research article macroscopic model and simulation analysis ...in airport terminal area ( figure )....
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Research ArticleMacroscopic Model and Simulation Analysis ofAir Traffic Flow in Airport Terminal Area
Honghai Zhang, Yan Xu, Lei Yang, and Hao Liu
National Key Laboratory of Air Traffic Flow Management, Nanjing University of Aeronautics & Astronautics, Nanjing 211106, China
Correspondence should be addressed to Honghai Zhang; [email protected] and Yan Xu; [email protected]
Received 31 March 2014; Accepted 2 July 2014; Published 25 August 2014
Academic Editor: Xiang Li
Copyright Β© 2014 Honghai Zhang et al.This is an open access article distributed under the Creative CommonsAttribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We focus on the spatiotemporal characteristics and their evolvement law of the air traffic flow in airport terminal area to providescientific basis for optimizing flight control processes and alleviating severe air traffic conditions. Methods in this work combinemathematical derivation and simulation analysis. Based on cell transmissionmodel themacroscopicmodels of arrival anddepartureair traffic flow in terminal area are established. Meanwhile, the interrelationship and influential factors of the three characteristicparameters as traffic flux, density, and velocity are presented. Then according to such models, the macro emergence of traffic flowevolution is emulated with the NetLogo simulation platform, and the correlativity of basic traffic flow parameters is deduced andverified by means of sensitivity analysis. The results suggest that there are remarkable relations among the three characteristicparameters of the air traffic flow in terminal area. Moreover, such relationships evolve distinctly with the flight procedures, controlseparations, and ATC strategies.
1. Introduction
Air traffic management in terminal area is a knotty problemfor controllers as this place is considered to be the air conges-tion, flight delay, and aviation accident-prone area. Research-ing on the fundamental operating features of terminal areatraffic flow and deducing the macro emergence of traffic flowevolution may contribute to revealing the parameters withmutual relations, as well as the mechanism of spatiotemporalevolution, in terms of the traffic flow characteristic elements.By these means, we canmove forward to exploring the objec-tive law in air traffic in order to enrich the air traffic flowtheory and to provide scientific basis for air traffic disper-sion, which may have very important theoretical value andpractical significance.
Traffic flow parameters are the physical variables thatrepresent traffic flow characteristics including qualitative andquantitative features of operating states [1]. Basic theoriesof vehicle traffic flow have developed for decades and manyresults have been made by scholars. Lighthill and Whithamproposed the simulated dynamic model of traffic flow afterresearching on the evolution pattern of traffic flowunder high
traffic density circumstances [2]. Meanwhile, Richards pro-posed a first order continuummodel of traffic flow, which hasbeen integrated as the LWR theory [3]. Biham studied urbantraffic flow based on a two-dimensional cellular automaton[4], while Daganzo researched into dynamic traffic problemswith a cellular transmission model [5β8]. Compared to thevehicle traffic, less research has been devoted to air trafficflow theory so far, not to mention that many studies focusingjust on modeling. A simplified Eulerian network model of airtraffic flowwas proposed byMenon et al. [9, 10], and Bayen etal. studied the liner control problems derived from Euleriannetwork model [11β13]. Laudeman et al. noted a quantitativemathematical model on dynamic density of air traffic flow[14]. Complexity model based on traffic flow disturbancewas advocated by Lee et al. [15]. Liu et al. proposed a one-dimensional cellular transmission model specifically appli-cable to air route [16]. Wang et al. studied the microscopicplane-following performance and built the air freeway flowmodel [17]. Primary discussion for the stability of air trafficflow operating system was advocated by Zhang and Wang,while some basic characteristics of air traffic flow were alsoinvolved in this paper [18]. Such research findings have
Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2014, Article ID 741654, 15 pageshttp://dx.doi.org/10.1155/2014/741654
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2 Discrete Dynamics in Nature and Society
made great foundations for further study on air traffic flowtheory. However, no detailed studies have investigated thecharacteristic parameters with their objective evolution lawof air traffic flow. In this paper, we will combine traditionalmathematical formula derivation with modern simulationtechniques. Amacroscopic model of air traffic flow in airportterminal area will be proposed, with which we will simulateand analyze the interrelation and influential evolvement lawof the characteristic parameters with the goal of providingtheoretical basis for scientific air traffic management.
2. Macroscopic Model
2.1. Definitions. There are many kinds of definitions forair traffic flow parameters, because of different researchintentions and different methods that can be used. Since wefocus on airport terminal area, the definitions of velocity,density, and trafficfluxon a segment of air route are as follows.
Trafficflux (π) is the number of aircraftpassing a referenceprofile of the observation segment per unit of time. π = π/π,whereπ is the observation time andπ represents the numberof aircraft passing in π.
Density (π) is the number of aircraft per unit length of theobservation segment. π = π/π, where πmeans length of theobservation segment, whileπ is the number of aircraft in π.
Velocity can be divided intomicro andmacro definitions.Microscopic definitions including instantaneous velocity (V)and average velocity (V) focus on some point or profile of theobservation segment; macroscopic ones including the spacemean velocity (Vπ ) and timemean velocity (Vπ‘) focus on somearea extents or time ranges:
V =ππ₯
ππ‘, V =
1
π
π
β
π=1
Vπ,
Vπ = π· β (1
π
π
β
π=1
π‘π)
β1
, Vπ‘ =1
π
π
β
π=1
π π
π‘1β π‘0
.
(1)
One flight (ππ) of all (π) uses π‘
πtime to pass the
observation segment, where the actual distance is π·. And π π
is the flying distance of ππin the time period π‘
1β π‘0.
Historically, the first macroscopic traffic flow model is acontinuity equation, called the Lighthill-Whitham-Richards(LWR) equation [2, 3]. Like the vehicle traffic flow, airtraffic flux (π) and linear density (π) may also satisfy thecorresponding relations for some given functions π(β ) andπ(β ) with location π₯ and time π‘, as follows:
π (π₯, π‘) = π (π (π₯, π‘) , π₯) ,
π (π₯, π‘) = π (π (π₯, π‘) , π₯) ,
(2)
ππ (π₯, π‘)
ππ₯+ππ (π₯, π‘)
ππ‘= π . (3)
On the right side of (3), π denotes the number of aircraftthat enter or exit the observation segment. With the basic
Figure 1: Radar track plot in airport terminal area (source: ZGGGTMA).
equation π = πV, where V denotes space mean velocity Vπ andassuming every flight has the same flying case, we can derive
V (π₯)ππ (π₯, π‘)
ππ₯+ππ (π₯, π‘)
ππ‘= V (π₯) π . (4)
There are two variables as density and velocity but onlyone equation in the LWR model. It cannot be solved asthe differential equation is not closed. In response to thisproblem, a balancing velocity-density functional relationship,as V(π₯) = V
π(π(π₯, π‘)), was introduced into LWR theory by
assuming traffic flow invariably in equilibrium state. To plugthis into the equation, a hyperbolic equation of density can bederived, as follows:
πππ(π (π₯, π‘))
ππ₯+ππ (π₯, π‘)
ππ‘= π . (5)
2.2. Arrival Traffic Flow Model. LWR model described thepropagation characteristic of the nonlinear density wave tofind the evolution rule of traffic shock wave and rarefactionwave by characteristics method or numerical simulation [19].However, air traffic flow differs from vehicle traffic, especiallyin airport terminal area (Figure 1). First, the arrival anddeparture traffic in terminal area will follow the designedSTAR/SID (standard terminal arrival route/standard instru-ment departure), which means that aircraft in every routeposition must act in accordance with the operational flightprogram, that is, flying within a certain scope of designedflight level and speed. It is usually a small scope and can bereassigned by controllers. As a result of that, some balancingvelocity-density functional relationships in vehicle trafficmaynot exist in air traffic flow. Second, the density of air trafficflow is generally much lower than the vehicle traffic in realoperations, which causes the interactions among aircraft tobe weaker compared to vehicles. Therefore, it will be difficultto make statistical fit of the relationships according to theavailable radar data.
To solve the partial differential equation of the LWRtheory, we use cell transmission model to discretize thecontinuity equation of macroscopic traffic flow. The methodof discretization is applying a series of interconnected one-dimensional cells to denote the air route and using thedifference equation of time discretization to describe aircraftpassing through every adjoining cell. It should be feasible to
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Discrete Dynamics in Nature and Society 3
Cell
Control
Enter Exit
Figure 2: One-dimensional cell transmission model of air trafficflow in single direction.
model any air traffic flow scenarios by using interconnectedcells, as shown in Figure 2, putting all the aircraft flying alongthe arrival route while in several flight levels onto the sameimaginary plane, hence the air traffic flow in one route can beapproximately seen as one-dimensional continuous flow. Inaddition, one air route will be divided into several segmentsby series of unit cells. To simplify the model we assume thatthere is a unified variable which we will introduce later torepresent different kinds of control measures in one cell, suchas speed control, maneuvering actions, or circling, that is,adjusting flows by changing speed or flight path of certainaircraft in the cell.
In the matter of arrival route, let πππ(π‘) be the number
of arrival aircraft in cell π at time π‘; then the changes in thenumber of arrival aircraft in one cell can be described by thefollowing difference equation of time discretization:
ππ
π(π‘ + 1) = π
π
π(π‘) + π
π[ππ
πβ1(π‘) β π
π
π(π‘)] . (6)
In the equation above,πππ(π‘ + 1) is the number of arrival
aircraft in cell π at time π‘ + 1, πππβ1(π‘) represents the flux of
arrival aircraft entering cell π from cell π β 1 at time π‘, whileππ
π(π‘) means the flux of arrival aircraft exiting cell π at time π‘,
and ππis the time step.
Note. The number, flux et al. mentioned in this section, isspecific to arrival aircraft, while departures will be includedin following parts.
Air traffic flow in air route segment still satisfies the basicequation π
π= ππVπ, where π
πand V
πstand, respectively, for
segment liner density and space mean velocity of cell π. Thetraffic flux of cell πwill be π
π(π‘) = πΌ
πππ. Coefficient πΌ
πis the rate
of outflow per unit time, which reflects the saturation levelof cell π. From Section 2.1., we know segment liner densityππ= ππ/Ξ©π, where Ξ©
πis the length of cell π. Since most
of the time arrival aircraft in airport terminal area are in adeceleration process, we assume aircraft entering cell π fromsome certain arrival route positions with an initial velocity V
π
that comes from the STAR and then uniformly deceleratingalong the cell (route) with a rate ππ
π, so we can get the space
mean velocity of cell π, as follows:
Vπ=
ππ
πΞ©π
Vπβ βV2πβ 2ππ
πΞ©π
. (7)
To plug (7) into ππ= ππVπ, we can derive
ππ
π(π‘) =
πΌπππ
πππ
π(π‘)
Vπβ βV2πβ 2ππ
πΞ©π
. (8)
However, since the aforementioned situation is an idealcondition, it is necessary to take control measures for part ofthe arrival aircraft into consideration, such as speed control,maneuvering actions, or circling, because of the existenceof traffic congestion, safety interval, and so forth in realconditions. For simplicity, we assume that there is a unifiedvariable to stand for different kinds of control measures,since whichever measures controllers took, the actual effectscaused by changing speed or flight path and so forth can all beseen as there to beππATC
πaircraft flying along the cell (route)
with a new constant velocity VATCπ
in cell π. To be clear, thevariable VATC
πhere represents the displacement velocity that
is lower than the STAR designed velocity generally.Considering these problems,we derived (9) after integrat-
ing control measures into (8), as follows:
ππ
π(π‘) =
πΌπππ
π[ππ
π(π‘) β π
πATCπ
(π‘)]
Vπβ βV2πβ 2ππ
πΞ©π
+VATCπππATCπ
(π‘)
Ξ©π
=πΌπππ
π
Vπβ βV2πβ 2ππ
πΞ©π
ππ
π(π‘)
β (πΌπππ
π
Vπβ βV2πβ 2ππ
πΞ©π
βVATCπ
Ξ©π
)ππATCπ
(π‘) .
(9)
After substituting πππ(π‘) in the difference equation (6) with
(9), we got
ππ
π(π‘ + 1) = (1 β
πΌπππ
πππ
Vπβ βV2πβ 2ππ
πΞ©π
)ππ
π(π‘)
+ (πΌπππ
πππ
Vπβ βV2πβ 2ππ
πΞ©π
βVATCπππ
Ξ©π
)ππATCπ
(π‘)
+ ππππ
πβ1(π‘) .
(10)
Letπ΄ππ= 1β(πΌ
πππ
πππ/(VπββV2πβ 2ππ
πΞ©π));π΅ππ= (πΌπππ
π/(Vπβ
βV2πβ 2ππ
πΞ©π)) β (VATC
π/Ξ©π); πΆππ= πΌπππ
π/(Vπβ βV2πβ 2ππ
πΞ©π);
we can have the simplified equation sets, as follows:
ππ
π(π‘ + 1) = π΄
π
πππ
π(π‘) + π΅
π
πππππATCπ
(π‘) + ππππ
πβ1(π‘) ,
ππ
π(π‘) = πΆ
π
πππ
π(π‘) β π΅
π
πππATCπ
(π‘) .
(11)
The aircraft in cell are taken as evenly distributed in themacroscopic traffic flow theory, so the number of aircraftshould be equivalent number. Let ππ
π(π‘) be the mean nose
interval of adjacent aircraft at the same direction in cell π;therefore the number of aircraft in the cell isπ
π(π‘) = Ξ©
π/ππ
π(π‘).
If πππ(π‘) becomes less than the control separation standard
denoted by ππATCπ
, that is to say, aircraft density exceeds thethreshold level, the exceeded aircraft should be arranged to
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4 Discrete Dynamics in Nature and Society
Segment k β 2
Segment k β 1 Segment k + 1
Segment k + 2
Segment k
Figure 3: Converging and diverging of air traffic flow.
take control measures. In addition, arrival aircraft in airportterminal area also have to obey the safety separation πsafe
π
according to STAR; that is, πππ(π‘) β₯ π
safeπ
.The safety separationis evaluated by air traffic security department and usuallygets more stringent compared to control separation as πsafe
π<
ππATCπ
. The number of controlled aircraft at time π‘ in cell πsatisfies
ππATCπ
(π‘) =
{{
{{
{
(1
ππ
π(π‘)β1
ππATCπ
)Ξ©πππ
π(π‘) < π
πATCπ
0 ππ
π(π‘) β₯ π
πATCπ.
(12)
We have discussed the one-dimensional cell transmissionmodel of air traffic flow before, while there may be multipleair routes in terminal area and some of them may cross witheach other, in the way, in the same imaginary plane, as shownin Figure 3. So the converging and diverging situations are asshown in Figure 3.
According to the conservation of number, in convergingsituation the number of aircraft satisfies
ππ= ππβ1+ ππβ2+ β β β + π
πβπ. (13)
On the contrary, in diverging situation we have
ππ+1= ππ+1ππ,
ππ+2= ππ+2ππ,
.
.
.
ππ+π= ππ+πππ.
(14)
The coefficient π stands for the proportion of traffic flowto different air route segments; meanwhile 0 β€ π
πβ€ 1,βπ
π=
1.
2.3. Departure Traffic Flow Model. The model of departureflow in airport terminal area consists of two phases: taking offfrom runway and flying along air routes. In the take-off phase,departure air traffic flow will make the most use of runwaytime slots based on the arrival priority, in order to guaranteea smooth arrival and landing process. Flight departure sched-ules are generated in stochastic cases, while the departureflights that disagree with the operation time interval ofrunway will be delayed to ground holding procedures untilenough time slots come [20β22]. In the flying stage, departureaircraft diverge in various directions from the runway center,
which differs from the arrival flow as there would be bothdiverging and converging situations in arrival processes.Under the condition of fully isolation between arrival anddeparture routes, flying stagewill dispensewith flow control ifthe runway interval problem is solved in the taking-off stage.And under the condition of semi-isolation between arrivaland departure routes, which means that some air routes maybe overlapped in several segments (in the same imaginaryplane), it is necessary to take both departure aircraft andarrival ones into consideration simultaneously to focus onthe average nose interval of all aircraft in the overlappedsegments (cells). If the average interval cannot meet safetyneeds, departure flow should be adjusted prior to the arrivals.
Let πground(π‘) be the number of aircraft holding fordeparture on ground in time π‘, and its change can bedescribed as follows:
πground
(π‘ + 1) = πground
(π‘) + ππ[ππ
plan (π‘) β ππ
π (π‘)] . (15)
In (15),πground(π‘+1) is the number of aircraft holding fordeparture on ground in time π‘ + 1. The demand of departureper unit time that produced by flight schedules is representedby ππplan(π‘), while π
π
π (π‘) represents the actual number of aircraft
taking off from the runway per unit time.The time step is stillππ, andπground(π‘) cannot be negative, so we can get
ππ
π (π‘)
=
{{{{{{
{{{{{{
{
πground
(π‘)
ππ
+ ππ
plan (π‘) πground
(π‘)
+ ππ[ππ
plan (π‘) β πΆπ
π (π‘)] < 0
πΆπ
π (π‘) π
ground(π‘)
+ ππ[ππ
plan (π‘) β πΆπ
π (π‘)] β₯ 0.
(16)
Consider maximizing the use of runway time slots; vari-able πΆπ
π (π‘) represents the maximum take-off rate of the run-
way, which depends on the time interval of runway operation.If the time interval of arrival landing aircraft gets large, influ-ence such as wake vortex on the runway will have no effect ontake-off aircraft, in which cases departure flows may take offby the standard time interval separately. Otherwise if arrivaltime interval gets intense, mutual interference betweendepartures and arrivals will be strong, so that departure flowsneed to make use of the interspace of the time slots under thecircumstances of an affected runway operation, as follows:
πΆπ
π (π‘) =
{{{
{{{
{
ππ‘π
π (π‘) β ππ
runwayπ
ππrunwayπ
β ππ‘ππ (π‘)
ππ‘π
π (π‘) < ππ
runwayπ
1
ππππ
ππ‘π
π (π‘) β₯ ππ
runwayπ
.
(17)
In (17), this ππ‘ππ (π‘) is the actual time interval of arrival
landing and ππππ is the standard take-off time interval of
departures, while ππrunwayπ
and ππrunwayπ
, respectively, standfor the irrelevant runway operation time interval and themodified operation time interval according to the mutualinterference between landing and taking off. Among this, the
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Discrete Dynamics in Nature and Society 5
irrelevant runway operation time ππrunwayπ
will be a criticalpoint where interference occurred.
Similar to the Arrival Model, we divide air routes intoseveral segments by a series of unit-interconnected cells inthe stage of flying along departure routes. Let ππ
π(π‘) be the
number of departure aircraft in cell π at time π‘ and its changingprocess can be described as (18). On the left side of theequationππ
π(π‘+1) is the number of departures in cell π at time
π‘ + 1; πππβ1(π‘) represents the flux of departure aircraft entering
cell π from cell π β 1 at time π‘, while πππ(π‘)means that the flux
of departure aircraft exiting cell π at time π‘, ππis the time step.
Since most of the time departure aircraft in airportterminal area are in an acceleration process, we assumeaircraft entering cell π from some certain departure routepositions with an initial velocity π’ that comes from the SIDand then uniformly accelerating along the cell (route) with arate πππ. So according to the samemodeling principle from the
Arrival Model, we can get the following similar equations:
ππ
π(π‘) =
π½πππ
π[ππ
π(π‘) β π
πATCπ
(π‘)]
βπ’2π+ 2ππ
πΞ©πβ π’π
+π’ATCπππATCπ
(π‘)
Ξ©π
=π½πππ
π
βπ’2π+ 2ππ
πΞ©πβ π’ππ
π(π‘)
β (π½πππ
π
βπ’2π+ 2ππ
πΞ©πβ π’
βπ’ATCπ
Ξ©π
)ππATCπ
(π‘) ,
(18)
ππ
π(π‘ + 1) = (1 β
π½πππ
πππ
βπ’2π+ 2ππ
πΞ©πβ π’π
)ππ
π(π‘)
+ (π½πππ
πππ
βπ’2π+ 2ππ
πΞ©πβ π’π
βπ’ATCπππ
Ξ©π
)ππATCπ
(π‘)
+ ππππ
πβ1(π‘) .
(19)
In the above equations, coefficient π½πstands for the rate
of departures outflow per unit time. The number of aircraftthat need to be arranged to take departure flow controlmeasures is denoted by the variableππATC
π(π‘). As mentioned
before, in circumstances of fully isolation between arrival anddeparture route segments there will be no extra control to thedepartures; that is, ππATC
π(π‘) = 0. On the other hand, in the
routes overlapped condition part of the departures may takeextra controls. We assume that these controlled departureflows would move with a new constant ATC velocity π’ATC
πin
cell π and be the same with variable VATCπ
in the ArrivalModel,in which both represent displacement velocity. Meanwhilethe simplification form is
ππ
π(π‘ + 1) = π΄
π
πππ
π(π‘) + π΅
π
πππππATCπ
(π‘) + ππππ
πβ1(π‘) ,
ππ
π(π‘) = πΆ
π
πππ
π(π‘) β π΅
π
πππATCπ
(π‘) ,
π΄π
π= 1 β
π½πππ
πππ
βπ’2π+ 2ππ
πΞ©πβ π’π
;
π΅π
π=
π½πππ
π
βπ’2π+ 2ππ
πΞ©πβ π’π
βπ’ATCπ
Ξ©π
;
πΆπ
π=
π½πππ
π
βπ’2π+ 2ππ
πΞ©πβ π’.
(20)
To determine the value of variableππATCπ(π‘) in overlapped
segments, we will bring the average nose interval (nottime interval) of arrivals noted by ππ
π(π‘), the average nose
interval of departures noted by πππ(π‘), and the average interval
between both arrivals and departures noted by ππ/ππ(π‘) in cell
π all into consideration. The fundamental aim of flow controlin this stage is to make the entire average interval of arrivalsand departures meet the ATC separation requirement πATC
π
by adjusting departure flows, as follows:
ππATCπ
(π‘)
=
{{{{{{{{
{{{{{{{{
{
Ξ©π
ππ
π(π‘)
ππ
π(π‘) β€ π
ATCπ
(1
πππ·πΈπ
π(π‘)β1
πATCπ
) β Ξ©πππ
π(π‘) > π
ATCπ,
ππ/π
π(π‘) < π
ATCπ
0 ππ/π
π(π‘) β₯ π
ATCπ.
(21)
The average interval of arrivals and departures isππ/ππ(π‘) =
ππ
π(π‘) β π
π
π(π‘)/[π
π
π(π‘) + π
π
π(π‘)].
3. Parameter Analysis
We focus on one single cell in the model as to deduce andanalyze the interrelationship of air traffic flow characteristicparameters including flight flux, linear density, and trafficvelocity. In the following sections, we discuss this problemfrom the two different aspects as arrival and departure, justlike the models we had established before.
3.1. Arrival Routes. To plug (12) into (9), we can have
ππ
π(π‘) =
{{{{{{{{{{
{{{{{{{{{{
{
πΌπππ
πΞ©π
ππATCπ(Vπβ βV2πβ 2ππ
πΞ©π)
+VATCπ(1
ππ
π(π‘)β1
ππATCπ
) ππ
π(π‘) < π
πATCπ
πΌπππ
πΞ©π
ππ
π(π‘) (Vπβ βV2πβ 2ππ
πΞ©π)
ππ
π(π‘) β₯ π
πATCπ.
(22)
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6 Discrete Dynamics in Nature and Society
According to πππ(π‘) = 1/π
π
π(π‘), we get
ππ
π(π‘)
=
{{{{{{{{{
{{{{{{{{{
{
VATCπππ
π(π‘)
+
πΌπππ
πΞ©π+ VATCπ(Vπβ βV2πβ 2ππ
πΞ©π)
ππATCπ(Vπβ βV2πβ 2ππ
πΞ©π)
ππ
π(π‘) >
1
ππATCπ
πΌπππ
πΞ©π
(Vπβ βV2πβ 2ππ
πΞ©π)
ππ
π(π‘) π
π
π(π‘) β€
1
ππATCπ
.
(23)
From (23), we can see that the basic parameters such asflux ππ
π(π‘) and density ππ
π(π‘) form a piecewise function and
the segment point is the reciprocal value of ATC separationππATCπ
to the arrival aircraft. Assuming that coefficient πΌπ,
standard deceleration πππ, initial speed V
π, and length Ξ©
πof
one cell are all constant values, the slope and intercept of therelation curve will be determined by ATC velocity VATC
πand
the position of inflection point will be determined by ππATCπ
.Let both sides of (23) be divided by ππ
π(π‘):
Vπ(π‘)
=
{{{{{{{{{{{
{{{{{{{{{{{
{
πΌπππ
πΞ©π+ VATCπ(Vπβ βV2πβ 2ππ
πΞ©π)
ππATCπ(Vπβ βV2πβ 2ππ
πΞ©π)
β 1
ππ
π(π‘)+ VATCπ
ππ
π(π‘) >
1
ππATCπ
πΌπππ
πΞ©π
(Vπβ βV2πβ 2ππ
πΞ©π)
ππ
π(π‘) β€
1
ππATCπ
.
(24)
If aircraft density comes lower than the critical value, themean velocity of traffic flow in one cell will stay constant;otherwise, there will be an inverse relation between velocityVπ(π‘) and density ππ
π(π‘). With an increase of density in cell π,
mean velocity will become lower gradually and the speed ofreducing just gets more and more slow and eventually tendsto the ATC velocity VATC
π. Using the basic formula of fluid
π = πV, we can get
ππ
π(π‘) =
{{{{{{{{{{{{{{{{{{{
{{{{{{{{{{{{{{{{{{{
{
πΌπππ
πΞ©π+ VATCπ(Vπβ βV2πβ 2ππ
πΞ©π)
ππATCπ(Vπβ βV2πβ 2ππ
πΞ©π)
β Vπ(π‘)
Vπ(π‘) β VATC
π
VATCπ< Vπ(π‘)
<πΌπππ
πΞ©π
(Vπβ βV2πβ 2ππ
πΞ©π)
βππ
π(π‘) β
[[
[
0,πΌπππ
πΞ©π
ππATCπ(Vπβ βV2πβ 2ππ
πΞ©π)
]]
]
Vπ(π‘) = VATC
π,
πΌπππ
πΞ©π
(Vπβ βV2πβ 2ππ
πΞ©π)
.
(25)
From (25), we can see that the mean velocity of entiretraffic flow in cell π ranges fromATC velocity VATC
πto standard
velocity designed by STAR. If Vπ(π‘) is equal to one of the
critical values flux πππ(π‘) may be arbitrary-sized data taking
from zero to the maximum flux value. With an increase ofmean velocity in cell π, the flux value goes down graduallyand the speed of reducing gets slow.When this mean velocitygoes up to the peak value, it will cause a jump of traffic flux.Variables ππATC
πand VATCπ
are still the order parameters of theinterrelationship between flux and velocity in cell.
3.2. Departure Routes. In the stage of flying along departureroutes, aircraft keep accelerating and diverging alongwith thevarying of orientation of air routes. There may be ππATC
π(π‘)
aircraft taken departure flow control measures in some of theoverlapped segments. As an exceptional case of this situation,flying stage will dispense with flow control in the case offully isolation between arrival and departure routes; that is,
ππATCπ(π‘) = 0. We will focus on the overlapped segments
and analyze the traffic flow characteristic parameters ofdepartures in detail. To plug (21) into (18) we can have
ππ
π(π‘) =
{{{{{{{{{{{{{{{{{{{{{
{{{{{{{{{{{{{{{{{{{{{
{
π’ATCπ
ππ
π(π‘)
ππ
π(π‘) β€ π
ATCπ
π’ATCπ
ππ
π(π‘)+ππ
π(π‘) β π
ATCπ
πATCπππ
π(π‘)
Γ(π½πππ
πΞ©π
βπ’2π+ 2ππ
πΞ©πβ π’
β π’ATCπ) π
π
π(π‘) > π
ATCπ,
ππ/π
π(π‘) < π
ATCπ
π½πππ
π
ππ
π(π‘) (βπ’
2
π+ 2ππ
πΞ©πβ π’)
ππ‘
π(π‘) β₯ π
ATCπ.
(26)
-
Discrete Dynamics in Nature and Society 7
Since πππ(π‘) = 1/π
π
π(π‘), πππ(π‘) = 1/π
π
π(π‘), substituting in
(26), we get
ππ
π(π‘)
=
{{{{{{{{{{{{{{{{{{{{{{{{{{{{
{{{{{{{{{{{{{{{{{{{{{{{{{{{{
{
π½πππ
π
βπ’2π+ 2ππ
πΞ©πβ π’ππ
π(π‘) π
π
π(π‘) + π
π
π(π‘)
<1
πATCπ
π’ATCπππ
π(π‘) +
1 β πATCπππ
π(π‘)
πATCπ
Γ(π½πππ
πΞ©π
βπ’2π+ 2ππ
πΞ©πβ π’
β π’ATCπ) π
π
π(π‘) <
1
πATCπ
,
ππ
π(π‘) + π
π
π(π‘)
>1
πATCπ
π’ATCπππ
π(π‘) π
π
π(π‘) β₯
1
πATCπ
.
(27)
From (27), we can see that flux πππ(π‘) and density ππ
π(π‘)
form a piecewise function. Similar to the arrival segments(cells), we assume that coefficient π½
π, standard acceleration
ππ
π, initial speed π’, and length Ξ©
πof one cell are all constant
values. The density of arrival aircraft denoted by πππ(π‘) in cell
π changes over time, which has an effect on the interceptvalue of this linear equation. The ATC velocity of departuresdenoted by π’ATC
πshows the slope while the reciprocal of
ATC separation πATCπ
is the critical value of flow density,
determining the position of inflection points. Then, lettingboth sides of (27) be divided by ππ
π(π‘), we got
π’π(π‘)
=
{{{{{{{{{{{{{{{{{{{{{{{{{{{
{{{{{{{{{{{{{{{{{{{{{{{{{{{
{
π½πππ
π
βπ’2π+ 2ππ
πΞ©πβ π’
ππ
π(π‘) + π
π
π(π‘)
<1
πATCπ
π’ATCπ+1 β π
ATCπππ
π(π‘)
πATCπππ
π(π‘)
Γ(π½πππ
πΞ©π
βπ’2π+ 2ππ
πΞ©πβ π’
β π’ATCπ) π
π
π(π‘) <
1
πATCπ
ππ
π(π‘) + π
π
π(π‘)
>1
πATCπ
π’ATCπ
ππ
π(π‘) β₯
1
πATCπ
.
(28)
From (28), we can see thatmean velocity π’π(π‘) and density
ππ
π(π‘) also form a piecewise function. Two extreme cases are
as follows: the entire density of arrivals and departures islower than theATCdensity that is denoted by 1/πATC
π; density
only considers arrivals that have already exceeded 1/πATCπ
.These two cases will cause mean velocity of departures tobe the standard velocity determined by π½
π, acceleration ππ
π,
initial speed π’, length Ξ©π, and departure ATC velocity π’ATC
π,
respectively. In a certain condition between such two extremecases the slope of linear equation is greater than or equal toπ’ATCπ
, while the intercept value is influenced by both π’ATCπ
andππ
π(π‘). According to the basic formula of fluid π = πV, we get
ππ
π(π‘) =
{{{{{{{{{{{{{{{{{{{{{{{
{{{{{{{{{{{{{{{{{{{{{{{
{
(1 β πATCπππ
π(π‘)) [π½
πππ
πΞ©πβ π’
ATCπ(βπ’2π+ 2ππ
πΞ©πβ π’
)]
πATCπ(βπ’2π+ 2ππ
πΞ©πβ π’)
β π’π(π‘)
π’π(π‘) β π’
ATCπ
π’ATCπ< π’π(π‘)
<π½πππ
π
βπ’2π+ 2ππ
πΞ©πβ π’
βππ
π(π‘) β
[[
[
0,π½πππ
πΞ©π
ππATCπ
(βπ’2π+ 2ππ
πΞ©πβ π’)
]]
]
π’π(π‘) = π’
ATCπ,
π½πππ
π
βπ’2π+ 2ππ
πΞ©πβ π’.
(29)
The formof (29) ismuch like (25) from the arrival parts. Ifmean velocity of departures lies between the standard velocityand the departure ATC velocity π’ATC
πthere will be an inverse
proportional function between departure flux πππ(π‘) and
mean velocity π’π(π‘). Moreover, the coefficient of this inverse
proportional function is determined by π½π, acceleration ππ
π,
initial speed π’, and cell lengthΞ©πall together and changes in
value of the ATC velocity π’ATCπ
will make the function curve
-
8 Discrete Dynamics in Nature and Society
POU
ArrivalDeparture
PositionDeparture flowArrival flow
Overlap
D5.1POU
D12.0IOO
D15.7IOO
AGVOS
TAN
RWY02L
DANZHU
YIN
LONGTANG
P101
OSIKA
BIPOP
MUBEL
D18.2TAN
GYA
Figure 4: STAR/SID routes to RWY02L ZGGG.
move horizontally. While if mean velocity of departures lieson either of the two critical positions departure fluxmay varyrandomly.
4. Simulation Experiment
4.1. Simulation Sample. Based on the simulation platformNetLogo [23β25] we focused on each cell in the macroscopicmodel and designed traffic inflow/outflow behaviors to beeach agent, so as to take control of the inflow/outflowvolumesamong all the cells. Applying the STAR/SID procedures torunway 02L of Guangzhou Baiyun International Airport (asshown in Figure 4) into the Netlogo system dynamic simula-tor (as shown in Figure 5) we can emulate the operation andevolution process of air traffic flow in the airport terminalarea.
The network of arrival and departure routes in terminalcan be separated into several single direction segments andconverging or diverging segments that can be further decom-posed into more unit cells. For the purpose of simplicity, weset one cell dimension to be equal to the length of the shortestsegment in the network and the other air segments to beintegermultiples of unit cell dimension. According to this, wemade the following Tables 1 and 2 of the terminal network.
Each yellow rectangle in Figure 4 represents one cell,which means a βstockβ of fluid in the simulator system. Eachgrey pipe represents flowing between adjacent cells, whichmeans βflowβ in system, and the direction of arrow is the sameas the flow direction. Each black valve represents controlmeasures, which means βstrategyβ in system, and the inflowand outflow of air traffic will be under control by givingthe certain valves some corresponding rules. Take typicaloverlapped segments βTAN=AGVOS (Segment9&19)β as anexample, the cell transmission model of this is like (30). Inthe model, ππ
π/ππ΄π
π/ππ΅π
π/ππΆπ
π/π, respectively, represents the
number of arrival aircraft and the simplified coefficients of
Table 1: Cell quantities of arrival routes to RWY02L ZGGG.
Segment Code QuantityMUBEL-GYA Segment1 3 cellsBIPOP-GYA Segment2 2 cellsOSIKA-GYA Segment3 3 cellsP101-GYA Segment4 2 cellsGYA-AGVOS Segment5 2 cellsGYA-D5.1POU Segment6 2 cellsLONGTANG-TAN Segment7 2 cellsDANZHU-TAN Segment8 2 cellsTAN-AGVOS Segment9 2 cellsAGVOS-D15.7IOO Segment10 1 cellAGVOS-D5.1POU Segment11 1 cellD5.1POU-D15.7IOO Segment12 1 cellD15.7IOO-RWY02L Segment13 1 cell
Table 2: Cell quantities of departure routes to RWY02L ZGGG.
Segment Code QuantityRWY02L-D12.0IOO Segment14 1 cellD12.0IOO-YIN Segment15 2 cellsD12.0IOO-TAN Segment16 1 cellTAN-D18.2TAN Segment17 2 cellsD18.2TAN-POU Segment18 3 cellsTAN-AGVOS Segment19 2 cellsAGVOS-D5.1POU Segment20 1 cellD5.1POU-POU Segment21 2 cells
cell π in Segment π. ππATCπ/π
denotes the number of arrivalaircraft taking control measures in cell π of Segment π. ππ
π/π(π‘)
means the number of arrival aircraft that exit cell π in Segmentπ per unit time. ππ
π/ππ΄π
π/ππ΅π
π/ππΆπ
π/πππATCπ/π
ππ
π/π(π‘), respec-
tively, represents the similar corresponding ones but speciallyfor departures. Coefficient π stands for the proportion oftraffic flow into some segment when diverging occurs in airroutes. π is the designed time step for the simulator system:
ππ
9/1(π‘ + 1)
= π΄π
9/1ππ
9/1(π‘) + π΅
π
9/1πππATC9/1
(π‘) + π [ππ
7/2(π‘) + π
π
8/2(π‘)] ,
ππ
9/1(π‘) = πΆ
π
9/1ππ
9/1(π‘) β π΅
π
9/1ππATC9/1
(π‘) ,
ππ
9/2(π‘ + 1) = π΄
π
9/2ππ
9/2(π‘) + π΅
π
9/2πππATC9/2
(π‘) + πππ
9/1(π‘) ,
ππ
9/2β10/1(π‘) = π
9/2β10/1[πΆπ
9/2ππ
9/2(π‘) β π΅
9/2ππATC9/2
(π‘)] ,
ππ
9/2β11/1(π‘) = π
9/2β11/1[πΆπ
9/2ππ
9/2(π‘) β π΅
9/2ππATC9/2
(π‘)] ,
(30)
-
Discrete Dynamics in Nature and Society 9
Figure 5: NetLogo system dynamic simulator.
ππ
19/1(π‘ + 1)
= π΄π
19/1ππ
19/1(π‘) + π΅
π
19/1πππATC19/1
(π‘) + πππ
16/1(π‘) ,
ππ
19/1(π‘) = πΆ
π
19/1ππ
19/1(π‘) β π΅
π
19/1ππATC19/1
(π‘) ,
ππ
19/2(π‘ + 1)
= π΄π
19/2ππ
19/2(π‘) + π΅
π
19/2πππATC19/2
(π‘) + πππ
16/2(π‘) ,
ππ
19/2(π‘) = πΆ
π
19/2ππ
19/2(π‘) β π΅
π
19/2ππATC19/2
(π‘) .
(31)
4.2. Simulation Design. In this simulation sample there are6 entry points for arrivals of the terminal network: MUBEL,OSIKA, BIPOP, P101, LONGTANG, and DANZHU. Weassume that the arrival rate of each entry point obeys the neg-ative exponential distribution [26]. To plug the average arrivalrate that comes from flight historical statistics as expectedvalue into distribution functions we can obtain the changesof traffic flux over time at each entry point as shown in Figure6. It should be noted that we use equivalent traffic flow in thissimulation and take each simulation time step as 1 minute.
It can be noticed that traffic flux changes significantlyover time at each entry point. When aircraft keep going toconvergent points, the peak of traffic wave may happen tomeet another one, in which situation traffic density nearbywill be too high to meet separation requirements and the
probability of unsafe events will increase accordingly. Instead,the trough of traffic wave may also meet another trough thatmay lead to a low traffic density and a large aircraft interval atconvergent points, thus reducing the time/space utilization oflimited airspace. Based on the situation we designed a βvalveβ.Agent aims at balancing traffic flow in the simulator system tomanage the inflow/outflow of adjacent cells. In addition, thisis much like the principle of βCutting peak and filling valleyβin real air traffic flow management [27, 28].
The basic strategy of arrival βvalveβ control is as follows:first determine whether the mean arrival aircraft interval islower than the arrival ATC separation ππATC
πin one cell at any
time, which means whether the stock of aircraft in one cellexceeds. If yes then the excessive number of aircraft shouldoperate with the assigned ATC velocity VπATC
πand the rest do
not change their speed, while if no then all aircraft can stillfollow the STAR. Since VπATC
πis lower than normal velocity,
the traffic flux will also be lower compared to no controlsituations. With outflow decreasing, the stock of aircraft inone cell will increase accordingly. At next time step inflowwill be added to the original cell stock. If this total value stillexceeds the standards, control measures should be taken likebefore. These cyclic steps keep going until sometime therecomes a small inflow and the total value added with cell stockgoes lower than the standards. In this circumstance all aircraftin the cell can operate with STAR and aircraft controlledbefore can exit cell normally. In the whole process, the total
-
10 Discrete Dynamics in Nature and Society
BIPOP
MUBEL OSIKA LONGTANG
P101 DANZHU
Time (m)
0
0 200Time (m)
0 200Time (m)
0 200
Time (m)0 200
Time (m)0 200
Time (m)0 200
1
Flow
(f/m
)
0
1Fl
ow (f
/m)
0
1
Flow
(f/m
)
0
1
Flow
(f/m
)
0
1
Flow
(f/m
)
0
1
Flow
(f/m
)
qs3qs1
qs2
qs7
qs8qs4
Figure 6: Changes of traffic flow over time at entry points.
number of aircraft in one cell cannot exceed the safety value atany time; that is, mean interval should never be smaller thanthe safety separation.
Take typical arrival segment βGYA-AGVOS (Segment5)βas an example andmaking a comparison of air traffic flux andstock in cells between before and after the βvalveβ control, itis easy to find that the inflow and outflow in this segmentbecome smooth and steady when there is the βvalveβ controland the stock of aircraft in each cell always meets the safetyrequirement as shown in Figures 7(a) and 7(b). The unit isflights/minute.
The departure in terminal area consists of two majorparts: taking off from runway and flying along departureroutes. We designed two successive processes in this simu-lation accordingly called the airport surface part and the airroutes part, which are complementary in departure process.Specifically, the airport surface part mainly consists of depar-ture flights schedule generation, runway time occupation,take-off slots allocation, and aircraft ground holding. Weassume that the generation of departure flights scheduleobeys a negative exponential distribution. The number ofscheduled departure flights per unit time is as shown inFigure 8(a). The runway time occupation and take-off slotsallocation are actually how the landing aircraft and taking-offaircraftmake themost use of runway slot resources within thelimited and dynamic runway operation capacity. Changes inthe number of landing and taking-off aircraft per unit timeare as shown in Figures 8(b) and 8(c). Since landing aircrafthave the priority, taking-off aircraft that are unable to go byflight schedules will be postponed to take ground-holdingprocesses. Changes in the number of ground holding aircraftover time are as shown in Figure 8(d).
In the process of flying along departure routes, there isno need to take control measures under the condition of full
isolation between arrival and departure routes. The compar-ison of departure flux and stock over time before and aftercontrol measures is as shown in Figures 7(c) and 7(d). Thebasic strategy of departure βvalveβ control is as follows: firstdetermine whether the mean arrival aircraft interval is lowerthan the ATC separation πATC
πin one cell at any time. If yes
then all the departures in this cell should take control mea-sures, which means passing the segment with constant ATCvelocityπ’ATC
π. If no then determinewhether themean interval
of both arrivals and departures is lower than ATC separationπATCπ
in one cell, and if yes departures must be adjusted toincrease the entire mean interval until it goes above πATC
π.
4.3. Result Analysis. According to the export data derivedfromNetLogo simulation platformwe got series of parameterscatter diagrams that could reflect the basic traffic flowcharacteristics through statistic and analysis, as shown inFigure 9. Based on this, we analyzed the mutual influencerelationship among traffic flux π, density π, and velocity Vof the arrival and departure routes in airport terminal area.Limited by space this paper only made detailed discussion onπ β π relation of arrivals and departures; meanwhile the restof Vβπ and πβV relations were listed in the form of statisticaldiagrams for reference.
Taking typical arrival segment, Segment5, as an example,we derived the basic tendency of traffic flux and densityrelationship for arrival routes in terminal area as shownin Figure 9(a1). From the diagram we can find that therelationship tendency consists of three main stages.
Stage I is the free flow state inwhich the number of aircraftin segment is very low and the mean interval exceeds arrivalATC separation ππATC
5, whichmeans all the aircraft can follow
STAR. The traffic flux is directly proportional to density in
-
Discrete Dynamics in Nature and Society 11
Segment5: GYA-AGVOS Segment5: GYA-AGVOS
Time (m)0 200
Time (m)0 200
0
1
0
2
Num
ber (
f)
Flow
(f/m
)
q51q52-10q52-11 Safe
Cell 51Cell 52
(a) Before arrival control
Segment5: GYA-AGVOSSegment5: GYA-AGVOS
q51q52-10q52-11
Time (m)0 200
Time (m)0 200
Num
ber (
f)
0
2
0
1
Flow
(f/m
)
Safe
Cell 51Cell 52
(b) After arrival control
Segment19: TAN-AGVOS Segment19: TAN-AGVOS
Time (m)0 200
Time (m)0 200
0
1
0
2
Num
ber (
f)
Flow
(f/m
)
q191q192
arr/dep1arr/dep2Standard
(c) Before departure control
Segment19: TAN-AGVOS Segment19: TAN-AGVOS
q191q192
Time (m)0 200
Time (m)0 200
0
2
0
1
Num
ber (
f)
arr/dep1arr/dep2Standard
Flow
(f/m
)
(d) After departure control
Figure 7: Comparison of traffic flux and stock over time before and after control measures.
segment and the proportionality coefficient is equal to themean velocity of segment. Stage II is the congestion flowstate in which the number of aircraft in segment increasesand then the mean interval is under arrival ATC separationππATC5
, which means part of the aircraft should be assigned totake control measures including decelerating, maneuveringor holding, and so forth. The relationship between trafficflux and density occurs a inflection point. The traffic flux insegment is still directly proportional to density; however thenew proportionality coefficient is equal to the space meanvalue of the controlled aircraft velocity and uncontrolledaircraft velocity; that is,π·ππ
5(ππATC5π‘π
5+ βππATC5
π=1π‘π)β1, see details in Section 2.1(1).
All of the control measures in the arrival process, includ-ing deceleration, maneuvering and holding, will lead to adecline in the displacement velocity. In the diagram it madethe tendency of flux and density relationship to be levelingoff.While the traffic flux will still increase with density, whichdiffers from normal vehicle traffic since after inflection pointthe vehicle traffic flux decreases with density. The reasonwhy this difference exists is that the car-following behaviorbetween adjacent vehicles has significant influences whencongestion occurs on the road while air traffic normallymaintains a larger safe separation. Apart from this, somecontrol measures as hold pattern make aircraft deviate fromoriginal air routes, which does not affect the other aircraft.Therefore, after the inflection point traffic flux will notdecrease but increase with a low slope. Stage III is the block
flow state in which the number of aircraft in segment exceedssafe value with the control adjustment in congestion stateand large amount of traffic converging continuously. Themean interval is under the safe separation πsafe
5and insecurity
factors surge that should be avoided as much as possible.From the above theoretical derivation we can find that
variables related to operational flight program including ini-tial velocity V
π, acceleration π
π, and lengthΞ©
πof segment (cell)
become constant values as the STAR/SID are established.This paper focuses on the rest of the variables especially flowcontrol variables including ATC separation πATC
πand ATC
velocity VATCπ
. Such variables will become order parametersthat influence the mutual relationships among three basic airtraffic flow parameters.
As shown in Figure 9(a2), when arrival ATC separationππATC5
rises to 25% the inflection point of traffic flux anddensity relationship moves forward and actually the horizon-tal axis value equals 1/ππATC
5. The rise of ATC standard will
lead to a decrease of free flow state in Stage I. More aircraftneed to take control measures; meanwhile the frequency ofhigh traffic density even block flow in Stage III also increases.But in whatever Stage I or Stage III, the slope of traffic fluxand density relationship maintains constant; that is, beforeinflection point the increase tendency coincides and afterinflection point the increase tendency still keeps parallel.As shown in Figure 9(a3), when arrival ATC separationππATC5
remains the same and arrival ATC velocity VATC5
turnsdown, the inflection point stays the same but aircraft pass
-
12 Discrete Dynamics in Nature and Society
Flight schedule
Time (m)0 200
0
3
Plan
Flow
(f/m
)
(a) Scheduling
Arrival landing
Time (m)0 200
0
1
q131
Flow
(f/m
)
(b) Landing
Departure taking off
0
1
Time (m)0 200
qd
Flow
(f/m
)
(c) Taking off
Ground holding
Time (m)0
0
6
200
Ground
Num
ber (
f)
(d) G-holding
Figure 8: Changes of traffic flux and stock over time in airport surface part simulation.
the segment with a lower mean velocity after this point.Therefore, traffic flux has a decelerated growth with thedensity. Conversely, if it needs to let traffic flux reach thelevel before changing of ATC velocity VATC
5, the density of air
segment should be higher, which may lead to an early arrivalof the block flow state in Stage III.The free flow state in Stage Ihas no change now since it is not affected by controlmeasures.
Taking typical departure segment, Segment19, as anexample, we derived the basic tendency of traffic flux anddensity relationship for departure routes in terminal area asshown in Figure 9(b1).The relationship tendency also consistsof threemain stages. Similar to arrival parts, Stage I is the freeflow state inwhich the traffic flux of departures is directly pro-portional to density in segment and the proportionality coef-ficient is equal to the mean velocity of segment from SID.Stage II is the congestion flow state which is unlike arrivalparts, since in departure routes we focus on the entire meaninterval of both arrivals and departures, not just departures.With the density of departures rising up in segment, theentiremean interval can still meet the ATC separation πATC
19if
arrivals density is small enough. Then the tendency of trafficflux and density relationship lies on the extension line ofStage I. Stage III is the block flow state in which the numberof arrivals and the number of departures increase simulta-neously. Accordingly the entire mean interval becomes lowerthan the ATC separation πATC
19. Considering the principle
of arrivals priority part of the departures will accelerateto leave the heavy-traffic segment in order to release thespace resources for arrivals. The new slope of traffic flux anddensity relationship equals the departure ATC velocity π’ATC
19.
The number of adjusted departures is determined by arri-vals, which is shown as a series of scatter values betweenthe tendency line of Stage III and extension line of Stage II.The vertical distance from tendency line of Stage III to thescatter values varies inversely to the density of arrivals, thatis, ((1 β πATC
19ππ
19(π‘))/π
ATC19)πΎ (πΎ is constant coefficient); see
details in (27).As shown in Figure 9(b2), after raising up 25% of the ATC
separation πATC19
, there appear more departure aircraft thatneed to take control measures.Thus the Stage III also appears
-
Discrete Dynamics in Nature and Society 13
0
0
0
0.05
0.1
0.2
0.3
0.15
0.25
0.35
0
0.05
0.1
0.2
0.3
0.15
0.25
0.35
0.01 0.02 0.03 0.04 0.05
0.005 0.01 0.02 0.030.015 0.025
0.06 0.07 0 0.02 0.04 0.06 0.08 0.1 0.12
I
I
I
II
II
II
III
III
III
Airspace5
Airspace5
Airspace5
Airspace19
3
4d5
5
4d19
5
4u19
3
4d19
3
4u19
u19
d19
5
4d5
1
5οΏ½5
1
5οΏ½5
q(nΒ·m
inβ1)
0
0.05
0.1
0.2
0.3
0.15
0.25
0.35
q(nΒ·m
inβ1)
0
0.05
0.1
0.2
0.3
0.15
0.25
0.35
q(nΒ·m
inβ1)
q(nΒ·m
inβ1)
0
0123456789
0.05
0.1
0.2
0.3
0.15
0.25
0.35
q(nΒ·m
inβ1)
οΏ½(k
mΒ·m
inβ1)
012345678910
012345678910
0
0.05
0.1
0.2
0.3
0.15
0.25
0.35
q(nΒ·m
inβ1)
οΏ½5
οΏ½5
5οΏ½5
k (nΒ·kmβ1)
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07
k (nΒ·kmβ1)
k (nΒ·kmβ1)0 0.005 0.01 0.02 0.030.015 0.025
k (nΒ·kmβ1)0 0.005 0.01 0.02 0.03 0.0350.015 0.025
k (nΒ·kmβ1)
k (nΒ·kmβ1)
0 0.02 0.04 0.06 0.08 0.1 0.12
k (nΒ·kmβ1)
0 0.01 0.02 0.03 0.04 0.05 0.06
k (nΒ·kmβ1)
0 0.01 0.02 0.03 0.04 0.05 0.06
k (nΒ·kmβ1)
5
4d
πΌ
5
5
4d5
3
4d5
da a
a
5
d5
00
0 1 2 3 4 5 6 7 8 9
2
4
6
8
10
12
14
024681012141618
024681012141618
0.005 0.01 0.02 0.030.015 0.025
k (nΒ·kmβ1)0 0.005 0.01 0.02 0.030.015 0.025
k (nΒ·kmβ1)0 0.005 0.01 0.02 0.030.015 0.025
k (nΒ·kmβ1)
οΏ½(k
mΒ·m
inβ1)
οΏ½ (kmΒ·minβ1)
00 1 2 3 4 5 6 7 8 9 10
οΏ½ (kmΒ·minβ1)0 1 2 3 4 5 6 7 8 9 10
οΏ½ (kmΒ·minβ1)
οΏ½(k
mΒ·m
inβ1)
οΏ½(k
mΒ·m
inβ1)
I
I
II
II
III
III
Airspace19
5
4d19
3
4d19
d19
5
4u19
3
4u19
u19
0
0.05
0.1
0.2
0.3
0.15
0.25
0.35
q(nΒ·m
inβ1)
0.05
0.1
0.2
0.3
0.15
0.25
0.35
q(nΒ·m
inβ1)
0
0.05
0.1
0.2
0.3
0.15
0.25
0.35
q(nΒ·m
inβ1)
οΏ½55οΏ½5
5οΏ½5
(a1) Arrival q β k (a2) Sensitivity analysis daATC5
(b2) Sensitivity analysis dATC19 (b3) Sensitivity analysis uATC19
(d3) Sensitivity analysis uATC19
(c3) Sensitivity analysis οΏ½ ATC5
(e3) Sensitivity analysisοΏ½ATC5
(a3) Sensitivity analysis οΏ½ATC5
ATC
ATC
ATC
ATCATC
ATCATC
ATC
ATC
ATCATCATC
da
a
5ATC
aATC
(c2) Sensitivity analysis daATC5
(e2) Sensitivity analysisdaATC5
(d2) Sensitivity analysisdATC19
(c 1) Arrival οΏ½ β k
(e 1) Arrival q β οΏ½
(d1) Departure οΏ½ β k
(b1) Departure q β k
ATC
ATC
ATC
ATCATC
ATC
ATCATC
ATC
ATCATC
ATC
ATCATC
οΏ½(k
mΒ·m
inβ1)
οΏ½(k
mΒ·m
inβ1)
1
5οΏ½5
ATC
a
a
3
4d5
ATCa
(a)
Figure 9: Continued.
-
14 Discrete Dynamics in Nature and Society
0
0.05
0.1
0.2
0.3
0.15
0.25
0.35
q(nΒ·m
inβ1)
0
0.05
0.1
0.2
0.3
0.15
0.25
0.35
q(nΒ·m
inβ1)
0
0.05
0.1
0.2
0.3
0.15
0.25
0.35
q(nΒ·m
inβ1)
0 2 4 6 8 10 12 14
οΏ½ (kmΒ·minβ1)(f1) Departure q β οΏ½
0 2 4 6 8 10 12 14 16 18
οΏ½ (kmΒ·minβ1)0 2 4 6 8 10 12 14 16 18
οΏ½ (kmΒ·minβ1)
(f2) Sensitivity analysis dATC19 (f3) Sensitivity analysisuATC19
3
4uATC19
5
4uATC19
5
4dATC19
3
4dATC19
uATC19
dATC19
Airspace19
I
II
III
(b)
Figure 9: Scatter diagrams of basic arrival/departure flow characteristic parameters in terminal area.
earlier while the slopes of tendency lines in congestion andblock flow states have no change. However, there are morescatter values between these two lines.The Stage II congestionflow state in which entire mean interval (high departure,low arrival) still exceeds the ATC separation will reduce itsfrequency-of-occurrence. More scatter values lie in the StageIII block flow state. Conversely reducing the ATC separationπATC19
departures may not need to be adjusted in most cases,which tend to be the Stage I free flow state.More scatter valuesof traffic flux and density liemore on the extension line of freeflow state.
Keeping the ATC separation πATC19
unchanged and chang-ing the departure ATC velocity π’ATC
19we can get new relation-
ship tendencies as shown in Figure 9(b3). The Stage I freeflow state keeps the same. But in Stage III block flow statethe variation rate of traffic flux with density increases withthe departure ATC velocity π’ATC
19, which means departure
flow has raised its outflow rate of heavy-traffic segment.Meanwhile the impact of arrivals increases by πππ
19(π‘)π’
ATC19
(πis constant coefficient); see details in (27), which is shown asthe scatter values spread a larger scope from center line of thefree flow state extension direction.
5. Conclusions
Based on CTM we have proposed the macroscopic modelof air traffic flow in airport terminal area and carried out aseries of simulation experiments with the NetLogo platform.Through both of the theoretical and practical discussions,we could generally reveal the basic interrelationships andinfluential factors of air traffic flow characteristic parameters.The research findings are as follows.
(1) TheCTM could accurately reflect themacroevolutionlaws of air traffic flow in terminal air route network.Meanwhile, it may also be applied to different kindsof air traffic scenes including airways, sectors, orairspaces by modifying conditions correspondingly.Moreover, it has a remarkable operational efficiencyin multiagent simulations.
(2) There are obvious relationships among the three cha-racteristic parameters as flux, density, and velocity of
the air traffic flow in terminal area. And such rela-tionships evolve distinctly with the flight procedures,control separations, andATC strategies.The air trafficflow characteristics may take the specific changesthrough flight procedure optimization, control sep-aration modification, or ATC strategy regulation,which could be part of the scientific basis for air trafficmanagement in airport terminal area.
(3) The default parameters we used in simulation experi-ments are from practical ATC rules. Automatic opti-mization of these parameters for desired traffic flowcharacteristics should be taken into consideration infurther studies. In addition, discussions in this paperfocus onmacro perspectives, thus the research resultsseem rough anyway. To obtain more detailed andelaborate traffic flow characteristics it is necessary tocombine such macro studies with micro perspectivesthat give full expressions to the following: overflyingor turning and so forth of individual behaviors andinteractive effects. It also should be an importantdirection for further study.
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper.
Acknowledgments
This research is supported by βthe National Natural ScienceFoundation of China (NSFC) no. 61104159β and βthe Fun-damental Research Funds for the Central Universities no.NJ20130019.β
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