435
1 INTRODUCTION
The dimensioning of bends in existing fairways for
preset turn angles and arc radiuses comes down to
thedeterminationoftheirsafewidths.Suchproblems
areusuallysolvedwhentheexistingwaterwaysareto
be modernized to allow navigation by ships with
largerparameters.Thismeans the conditions
of safe
ship operation in a specific waterway are to be
changed. Although in such situations the existing
infrastructure of the area restricts the possibility of
changing turn anglesand radiuses of bend arcs, the
problemcanbesolvedusingsimulationorempirical
methods.
Simulation methods inwhich the simulation
experiment is conducted on full mission bridge
simulators (FMBS) following specific research
procedures [Gucma S. et al. 2015] enable the
determination of bend widths with high accuracy.
However, thesemethods are relatively cost‐
intensiveduetotheneedtocarryoutmany(reliable
samplesize)simulatedpassagesinrealtime,executed
byproperly qualifiedfairwaypilots [GucmaS.etal.
2008].
Empirical methods of determining safe bend
widthsaremuchlesscostlyapproximatemethods.An
analysis has been made to examine the accuracy of
safewidthsoffivebendsontheŚwinoujście‐Szczecin
fairway,determinedbyusingthesemethods:
PIANC[PIANC2014]
Spanish[PuertosdelEstado2007]
Japanese[JapanInstituteofNavigation2003]
USACE[USACE2006,USACE2008]
Canadian[CanadianCoastGuard1999]
PolishINM[GucmaS.2001]
PolishMTEC[GucmaS.etal.2015]
Theanalysishasshowna relativelylowaccuracy
of
determiningsafefairwaywidthsdependingonthe
parametersofthebendandofthemanoeuvringship
[GucmaS.etal.2017].
In empirical methods such as MTEC (Marine
Traffic Engineering Center) and INM (Institute of
MarineNavigation)thewidthofasafemanoeuvring
areainbendshastwocomponents:manoeuvring
and
Kinematic Method of Determining Safe Fairway Bend
Widths
S.Gucma,J.Dzwonkowski&M.Przywarty
M
aritimeUniversityofSzczecin,Szczecin,Poland
ABSTRACT:Thisarticlepresentsadedicatedkinematicmethodofdeterminingasafefairwaybendwidthwith
aspecificturnangleandarcradiusasthefunctionofshipparametersandprevailingnavigationalconditions
onthefairway.Theassumedapproachtakesintoconsiderationmonoeuvringandnavigational
componentsof
thesafefairwaybendwidth.Themethodisbasedonananalysisoftheresultsofnumericaltestsconductedon
amodelrepresentingallphysicallypossiblemovementsofshipʹscentreofgravityinthebend.Thedeveloped
methodwasinitiallyverifiedontheIńskiebend,part
oftheŚwinoujście‐Szczecinfairway.
http://www.transnav.eu
the International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 14
Number 2
June 2020
DOI:10.12716/1001.14.02.22
436
navigational. These components depend on the
parameters of the bendand the ship manoeuvring
therein, while they do not depend on the fairway
coordinatesofthebend.Foraspecificbendandship,
the components assume constant values, while the
manoeuvring component is an empirically
determinedquantity,andthenavigational
component
isprobabilistic(shippositioningerrorinthebend).
Asthesimulationmethodiscostly,andempirical
methodsarenotaccurateinthedeterminationofsafe
bend widths for modernization or changed ship
operational conditions, a kinematic method for the
determinationofthemanoeuvringcomponentofsafe
fairway bend widths
with a known turn angle and
bend arc radius has been developed. The method is
basedonananalysisoftheresultsofnumericaltests
conducted on a model representing all physically
possiblemovementsofshipʹscentreofgravityinthe
bend. The model is analytical, solved by numerical
procedures
[Dzwonkowski J. 2018; Dzwonkowski J.,
PrzywartyM.2017].
Thedimensioningofbendsinexistingfairwaysfor
preset turn angles and arc radiuses comes down to
the determination of their manoeuvring areas. The
safe manoeuvring area in the bend must meet the
basicconditionofnavigation:




1
1
(,) ()
i
i
ik
xy k
ik
pxy t
dDt
ht T

D
where
d
ik(1‐α)–safeareaini‐thbendfork‐thshipperforming
a simulated manoeuvre specified at the level of
confidence1‐α;
Di(t) – navigable area of i‐th bend (the condition of
safedepthatinstanttissatisfied);
h
xy(t)–waterdepthatpoint(x,y)atinstantt;
T
k–maximumdraftofk‐thship;
∆ik(1‐α) – underkeel clearance in the i‐th bend
determinedfork‐thshipatthelevelofconfidence1‐
α.
In pilot navigation, shipʹs position in the bend is
determined in the path coordinates [Gucma S. et al.
2015]. In the kinematic method of determining safe
bendwidthsthe
followingfairwaycoordinatesystem
isadopted:
thexaxiscorrespondstotheadoptedcentrelineof
thefairway;
the y axis is perpendicular to the tangent of the
fairwaycentreline at a given point. The adopted
direction of y‐axis outward is (+) and inward is
(–).
Further
in the article the the width of the safe
manoeuvring area of the bend at j‐th point of the
fairwaycentrelineataspecific confidencelevelwill
bedenotedasd
(1‐α)(j).
Inthefairwaybendforone‐waytrafficthewidth
of a safe area at the confidence level (1‐α) is
determinedbyrelation:





11 1
2
mn
djd jd



where
d
(1‐
)(j)–safewidthforj‐thpointofthebendatthe
confidencelevel(1‐α);
d
m(1‐α)(j)–manoeuvringcomponentofsafewidthfor
j‐thpointofthebendattheconfidencelevel(1‐α);
d
n(1‐α) – navigational component of safe width of a
givenbendattheconfidencelevel(1‐α).
The manoeuvring component of the safe
manoeuvring area width is calculated using a
speciallydevelopedkinematicmethodofdetermining
themanoeuvringcomponentofsafebendwidths.
In the developed method, the manoeuvring
componentof
safebend widthisdivided intoasafe
width of the manoeuvring area of shipʹs centre of
gravityinthebendandanadditionalmarginforship
parametersanddriftangle:


11
ΔΔ
zw
mms
djd jdd



where
d
ms(1‐
)(j) – widthofthemanoeuvringareaofshipʹs
centreofgravityatj‐thpointofthebendatthelevel
ofconfidence(1‐α);
∆d
z – additional margin of the manoeuvring
component of the bend width taking into account
shipʹs parameters (Lc, B) and its drift angle on the
externalpartofthebend;
∆d
w – additional margin of the manoeuvring
componentofthebendwidthtakingintoaccountthe
shipʹs parameters and its drift angle on the internal
partofthebend.
Thesafewidthof themanoeuvringareaof shipʹs
centreofgravityinthebendisdeterminedusingthe
kinematic
methodpresentedinthisarticle.
The additional margin of the manoeuvring
component can be determined by either of the two
methods:
thedriftanglemethod
∆d
z=
c
L
sinα
2
+
B
cosα
2
∆d
w=
B
2
empiricalmethod[PuertosDelEstado2007]
∆d
z=
22
2R
c
KL
∆d
w=
B
2
where
B–breadthoftheship;
α–driftangle;
R–radiusofshipmovementinthebend;
K
– coefficient dependent on the depth h to draft T
ratio(forh/T≤1.2K=1/2;forh/T≥1.5K=2/3).
437
Thenavigationalcomponentofsafebendwidthis
the directional error of the bow or stern position
(point farther from the observer on the bridge)
determined at a specified confidence level. It is the
directional error perpendicular to the centre line of
thefairway,equalto:
d
n(1‐
)=

1yD
p

=


2
1
2
1
57,3
D
KR
y
mL
p






[m]
where

1
yD
p

– directional error of shipʹs bow at the
confidencelevel(1‐α)[m];

1y
p

– directional error of shipʹs position
determination(observerʹs position) at the confidence
level(1‐α)[m];

1
KR
m

– headingdeterminationerrorinabendat
theconfidencelevel(1‐α)[°].
Forlargeships(L
c≥150m)withthesuperstructure
aftitisassumed[GucmaS.etal.2015]that:

0.95
KR
m
=±2°(PilotNavigationSystem)

0.95
KR
m
= ±4° (other position determination
systems)
LD≈0.75Lc
For various methods of ship position
determination, types of fairway (inner or outer) and
theaidstonavigation(markedfairwayornot,types
of seamarks), directional errors of the bow/stern
position

1
yD
p

are determined using dedicated
algorithms[GucmaS.etal.2017].
2 THEKINEMATICMETHODOFDETERMINING
THEMANOEUVRINGCOMPONENTOF
FAIRWAYSAFEBENDWIDTHS.
The kinematic method for determining the
manoeuvring component of safe areas in a fairway
bend makes use of a ship movement model, whose
concept requires that multiple
simulations are made
of the shipʹs centre of gravity passage through the
bend, divided longitudinally into sectors and
transverselyintosegments.Themodelrepresentsthe
entirephysicallypossibleshipmovementinthebend.
Paths of shipʹs centre of gravity consist of,
numerically calculated in each sector, circle arcs
or
sections that are further regarded as separate
manoeuvring events (Fig. 1). Thus created sets of
manoeuvring events are subject to further analysis,
searching for the distribution of movement density
and the tolerance of rudder settings applied
[DzwonkowskiJ.2018].
Figure1. Paths of shipʹs centre ofgravityin the kinematic
modelofshipmovement.
The method requires the introduction of the
followingassumptionsanddefinitions:
1 The research is retrospective and starts from the
position, in which the shipʹs centre of gravity
completes negotiating the bend, which in the
analysisofthepathsfacilitatestheachievementof
thegoal.
2 Themovementmodelin
themethodreferstothe
centreofgravity,whichresultsinthefollowing:
width of shipʹs swept path differs from the
width of the path followed by the centre of
gravity by an additional value depending on
driftandshipʹslengthandbreadth,
rateof
turn(ROT)ofthecentreofgravityisthe
angular speed of movement along an arc of
circleradiusr.
3 Internalandexternalboundariesofthefairwayare
definedinthemodelbyarcsofconcentriccircles.
The fairway boundary line may in reality have
another shape, but it has
to lie outside the arcs.
Theproblemconsideredhereinreferstoapartof
the fairway, a single bend (turn), whose
boundaries can be defined by arcs of two
concentriccircles.
4 Forsetsofeventstodescribetheentirephysically
possiblemovementandthemovementmodeltobe
similar
to reality, the arcs along which the shipʹs
centre of gravity moves have to result from the
maximumruddersettingsthecaptainorpilotuse.
Inaddition,theshipwillnotturninthedirection
oppositetothe onefollowed,thereforethe centre
ofgravityROT0=0.Besides,for
themovementin
the bend to be considered as safe, the maximum
rudderanglehastoenablethesamechangeinthe
position of the centre of gravity from the axis
towards the external and internal boundaries.
Hence,ROT2istwicegreaterthanROT1.
5 The other requirement for a
set of manoeuvring
events to describe the whole physically possible
movement is to determine the magnitude of the
sectorsothatitcorrespondstoaminimumtimeof
makingdecisionsconcerningruddersettings[PRS
2013].Thetimewasdefinedas10seconds,which
corresponds to classification society requirements
for the rudder
to move fromʹmidshipsʹ to the
recommended [PIANC 2015] maximum value of
rudderanglein a bend, i.e.20°.Thereductionof
sectorlengthisrestricted bytheassumedtime of
numericalcomputing,becausethenumberofarcs
in a sector increases at least 2.5 times, so that in
sector
13, for instance, 100,000 arcs have to be
calculated.
The input data for the movement model are: the
positionandthecourseovergroundattheendofthe
438
bend, outside and inside fairway boundaries, sector
andsegmentsizes,longitudinalspeedandROT.
The outcome of calculations using the kinematic
modelaredifferentsetsofarcs(manoeuvringevents)
ineachsector.Theproposedmethodfordetermining
a safe manoeuvring area for the tested ship in the
bendofknown
arcradiususesthefollowingsets(Fig.
2):
successful manoeuvring events (SME), i.e. those
thatdonotgooutsidethefairway,
non‐rectilinear manoeuvring events, i.e. all those,
whoseROTisdifferentfrom0.
Figure2.Distributionsofmanoeuvringeventsinasector.
Along the external part of the bend where the
shipʹs centre of gravity can move 95% of the field
under the curve of SME distribution defines the
externalboundaryofthesafemanoeuvringareaofthe
shipʹs gravity centre at the confidence level (1‐α) =
0.95.
Alongthe
internalpartofthebendthattheshipʹs
centreofgravitycanmovetheboundary of the safe
manoeuvringareaisdeterminedbytheratioofnon‐
rectilinear events to SME in each segment. The
segmentinwhich the shipʹs centre of gravity moves
non‐rectilinearly at 95%
probability makes up an
internal border of the safe manoeuvring area of the
shipʹscentreofgravity,determinedattheconfidence
level(1‐α)=0.95.
Thesafemanoeuvringareaoftheshipʹscentreof
gravityinabendatthelevelofconfidence(1‐α)=0.95
isdeterminedby
conductingnumericalresearchand
ananalysisoftheresultsforfourcases,i.e.forboth
directions of passing through the bend and for two
positions of the end of the turn.These positions are
definedbytheboundariesofthemanoeuvringareas
ofadjacentstraightsectionsofthefairwaydetermined
attheconfidencelevel(1‐α)=0.95.
The safe manoeuvring area of shipʹs centre of
gravity in the bend is determined by summing up
four component manoeuvring areas of the shipʹs
centreofgravitycalculatedatthelevelofconfidence
(1‐α)=0.95(Fig.3):
    
11111ms pz pw lz lw


ddddd
where

1pz
d
–manoeuvringareaofthegravitycentreofa
shipproceedinginthebendʹtotherightʹ,completing
themanoeuvreatapointofexternalboundary(Z1)of
the adjacent manoeuvring area of a straight fairway
section,(1‐α)=0.95;

1pw
d
–manoeuvringareaofthegravitycentreofa
shipproceedinginthebendʹtotherightʹ,completing
themanoeuvreata pointofinternalboundary(W1)
oftheadjacentmanoeuvringareaofastraightfairway
section,(1‐α)=0.95;

1lz
d
–manoeuvringareaofthegravitycentreofa
ship proceeding in the bendʹto the leftʹ, completing
themanoeuvreatapointofexternalboundary(Z2)of
the adjacent manoeuvring area of a straight fairway
section,(1‐α)=0.95;

1lw
d
–manoeuvringareaofthegravitycentreofa
ship proceeding in the bendʹto the leftʹ, completing
themanoeuvreata pointofinternalboundary(W2)
oftheadjacentmanoeuvringareaofastraightfairway
section,(1‐α)=0.95;
Figure3.Thediagramofmanoeuvringareasofthegravity
centreofashipproceedinginthebendtotheright
d
pz(1‐α)anddpw(1‐α).
Taking into account the movement of a specific
shipalongtheexternalandinternalboundariesofthe
bend, the component manoeuvring areas allow to
determine a safe manoeuvring area of the shipʹs
center of gravity within this bend at the assumed
confidencelevel(1‐α)=0.95(Fig.4).
With
additional margins of the manoeuvring
componentofthebendwidthforshipparametersand
drift (∆d
z and∆dw), we determine the safe
manoeuvringareaofthebend,andaftersummingup
thenavigationalcomponentsweobtainthe
safeareaofthebendfortheexaminedship.On
this basis we can determine the safewidth at j‐
thpointofthebendatanappropriate
levelofconfidence
d(1‐α)(j).Itshould be notedatthis
point that thedetermination ofasafearea of the
bendbythekinematicmethodisdoneatthe
confidencelevel1‐α =
0.95foramaximumshipexpectedtosailthrough
thegivenfairway.
The allowable wind speed expected in the
conditionsof
safefairwayoperationisaccountedfor
by choosing a specific drift angle while defining
margins∆d
zand∆dw.
439
Figure4.Asafemanoeuvringareaofshipʹscentreofgravity
inthebend.
3 SHIPRATEOFTURNINFAIRWAYBENDS.
The rate of turn, i.e. angular speed at which ships
negotiate fairway bends depends primarily on the
parametersofthe bend, althoughotherfactorscome
intoplay:
shipsizeandmanoeuvringcharacteristics,
rudderangle,
initiallongitudinalspeedoftheship,
externalconditions(wind,waves,current).
Thepresentedmethodallowsthedeterminationof
the ROT in the fairway bend for ships of different
sizesandfordifferent inputparameters.To simplify
the method and extend its universality it was
assumedthatshipʹsROTinthebenddependsonship
lengthandblockcoefficientofthehull.Additionally,
theimpactofinitiallongitudinalspeedandrudder
angleweretakenintoaccount.ThereductionofROT
due to shallow water was regarded as negligibly
small and was not taken into account [Nowicki A.
1999]. Due to the variability of external conditions
(wind,
waves, current) it was decided not to take
themintoaccountinthedeterminationofshipʹsROT
inthebend.The shipʹs ROT can be calculated using
thefollowingrelationship:
NOM SOG
R
OT ROT ROT ROT

where
ROT
–shipʹsROTinfairwaybend;
ROT
NOM – nominal ROT determined from
manoeuvringdata;
ROTSOG–changeinROTduetootherthannominal
initiallongitudinalspeedoftheship;
ROTα–changeinROTduetothechangeinrudder
angle.
TheimpactofselectedfactorsonshipʹsROTinthe
bend was defined upon detailed review of some
publications [Eloot K. et al. 2018; Fossen T. 2011;
Nowicki A. 1999]and gathered manoeuvring data
from ships of various
types and sizes. First, the
relationship was determined between the ship size
(length) and ROT read out from available
manoeuvring data for a 90 degree turn (Fig. 5). To
assess the dependence between chosen factors
polynomial regression was used. The analysis was
made for ships with various sizes, different loading
state
and for longitudinal speeds corresponding to
fullspeedahead,atmaximumrudderangle(35°‐45°).
The ships were divided into two groups by block
coefficient criterion: large (Cb≥0.75) and small
(Cb<0.75). Nominal ROT values for ships of various
sizesandblockcoefficientsweredeterminedthrough
ananalysisofthe
gathereddata(Table1).
Figure5. ROT of ships of different lengths and block
coefficients.
Table1.NominalROTinfairwaybends
_______________________________________________
shiplengthROTNOM[deg/min]
[m]Cb≥0.75  Cb<0.75
_______________________________________________
60101.362 108.722
8091.626 103.846
10082.6199.05
12074.314 94.334
14066.738 89.698
16059.882 85.142
18053.746 80.666
20048.3376.27
22043.634 71.954
24039.658 67.718
26036.402 63.562
28033.866 59.486
30032.0555.49
_______________________________________________
AROTchangethatresultsfromotherspeedthan
thenominalinitiallongitudinalspeedoftheshipwas
calculated from gathered manoeuvring data and
literature review [Przywarty M. et al. 2013]. It was
found that no significant differences exist for ships
withlargeandsmallblockcoefficients,sotheanalysis
was
combinedforbothshipgroups(Fig.6).
440
Figure6. Change in ROT due to a change in shipʹs initial
longitudinalspeed.
Basedonananalysisoftheresultsitwasassumed
that the percentage change of ROT is equal to the
percentagechangeofinitiallongitudinalspeed.Itcan
therefore be calculated by using the following
relationship:
NOM
SOG NOM
NOM
SOG SOG
ROT ROT
SOG




where
ROTSOG–changeinROTduetospeedotherthanthe
initial longitudinal speed adopted for the
determinationofROT
NOM;
SOG
–initiallongitudinalspeedoftheship;
SOG
NOM – initial longitudinal speed adopted for the
determinationofROT
NOM;
ROT
NOM–nominalROT.
The additional decrease in ROT due to less than
themaximumrudderanglecanbecalculatedthus:

NOM SOG
ROT k ROT ROT

Thevalues of the coefficientkdetermined on the
basis ofliterature review [Fossen T. 2011], [Nowicki
A.1999]arepresentedinTable2.
Table2. Value of the coefficient k for different rudder
angles.
_______________________________________________
Rudderangle[deg]k
_______________________________________________
>=350
300.09
250.24
200.29
150.39
100.5
50.57
_______________________________________________
4 THEDETERMINATIONOFSAFEWIDTHSIN
THEIŃSKIEBENDOFTHEŚWINOUJŚCIE‐
SZCZECINFAIRWAYUSINGTHEKINEMATIC
METHOD
The research covered the Ińskie bend, part of the
fairwaylinkingtheportsofSzczecinandŚwinoujście
(51.8kmto52.9km).Safemanoeuvringareasofthat
bendweredeterminedforabulkcarrierhavingthese
parameters:
displacement47.000t;
Lc=195m,B=29.0m;T=11.0m;
lateralwindage1200m2;
powerofmainengine8,500kW.
Thebulkcarrierwasadoptedasaʹmaximumshipʹ
tosail
throughtheŚwinoujście‐Szczecinfairwayafter
its upgrade, for which simulation tests were carried
out[Analiza…2015;GucmaS. 2016]. The test results
servedfortheverificationofthekinematicmethodof
thedeterminationofsafebendwidths.Thealgorithm
of the kinematic method requires that the following
inputdataare
definedandassumed:
1 Determinationofapreliminarywidthofthebend
by the MTEC method. Preliminary bend width
0,95
w
d
calculated at the confidence level 0.95
accounts for the manoeuvring and navigational
components. The width defines preliminary
boundaries of the bend used in the kinematic
method.
2 Determinationofthewidthofsafemanoeu
vringareasinstraight
fairwaysectionsadjacent to the examined bend
attheconfidencelevel(1‐α)=0.95.
These
widthsarealsodefinedusingthe MTECmethod.
Thewidthsdefineexternalpoints(Z1andZ2)and
internalpoints(W1andW2)inwhichthecentreof
gravitycompletesthemovementalongthebendin
bothdirections.
3 Segments3meterswideareadopted(accuracyof
thePNSsystem).
4 Determinationof the sector magnitude that
depends on the timeofchanging the rudder and
longitudinalspeedsettingsoftheship, whichare
asfollows:
4m/s–halfaheadsettingofthemainengine,
3m/s‐slowaheadsettingofthemainengine,
5 Adopted maximum number
of manoeuvring
eventswas equalto100,000,ensuringasufficient
accuracyoftheresults.
6 AdoptionofmaximumrateofturnROT2,reached
bytheship’scentreofgravityinthebend:
for the bend exit position of the ship at the
internal boundary, ROT2 results from the
rudder
angle20°,
for the bend exit position of the ship at the
external boundary, ROT2 results from the
rudderangle35°.
The developed kinematic method was used to
determinesafewidths(safearea)oftheIńskiebend,
in the currently modernizedŚwinoujście – Szczecin
fairway,forabulk
carrierLc= 195m,B= 29.0m,T=
11.0m(Fig.7).Thefigurealsodepictssafewidthsin
the Ińskie bend determined by the computer
simulationmethodandthedeterministic‐probabilistic
MTECmethod[Analiza...2015].
Thefollowingconclusionscanbedrawnfromthe
results:
1 Safewidthsofthe
benddeterminedbyasimulated
experiment conducted on a Konsberg‐made full‐
mission bridge simulator at the Maritime
University of Szczecin, with a participation of
highly qualified pilots, who executed a reliable
number of bend passages, can be regarded as
modelonesduetotheirhighaccuracy.
441
2 Thekinematicmethod,likethesimulationmethod,
estimatesthesafebendwidthasafunctionofthe
turn angle. The MTEC method determines a safe
bendwidthasaconstantquantity.
3 The kinematic method overestimates safe bend
widthbyroughly30%comparedtothesimulation
method, which is
related to the human factor
involvedinthesimulationmethod.
4 The kinematic method is more accurate than the
MTECmethodbyapproximately40%.
Figure7. Safe width of the Ińskie bend,Świnoujście‐
Szczecinfairway, determinedbythe kinematic, simulation
anddeterministic‐probabilisticMTECmethods.
5 CONCLUSIONS
The article presents a kinematic method of
determiningsafewidthoffairwaybends.Themethod
enables determination of the manoeuvring
componentofthesafebendwidthbasedonmultiple
simulationsofthepassageofshipʹscentreofgravity
through the bend, which represents the whole
physically possible movement
of ships within this
fairwaysection.
Using the developed kinematic method, the
authors have determined safe widths of the Ińskie
bendintheŚwinoujście‐Szczecinfairway,undergoing
modernization at present, for a specific bulk carrier
havingL
c=195m,B=290m,T=11.0m.Theresults
havebeencomparedtothesafebendwidthsobtained
by two alternative methods: simulation and
deterministic‐probabilistic MTEC method. The
analysisoftheresultsindicatesthat:
1 Thekinematicmethod,likethesimulationmethod,
estimatesthesafe
bendwidthasafunctionofthe
turn angle. The MTEC method determines a safe
bendwidthaconstantquantity.
2 The kinematic method is more accurate than the
deterministic‐probabilisticMTECmethod.
3 The kinematic method overestimates safe bend
widthbyroughly30%comparedtothesimulation
method, which is
related to the human factor
involvedinthesimulationmethod.
4 Thecostsofthekinematicmethodaremuchlower
than those of the simulation method and
comparabletotheMTECmethod.
Giventhemoderatecostsandhighaccuracyofthe
kinematicmethoditshouldberecommendedforuse
in
the preliminary design of marine traffic
engineeringsystems.
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