International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 5
Number 3
September 2011
377
1 INTRODUCTION
An oceangoing ship is affected by external forces
such as wind, ocean currents, and waves. Weather
routing techniques are usually applied in cases
where the magnitude of those external forces are
very large and may pose a danger to ships. In addi-
tion, optimal navigation is required from an econom-
ical point of view. Although there have been many
efforts to develop route optimization techniques,
most cases do not take ship maneuvering dynamics
into account. Recent improvements in computing
performance have made it possible to carry out a
large amount of computation, so it is even possible
to take ship motion dynamics into consideration. In
this study, a new weather routing method was devel-
oped that takes ship maneuvering motions, current,
wind, and waves into account by time domain com-
puter simulation.
An MMG-type mathematical model of ship ma-
neuvering motions is introduced for the dynamic
calculation. The model includes many variables such
as sway, yaw, propeller thrust and torque, rudder
force, and fuel consumption. The maneuvering mo-
tions are solved by differential equations of motion
for every moment throughout the voyage. Moreover,
the optimal navigation route is determined by mini-
mizing the fuel consumption through Powell’s
method. In this paper, the mathematical models of
the ship maneuvering navigation model are first
shown. Next, methods for calculating the current,
wind, and waves from the database are introduced
and demonstrated. Then, several kinds of computer
simulations for weather routing are shown. Finally,
the applicability and future works are discussed.
2 TARGET SHIP AND ROUTE
A container ship was chosen as the subject ship of
this study because it is one of the principal means of
marine transportation. Moreover, container ships
consume more fuel than other marine vehicles be-
cause they are run faster to accommodate tight cus-
tomer schedules. The specifications of the subject
ship are shown in Table 1.
Table 1. Specifications of subject container ship.
_______________________________________________
Length overall Loa 299.85 m
Length between perpendiculars Lpp 299.85 m
Breadth molded Bmld 40.00 m
Depth molded Dmld 24.30 m
Draft designed d 14.02 m
Propeller diameter Dp 9.52 m
Propeller pitch Pp 7.25 m
Lateral projected area A
AL
8284.25 m
2
Transverse projected area A
AT
1052.18 m
2
Gross tonnage GT 75,201 t
Service speed Vs 25.0 kt
_______________________________________________
Advanced Navigation Route Optimization for
an Oceangoing Vessel
E. Kobayashi, T. Asajima & N. Sueyoshi
Kobe University, Kobe, Japan
ABSTRACT: A new weather routing method is proposed that accounts for ship maneuvering motions, ocean
currents, wind, and waves through a time domain computer simulation. The maneuvering motions are solved
by differential equations of motion for every moment throughout the voyage. Moreover, the navigation route,
expressed in terms of a Bézier curve, is optimized for minimum fuel consumption by the Powell method. Alt-
hough the optimized route is longer than the great circle route, simulation results confirm a significant reduc-
tion in fuel consumption. This method is widely applicable to finding optimal navigation routes in other areas.
378
The intercontinental route between Yokohama,
Japan, and San Francisco, USA, as shown in Figure
1, was used as the subject route in this study because
it is a very important trade route for Japan and
weather conditions along the route are sometimes
rough.
Figure 1. Subject transportation route between Yokohama and
San Francisco.
3 WIND, WAVES, AND CURRENT
ESTIMATION
Ocean surface current data from the National Ocean-
ic and Atmospheric Administration (NOAA) of the
United States Department of Commerce were used
in this study. The data were five-day averages for a
1.0° × 1.0° mesh in the area between 0.5°E and
0.5°W and 59.5°N and 59.5°S. Sample data from 26
December 2008 are shown in Figure 2. The arrows
show the eastbound and westbound current vectors,
respectively.
The predicted data for wind and waves were de-
rived from global prediction data by NCEP (Nation-
al Center for Environmental Prediction); they in-
clude several parameters for wind wave information,
such as wind direction, wind velocity, and signifi-
cant wave height every 3 h; the data were updated
every 6 h. The range of the data were between longi-
tudes of 0°E and 1.25°W and latitudes of 78°N and
78°S; the mesh size was 1.25° and 1° for longitude
and latitude, respectively. The target data times of
the voyage simulation starting in this study were
midnight on 6, 9, 22, and 25 December 2008. The
current, wind, and wave data were updated every 3
h.
Figure 2. Sample data of ocean surface current.
4 MATHEMATICAL MODEL OF SHIP
MANEUVERING
4.1 Coordinate system and basic equations
A body-fixed coordinate system whose origin is lo-
cated at the ship center of gravity was adopted to
express ship maneuvering motions (Figure 3).
Figure 3. System coordinates in ship maneuvering motion.
Basic equations for ship maneuvering motions in
the longitudinal, lateral, and yaw directions are ex-
pressed in the following equations:
−−
+−−
++−+
ARPHzzzz
ARPH
xycc
xy
AWARPH
vrxycc
vryx
+N+N+N=Nr)+J(I
+Y+Y+Y=Y
)r+mmψ)(ψ+v(u
)ur+(m+mv)(m+m
+R+X+X+X=X
)Xmψ)(mψ-v(u
)vrXm(mu)m(m
sincos
cossin
00
00
(1)
where
m
= mass of a ship;
x
m
,
x
m
= added masses in
the x and y directions, respectively;
zzzz
JI ,
= mass
moment of inertia and added mass moment of inertia
around the z axis;
vu,
= ship speed components in x,
y coordinates;
00
,
CC
vu
= current velocity components
in x, y coordinates;
r
= rate of turn;
ψ
= yaw angle;
v,u
= time differentiation for
vu,
;
,
HHH
,N,YX
PPP
,N,YX
,
RRR
,N,YX
,
A AA
,N, YX
= longitudinal force
in the x direction, lateral force in the y direction, and
yaw moment in the z axis acting on the hull, propel-
ler, rudder and wind, respectively;
AW
R
= added re-
sistance by waves; and
vr
X
= hydrodynamic deriva-
tive.
4.2 Hull force
The hull forces and moment acting on the ship were
expressed by the polynomial of motion variables u,
379
v, and r in the abovementioned basic equations for
maneuvering motion based on the model test results
as follows:
′
′
′
+−
HH
HH
HH
NdUL = N
YLdU = Y
XLdUR = X
22
2
1
2
2
1
2
1
2
1
ρ
ρ
ρ
(2)
′′
+
′′′
+
′′′
+
′′
+
′′
+
′′
=
′
′′
+
′′′
+
′′′
+
′′
+
′′
+
′′
=
′
′′
+
′′′
+
′′
+
′′
=
′
3223
3223
422
1
rNrvNrvNvNrNvNN
rYrvYrvYvYrYvYY
vXrvXrXvXX
rrrvrrvvrvvvrvH
rrrvrrvvrvvvrvH
vvvvvrrrvvH
(3)
where
R
is ship resistance;
HHH
NYX ',','
1
are non-
dimensional forces and moments acting on a wetted
hull due to swaying and yawing motions in the the x,
y, and yawing directions, respectively; and
rrrrrvv
NXX ',....,','
are the hydrodynamic derivatives.
4.3 Propeller force and fuel consumption
In this study, the lateral force and moment due to the
propeller were neglected because they were negligi-
ble under straight-going conditions in the ocean. The
longitudinal propeller force is expressed as follows:
0,0,)1( ==−=
PPP
NYTtX
(4)
where
t
is the propeller thrust reduction factor and
T
is propeller thrust. Moreover, the propeller thrust
is expressed as follows:
=
++=
=
P
P
T
tP
nD
u
J
JcJccK
KDnT
2
210
42
ρ
(5)
where
n
is the number of propeller revolutions;
T
K
is the propeller thrust coefficient;
J
is the advance
coefficient;
210
,, ccc
are the propeller characteristics
coefficients; and
P
u
is the propeller inflow velocity.
The propeller torque
Q
is expressed as follows:
++=
=
2
210
52
JdJddK
KDnQ
Q
QP
ρ
(6)
where
Q
K
is the propeller torque coefficient; and
210
,, ddd
are the propeller open characteristics coeffi-
cients.
Then, the main engine power is calculated as fol-
lows:
=
=
nQDHP
t
DHPBHP
π
η
2
/
(7)
where
BHP
represents the brake horsepower of the
main engine,
DHP
is the delivered horsepower, and
t
η
is the transmission efficiency.
Finally, the fuel oil consumption per unit time
FOC∆
is calculated by the equation below:
FOCRBHPFOC ×=∆
(8)
where
FOCR
is the fuel oil consumption rate.
Then, the total fuel oil consumption during the
voyage
FOC
is expressed as follows:
dtFOCFOC
∫
∆=
(9)
where
FOCR
is the fuel oil consumption rate.
4.4 Rudder force
The rudder forces and moment are expressed as fol-
lows:
+−=
+−=
−−=
δ
δ
δ
cos)(
cos)1(
sin)1(
NHHRR
NHR
NRR
FxaxN
FaY
FtX
(10)
where
RRR
N,YX ,
are nondimensional forces and
moments on the rudder in the x, y, and yaw direc-
tions, respectively;
N
F
= rudder normal force;
δ
=
rudder angle;
HHR
xat ,,
= interaction coefficients be-
tween the hull and rudder; and
R
x
= coordinate of
rudder position. The abovementioned rudder normal
force is expressed as follows:
)sin(
2
1
2
eRRN
fUAF
δρ
α
=
(11)
where
R
A
is the rudder area;
R
U
is the rudder inflow
velocity;
α
f
is the rudder normal force coefficient;
and
e
δ
is the effective rudder angle. For the value of
α
f
, the following empirical formula is used:
λ
λ
α
+
=
25.2
13.6
f
(12)
where
λ
is the aspect ratio of the rudder as ex-
pressed by
hbf /=
α
.
On the other hand, the rudder inflow velocity
R
U
is expressed by
22
RRR
vuU +=
(13)
where
R
u
and
R
v
are the velocity components in the
x and y directions, respectively. Here,
R
u
is ex-
pressed by the following equation as a function re-
lated to propeller thrust:
2
81
J
K
uu
T
PR
π
κ
ε
+=
(14)
where
ε
and
κ
are empirical or experimental coef-
ficients for the propeller flow acceleration;
P
u
is the
propeller inflow velocity, and is expressed as
uwu
P
)1( −=
by the use of the wake fraction coeffi-
380
cient
w
; and
T
K
and
J
are the thrust coefficient
and advance constant explained above, respectively.
Moreover,
R
v
is expressed as follows:
( )
rLlvv
PPRR
'+−=
γ
(15)
where
γ
and
R
l'
are empirical factors,
v
is the lat-
eral velocity component, and
r
is the rate of turn.
The effective rudder angle
e
δ
is expressed as fol-
lows:
RRe
uv−=
δδ
(16)
where
R
v
and
R
u
is the longitudinal and lateral rud-
der inflow velocity components.
4.5 Rudder control
An automatic rudder control algorithm was intro-
duced to perform ship maneuvering simulations dur-
ing the voyage for the ship passing through desig-
nated points and courses as follows:
rccyc
210
~~~
−∆−∆−=
ψδ
(17)
where
210
~
,
~
,
~
ccc
are control coefficients;
y∆
is the de-
viation from the designated route,
ψ
∆
is the devia-
tion from the designated heading angle, and
r
is the
rate of turn.
4.6 Wind force and moment
The forces and moment by wind are expressed as
follows:
)(θCAVρ =N
)(θCAVρ =Y
)(θCAVρ =X
ANATAAA
AYALAAA
AXATAAA
2
2
2
2
1
2
1
2
1
(18)
where
AAA
N,YX ,
are nondimensional forces and rep-
resent the moment due to wind in the x, y, and yaw
directions, respectively;
H
ρ
is the density of air;
A
V
is the relative wind velocity;
T
A
is the transverse
projected area;
L
A
is the lateral projected area;
A
θ
is
the relative wind direction; and
NAYAXA
CCC ,,
are the
wind force and moment coefficients in the x, y, and
yaw directions, respectively.
Wind force characteristics such as
NAYAXA
CCC ,,
were calculated by using Fujiwara’s (Fujiwara 2001)
method, as shown in the following figure.
Figure 4. Wind force and moment characteristics obtained by
Fujiwara’s method.
4.7 Added resistance due to waves
The added resistance due to waves was estimated by
following simple reliable equations (Sasaki 2008)
because the significant wave height in most of the
navigated sea area was expected to be less than 3 m:
( )
( )
( )( )
−+
+
2
2
5.0
228.0
21
121
1
1
2
1
pfpf
fcp
S
nB
fcpWnBAW
CC
=B
gB
V
=F
BBHFCρg =CR
(19)
-1.5
-1
-0.5
0
0.5
1
1.5
0 30 60 90 120 150 180
Ψ[deg]
C
XA
[-]
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 30 60 90 120 150 180
Ψ[deg]
CYA[-]
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0 30 60 90 120 150 180
Ψ[deg]
CNA[-]
381
where
AW
R
= added resistance due to waves;
ρ
=
density of seawater;
g
= gravity acceleration;
nB
F
=
Froude number by breadth of the ship;
W
H
= signif-
icant wave height;
B
= breadth of the ship;
fcp
B
=
bluntness coefficient;
S
V
= ship speed;
pf
C
= fore
part of the prismatic coefficient. Moreover, the
abovementioned coefficients
21
,CC
are expressed as
follows:
46.0
1
=C
(20)
0.2
2
=C
, when
3.0> B
fcp
(21)
( )
fxp
B =C ++ 3.0600.2
2
, when
3.0≤ B
fcp
(22)
5 OPTIMAL METHOD
Although there are several methods such as the gra-
dient method, which uses first-order differentiation
of a function, or the Newton method, which uses se-
cond-order differentiation, they require procedures
for differentiation. However, the gradient could not
be obtained analytically in some complex subjects
such as route optimization in this study. Therefore,
Powell’s method, which does not use gradients, was
used in this study. Moreover, the following cost
function was used in this study:
FOCJ =
(23)
The value
FOC
is the integrated fuel consump-
tion during a voyage route as defined by a Bézier
curve, which is an ‘N - 1’th order curve defined by
‘N’ control points.
In the optimal procedure using Powell’s method,
some variables are changed through iterative calcu-
lations until convergence. Therefore, the route
should be expressed by several variables.
Although there are several methods to express a
curve with several variables such as a trigonometric
function and multi-degree polynomials, the Bézier
curve was chosen because it can be used to create a
smooth curve suitable for navigation route expres-
sion. A flowchart of this optimal calculation is
shown in Figure 6. Calculations were repeated au-
tomatically until the results were confirmed to con-
verge.
Figure 5. Bézier curve treatment for the expression of an
oceangoing route. (temporary figure)
Figure 6. Flowchart of optimal calculation
6 RESULTS OF SIMULATION
The optimized westbound and eastbound routes for a
voyage on 9
December 2008 using this method and
the initial condition of the great circle route, i.e.,
minimum distance, for optimal calculation are
shown in Figure 7.
Initial route (great circle
)
Seeking new object route
Ship maneuvering simulation
Calculation of cost
fi
Divergence judg-
Optimal route
yes
no
382
Figure 7. Optimized transportation routes between Yokohama
San Francisco.
Moreover, the fuel consumption, voyage distance,
and voyage time of the initial optimized routes are
shown in Figures 8–10.
Although the voyage distance of the optimized
route was sometimes over 200 hundred miles larger
than the original great circle route, as shown in Fig-
ure 9, the fuel consumption for the optimized route
was 10–50 tons less than that of the great circle
route. On the other hand, the travelling times of the
original and optimized routes were almost the same,
as shown in Figure 10. Thus, the optimal route was
concluded to be more economical. Moreover, by
comparing the westbound and eastbound legs of the
great circle route, the fuel consumptions of the two
were found to be slightly different. This suggests
that fuel consumption is affected by weather condi-
tions such as wind and waves.
Eastbound Eastbound Westbound Westbound
Great Circle Optimized Great Circle Optimized
Figure 8. Comparison of fuel consumption between great circle
and optimized routes.
Eastbound Eastbound Westbound Westbound
Great Circle Optimized Great Circle Optimized
Figure 9. Comparison of voyage distance between great circle
and optimized routes.
Eastbound Eastbound Westbound Westbound
Great Circle Optimized Great Circle Optimized
Figure 10. Comparison of voyage time between great circle and
optimized routes. (to be changed, temporary figure)
7 CONCLUSIONS
In this study, advanced weather routing using a op-
timized method to minimize fuel consumption was
demonstrated. The target ship of this study was a
large container ship that traverses the Pacific Ocean
between Yokohama and San Francisco. The route
was expressed in term of a Bézier curve, and Pow-
ell’s method (Fletcher and Powell 1963) was intro-
duced to optimize the route. The results of this study
can be summarized as follows:
1 A new weather routing method is proposed that
uses an optimized Powell’s method. This optimal
1700
1900
2100
2300
2500
2700
2900
FOC(t)
4300
4350
4400
4450
4500
4550
4600
4650
4700
4750
distance(mile)
200
205
210
215
220
225
230
航海時間
time(hour)
Travel
383
method can be applied to seeking an optimal nav-
igating route.
2 Moreover, the route was expressed by a Bézier
curve. The method was shown to be widely appli-
cable to expressing a voyage route.
3 Through computer simulations, a significant re-
duction in fuel consumption was obtained by
tracking the optimized route, even though the dis-
tance was longer than that for the great circle
route.
4 This method is widely applicable to seeking op-
timal navigation routes in other areas.
REFERENCES
Fletcher, R. and M.J.D. Powell 1963. “A rapidly convergent
descent method for minimization,” Computer Journal, Vol.
6, pp 163-168.
Fujiwara, T., Ueno, M. and Nimura, T. 2001. An estimation
method of wind forces and moments acting on ships, Mini
Symposium on Prediction of Ship Manoeuvring Perfor-
mance, 18 October, 2001. pp. 83-92.
Sasaki, N., Motsubara, T. and Yoshida, T. 2008. Analysis of
speed drop of large container ships operating in sea way,
Conference Proceedings, The Japan Society of Naval Ar-
chitects and Ocean Engineering, May 2008 Volume6, pp9-
12.
