analyze the sector boundaryflow rates determinedby the
game.Thesescenarios are describedas follows:
1. The IX and ~ of the time series modelthat restrains the
maximum
flow rate of aircraft throughmf~is set to 8 and
0.2 respectively. The meanand variance of the Gassuain
error associated with modelis set to 0 and 0.111. The
flow rate throughall other inboundmeter fixes is set to
zero for both sectors. Because of this setting, one
observesat most(8+_2) aircraft entering per 20 minutes
into the sector Sc’ and at most(8_+2)aircraft are leaving
per 20 minutes into the sector Si G from Sq which
happensto be the negotiated sector boundaryflow rate
from Sc~ to Sc2. The sector boundary flow rate from
Sc2 to Sci is zero. In the next stage gamewhichoccurs
after (AT/Simulationspeed) seconds(i.e., 3600/20=
seconds), the maximum
flow rate into the sector q will
the sector S~. This holding is due to the flow rate
restriction set for the outboundflow rate throughmeter
fixes mfgand mJ~.The IX and dp of the time series model
that sets this flowrate for every stage gameare 3 and 0.2
respectively.
3. In the third scenario, IX and ~b of the time series model
that restrains the maximum
flow rate of aircraft through
mf~and mf~are both set to 4 and 0.2 respectively and the
tx and dp of the time series modelthat restrains the
maximum
flow rate of aircraft through mf~are set to 3
and 0.2. In this scenario, one mayobserve a non-zero
negotiated sector boundaryflow rate being set from both
sectors. As the gamecontinues, one mayobservethat the
negotiated sector boundaryflow rate from the sector
from Sq to S~ will keep on rising higher than the
negotiated sector boundaryflow rate from the sector
from S~ to SIci . One mayalso observe that there will
be aircraft holding in sector S~ prior to sector boundary
meterfix mf~and on the segmentconnectingmf~and cole
in order to maintainthe lowsector boundaryflow rate at
mf~. That means,SIc2 will witness someadvisories due to
the sector boundaryflow restriction, whichit wouldnot
have occured if there were no aircraft injected to the
sector Sq and/or east-west airway intersecting west-east
airwayat controlpoints ctrl~ andctrlc.
4.In the fourth scenario, aircraft enter through all the
inboundmeter fixes. Ix and dp of the time series model
that restrains the maximum
flow rate of aircraft through
mf~and mfc are set to 6 and 0.2 respectively and the IX
and d~ of the time series model that restrains the
maximum
flow rate of aircraft through mf~and mj) is set
to 3 and 0.2. Becauseof the high inflow rate of aircraft
into the sector Sq in addition to someaircraft from S~
to Sc~, one mayobserve that the flow rate from S~ to
Sc~ is muchless than the sumof the inflow rate through
be at most(8 + 0.2x(8+_2)+_2)aircraft per 20 minutes.
2. The IX and d~ of the time series modelthat restrains the
maximum
flow rate of aircraft throughmf~and mf~are set
to 5 and 0.2 respectively. The meanand variance of the
Gassuain error associated with both modelsis set to 0
and 0.111. In this scenario, one maysometimesobserve
holdingat the mergingcontrol point ctrlb. Asa result, the
sector boundaryflow rate from SG to Sc’ will be less
mf~and mJ~Similar to the third scenario, the sector
boundary flow rate through meter fix mf~ does not
increase with stage gameas the flow rate through meter
fix mfy increases. One mayargue that the numberof
advisories incurred by the sector S~ due to low flow
rate set at meter fix mf~will decrease as a result of
increasing the flow rate. However,by increasing flow
rate, the sector Sc~ will try to increase its flow rate
than the sumof the inflow rate through meter fixes mfo
and mf~. The maximum
inflow rate through one of these
meter fixes will be (5+_2). In the next stage game,the
maximum
inflow rate will be (5 + 0.2x(5+_2)+_2) and
on. Note that during gameexecution, one mayobserve
holding prior to the outboundmeterfixes mfgand mj~in
through mfy. Consequently, the sector S~ will incur
additional advisories for holding aircraft prior to the
outboundmeter fLx mfgand mJ~because of the low flow
rate restriction at the meterfix mfgand m/~comparedto
79
the sector boundaryflow rate at mfr In summary,the
sector S~ does not gain by increasing the flow rate at
mf~but the sector Sct gains by lowflow rate set at the
meterfix mf~.
Observethat the metering times (i.e., time at which the
aircraft enter or scheduled to enter) are not calculated
basedon miles-in-trail restriction in order to maintainthe
flow rate. The metering times are calculated based on a
movingaverage methodso that the maximum
flow rate is
the set flowrate. In other words,in any givenflowrate unit
(e.g., 20 minutes), the numberof aircraft crossing a meter
fix does not exceedthe flowrate set for that meterfix. This
is an importantmigrationfromcurrent techniqueof setting
metering times that creates an unnecessary number of
advisories. The un-equalintervals of the scheduledarrival
and departure times of aircraft demonstratethat we do not
follow miles-in-trail restriction. However,we observed
minimum
separation distance betweentwo aircraft. Wealso
noticed that the average delay occurredin a particular AT
period for both sectors are low while the density remains
reasonablyhigh.
Animportantand intuitive observation is that computation
time for optimization does not depend on the numberof
aircraft, but their expectedPSTAat the time of entry that
maylead to holding and thus require the need to create
morebuckets. Therefore, if the initial bucket contains
PSTAsof the aircraft that lead to conflict (minimum
separation distance violation) at every control point
betweenevery two aircraft then the computationtime may
increase exponentially. However,the likelihood of such a
scenariois not high in real world.
Alsoobservethat the time (i.e., the numberof optimization
iterations) it takes for the sectors to convergeto a solution
0aSTAsat the sector boundary&the flow rate) dependson
the following- (1) parametersof the correction model, (2)
capacity constraints, (3) outboundflow rate constraint, and
(4) the numberof control points involving east-west
airwaysintersecting west-east airways. In our experimental
set up, since we did not consider capacity and outbound
flow rate constraints in our optimization,and since the right
sector does not haveany control points involving east-west
airway intersecting west-east airways, our optimization
convergesafter every two iterations. This is because the
fight sector accepts any flowrate that the left sector sends.
Lastly, we believe that the time for convergencedepends
on howmanysectors are negotiating concurrently, and is
importantaspect to considerin multi-sector negotiation.
Conclusions
In this paper we developed a theoretical frameworkfor
sector metering based on a Bayesian game with
communication, and implemented the optimization and
simulationin OpenCybele
agent infrastructure. For specific
80
scenarios, we also demonstratedhowthe sector boundary
flow rates are negotiated after every ATperiod depending
on the rate at whichaircraft are injected and the numberof
aircraft present in the airspace from the previous AT
period. The sector boundaryrates are not calculated based
on miles-in-trail restriction. Based on our preliminary
observations, discussions with Subject Matter Expertise
(SME) and Aviation Consultants regarding these
observations we conclude that our the approach provides
an innovative and feasible solution to collaborative sector
metering to reduce delays, improve efficiency and
controller workload.
Future workincludes the extension of the two-sector game
and the developmenta DSTfor multi-sector metering using
more realistic sector/center geometries, traffic flow
interactions and ETMS
data to tune the time series and
optimization models. Issues related to convergenceof the
game,stability and sensitivity of the algorithms, and realtime issues will be addressed. Highfidelity modelsfor the
sector utility will be developed.
Acknowledgments
This research is supported in part by the NASASBIR
Phase I contract NAS2-01025.The opinions presented are
those of the authors and do not necessarily reflect the views
of the sponsors.
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