109
1 INTRODUCTION
Different types of traffic flow exist in busy ports
aroundtheworld(Figure1).Marinetrafficisspecial
and different from other types. Ships are not
unlimited in their navigation, and deep waterways
exist across the port. These deep waterways are
generally associated with the principal water flow
pa
thofthetidalsystem.Insomecases,thepositionof
theprincipalwaterwayvariessignificantly,seasonto
season, and migrates across the port region. Marine
traffic within ports is frequently diverse as the
shallow, relatively sheltered waters permit the safe
navigation of small craft (leisure vessels, fishing
boats, fast launches) across the port, while larger
vessels (coast
al cargo vessels and ocean‐going
carriers)inhabitthedeeperwaterways.Thecontinued
growth of port traffic increases the congestion of
waterways.
Thisconceptualpaperrepresentsourfirstattempts
to investigate the traffic characteristics of marine
trafficflowinordertodevelopamarinet
rafficmodel.
As macroscopic models have been advanced in
different traffic disciplines, the proposed
establishment of marine traffic flow models is
expectedtocomplementtheexistingliteratureofland
traffic.
Figure1.Marinetrafficflow.
A Marine Traffic Flow Model
T.L.Yip
C.Y.TungInternationalCentreforMaritimeStudies,DepartmentofLogisticsandMaritimeStudies,TheHongKong
PolytechnicUniversity,HongKong
ABSTRACT: A model is developed for studying ma
rine traffic flow through classical traffic flow theories,
whichcanprovideuswith a betterunderstandingof thephenomenon oftrafficflow ofships.On onehand,
marinetraffichasitsspecialfeaturesandisfundamentallydifferentfromhighway,airandpedestriantraffic.
The existing t
raffic models cannot be simply extended to marine traffic without addressing marine traffic
features.Ontheotherhand,existingliteratureonmarinetrafficfocusesononeshiportwoshipsbutdoesnot
addresstheissuesinmarinetrafficflow.
http://www.transnav.eu
the International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 7
Number 1
March 2013
DOI:10.12716/1001.07.01.14
Ship’sdomain
–
theareafree
fromnavigationalobstacles
Watercurrent
110
The paper is organized as follows. Section One
provides the background of ship navigation and
summarise characteristics of marine traffic. Section
Two reviews previous studies on the traffic theory
and ship navigation modelling. Section Three
develops a new model for marine traffic and
especially marine traffic characteristics are
incorporatedin
themodel.SectionFouranalyzesand
understands the marine traffic model. Section Five
concludesthepresentstudy.
2 LITERATUREREVIEW
Systematic studies of traffic flow have been
conducted for more than five decades. A basic
building block is the kinematic waves in traffic
(Lighthill and Whitham 1955; Richards 1956), which
relates the continuum traffic flow, the traffic speed
andthetrafficdensity.Thefocusofthepresentstudy
isthemodelsofmarinetrafficflow.
Highway traffic has attracted considerable
attention for decades, for example, Gazis (2002).
Many highway traffic models assume the
homogenous vehicles are not applicable to marine
traffic.
The heterogeneity is recently considered in
highwaytrafficresearch,forexample,Wong&Wong
(2002), Park et al., (2010). They however did not
considerasawholewheretheexistingmodelscanbe
usedasmarinetraffic.
In the air traffic control, Andersson et al. (2003)
proposed a novel optimisation
approach to analyse
collaborative airport arrival planning. Ship
manoeuvring simulators are common in many
maritimecountriesand generally operateinthetime
domain.Theiruserangesfromthefullmissionbridge
simulator to PC‐based simulator. Existing Traffic
Alert and Collision Avoidance System (known as
TACSII)isusedtodetect
thealtitudesofaircraftand
then resolve (altitude crossing) encounters in the
verticaldomain.Ifanencounterisidentified,TACSII
will command one aircraft to climb and theother to
descend. However, ships can only manoeuvre
horizontally and ships have different
manoeuvrability. Different from air traffic control,
VTS is
only an advisory service for ships; ship
mastersareresponsibleforaship’scourse,speedand
safety.
Onthetrackofpedestrianflow,somemodelshave
been developed (see Hughes, 2003). The models of
pedestrian flow have three common assumptions.
First the speed of pedestrian walk is determined
solely by the
density of surrounding pedestrians.
Second,allpedestriansarethesame,similartoafluid
particle in flows. Third, pedestrians avoid extreme
densities, and so the model is mathematically
convenient.
Previousresearchmaynotbeapplicabletomarine
traffic flow, as existing studies do not take into
considerationthedifferencesbetween
ships.Trafficin
previous models is considered as continuous flow
and not as single ships with their individual
characteristics of type, dimensions and velocity.
Marine traffic is over moving water current. Real
marine traffic is not consisted of ships of equal size
moving with equal manoeuvrability. The depth of
waterhas
considerableinfluenceontherateofship’s
turnwhichmaybeobtainedatagivenrudderangle.
Ifnavigationinconfinedto watersthat requirelarge
alternations of course, the turning manoeuvres must
be commenced in due time with the knowledge of
how much room the ship needs to carry
out the
alteration ofcourse. This will,especially withregard
tolargeships,necessitatelongerresponsetime,larger
reaction zone ahead, and technically a higher
relaxationeffect.
Ship‐ship collision models have been developed
on the basis of geometrical distribution and/or
encounter‐to‐collision. Pedersen (2002; 2010),
Montewkaetal.(2010),
Debnath&Chin(2010),Tan&
Otay(1999),Seongetal.(2012)developedgeometrical
collision probability models that describe the
geometrical probability model of collision. Fowler &
Sorgard (2000) estimated the collision based on
encountersbyassumingthe trafficis independentor
uncorrelated. USCG (1999) found different types of
encounters
have different relative significance, with
crossings more hazardous than head‐on encounters,
whichareinturnmoreriskpronethanover‐takings.
These assumptions are applicable only when the
traffic density is low. In reality, ships may change
speed or direction so as to avoid possible collisions,
e.g. see Merrick et
al. (2002).The crossing traffic
modelscoveronlyacrossingsituationoftwovessels.
In particular, in heavily trafficked ports, like Hong
Kong,threeorevenmoreshipsmayapproachanarea
atthesametime.Inthiskind ofsituation,acollisionis
more difficult to avoid when the
actions of several
other vessels need to be observed.Hu et al. (2010)
usedAIStodeterminethecongestionlevelofmarine
trafficinrestrictedwaters.Theirfindingsareuseful
todevelopmacroscopicmarinetrafficmodels.
3 TRAFFICMODEL
3.1 Macroscopicmodel
Letusestimatehowthewatercurrentmanifests itself
inthemarinetrafficflowproblem(Figure1).Thelack
ofexperimentaldatadoesnotallowthemarinetraffic
flowtobeformulatedmathematically.Basedonsome
analogiesbetweenmarinetrafficandlandtraffic(e.g.
Payne, 1971), the one‐dimensional marine traffic
modelisgivenby
0
g
v
x
t
, (1)
x
CvV
x
v
v
t
v
ww
g
g
g
2
0
)(
, (2)
where
t is time,
x
is horizontal coordinate,
is
trafficdensity,
g
v
isaveragetrafficvelocityoverthe
ground in the
x
direction,
w
v is average traffic
velocity through water.
)(
w
V is the characteristic
through‐water speed determined by speed‐density
relationship (should be determined from field
survey). We have to emphasise that both the
111
relaxation term and anticipation term in Eq. (2)
depend on the average velocity (speed) through
water, rather than mean velocity (speed) over the
ground.
Assuming the ship draft is deep, ship’s velocity
over the ground
g
v
is the velocity through water
w
v plusvelocityofwatercurrent
c
v ,suchthat
cwg
vvv
. (3)
Consider a uniform water current
c
v constantof
t and
x
,Eq.(1)and(2)willbecome:
0
w
v
x
t
, (4)
x
CvV
x
v
v
x
v
v
t
v
www
c
w
w
w
2
0
)(
. (5)
Comparing Eq. (4) and (5) against the known
Payne trafficflow model (Payne, 1971), the presence
of water current adds an extra acceleration term
x
v
v
w
c
in the dynamic equation. The effect of extra
acceleration term increases the apparent acceleration
ofmarinetrafficinthemagnitudeof
c
v ,iftheships
navigatealongthedirectionofwatercurrent.
Similarly for the two‐dimensional marine traffic,
thegoverningequations(4)and(5)become:
0
ww
v
y
v
xt
, (6)
www ww
wwc
vvu vu
vvv
txy xy
2
0
()
ww
VvC
x
y
. (7)
However, several major components should be
integrated in the marine traffic models. As the
manoeuvrabilityofashipisrelatedtoitslengthand
watercurrent,therelaxation
),,(
0 ssw
TLvfC , (8)
in which
ss
TL , are average ship length and ship
type, respectively. The water current
c
v varies
acrossthewaterway,
)( yvv
cc
( Wy 0 ), (9)
andthusmostvariablesvaryacrossthewaterway.
3.2 Shipdomain
A critical step is to determine the density‐speed
relationshipofships,whichcanbeestimatedbyship
domains.Theshipdomainthatthenavigatorwantsto
keepclearofothershipsisdefinedastheshipdomain
(Goodwin,1975).
As shown in Figure 2, the ship domain occupied
by stationery ship can be calculated in a rectangular
form B x L, where B and L arethe bea
m and length
overall of the ship. When a ship is navigating, a
greater ship domain is required, that is W x D. Both
thetermsWandDcanbeexpressedasafunct
ionof
thenavigatingspeedthroughwater
w
v .
Figure2.Rectangularshipdomain.
Differentfromothertrafficflows,theWandDare
also dependent of water flow, since ships tend to
navigate with more clearance when the water speed
c
v is large. The length of ship domain of ship
movingoverwater,d,canbeexpressedasthesumof
twoterms:thenavigatinglength(whichisafunction
of the navigating velocity) and the watch distance
(whichisa function ofvelocity, visibility,traffic and
local and psychological factors). The navigating
lengt
h is the distance of the ship navigating over a
certain unit of time (e.g. 5 seconds). The watch
distance is the distance required by the ship for
steeringandreaction.
4 ANALYSIS
The asymptotic method of homogenisation will be
applied to deduce the traffic flow equations for
differentmecha
nisms.Thehomogenisationmethodis
basedontheasymptotictechniqueofmultiplescales,
e.g. Ng & Yip (2001). Consider three time scales are
associatedwiththemarinetrafficflowina horizontal
waterway(Figure1):
0
t forlongitudinal navigation;
and
1
t for longitudinal collision‐avoidance
movements. The model assumes “no conflict” or
minimum space is preserved between ships. As the
width of waterway is shorter than the length,
01
tt
. When ships navigate downstream, ships of
differentspeeds,manoeuvrability,etc.willspreadout
along the length of the waterway. By and large, we
expectthatthetimeforshipstospreadislongerthan
the time for ships to remove conflicts. With these
assumptionsoftwotimescale
01
tt t
,thetraffic
flow problem can be decomposed into two simpler
subproblems(
0
t subproblem;
1
t subproblem),Ng&
Yip(2001)refers.
The original derivative becomes, according to the
chainrule:
d
w
D
W
112
01
tt t
(10)
Thetrafficdensity
01
(11)
Thevelocityis:
01 0 1
()()()
www
vV V V
(12)
TheterminEq.(1)becomes:
01 01
() ( ) ( )
ww
VV
00 01 10
() () ()
www
VVV
(13)
WithEq.(10)‐(13),Eq.(4)becomes:
01
01
0v
tt x
(14)
Atthezero‐thorder
0
()O
,
0
00
0
()0
w
V
tx
(15)
Atthefirstorder
()O
,
0
1
0110
01
() () 0
ww
VV
ttx
(16)
Similarly,Eq.(2)at
0
()O
becomes:
2
00000
0
00
()
w
vvV C
v
tx x
(17)
Eq.(2)atthefirstorder
()O
is:
00
11
01
01
vv
vv
vv
tt x x
2
10
1
0
()
w
VC
x
(18)
The model, Eq. (15)‐(18), is then written in two
vectorforms:
00
00 0
0
()
QQ
A
QS
tx
(19)
11
11 10 0
0
() (, )
QQ
AQ Sv
tx
(20)
where
01
,QQ aretheconservativevariables,
01
,AA
are the fluxes, while
01
,SS
are the source terms,
suchas:
0
0
0
Q
v
,
1
1
1
Q
v
,etc. (21)
Eq. (19) and (20) are in the quasi‐linear matrix
form of the governing equations and can be solved
numericallybythemethodofcharacteristics.
The two dimensional problem, Eq. (6)‐(9), can be
solvedsimilarly.
5 CONCLUSIONSANDDISCUSSION
Thisconceptualpaperisthefirstattemptto
developa
marinetrafficforstudyingthedynamicbehaviourof
ships. The study uses the classical traffic model to
consider two special marine traffic characteristics (a)
thewatercurrent,and(2)theshipdomainconcept.
Sincethemarinetrafficismoresophisticatedthan
the classical (land) traffic, the marine traffic
model
will enable a new and richer insight in traffic
behaviour in general. In the marine traffic
engineering,themarinetrafficflowtheoryandtraffic
control schemes are under development using
different approaches based on classical traffic flow
theory and ship manoeuvring characteristics. They
can provide a supportable foundation for vessel
trafficcontrol.
Future work of this research is to conduct
computational experiments and to develop control
strategies of marine traffic for different scenarios.
Computationalexperimentscanbefurtherconducted
toevaluatethe overall controlstrategiesapplied toa
combinationoftrafficmix.
ACKNOWLEDGEMNTS
This research was supported by a grant
from by the
GeneralResearchFundingsponsoredbytheResearch
GrantsCouncil(ReferenceNo.PolyU5300/12E).
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