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1 INTRODUCTION

In the maritime industry, as the increasing regulatory

pressure and the economic competition, ship energy

efficiency becomes one of the most controversial

topics in every shipping company’s agenda, in

addition to the traditional safety issues. To ensure

better economic benefit as well as reduce negative

environmental impact from shipping emissions, a

voyage optimization system is recognized as an

efficient measure by the shipping industry (DNV

2015). A voyage optimization system is used to plan a

ship’s courses/schedules in order to reach her

destination with certain pre-defined objectives, e.g.,

the expected time of arrival with minimum fuel

consumption and air emissions, etc. (Bowditch, 2002).

The core element of such a voyage optimization

system is to implement a proper optimization

algorithm, which generates optimal ship routing

based on the sea weather conditions along the ship’s

sailing area, the ship’s characteristics and operational

capabilities, as well as some constraints for a

particular voyage, etc. (Wang et al. 2018). Many route

optimization algorithms have been implemented and

used in the shipping market and maritime research

community, e.g., the modified isochrone method

(Hagiwara, 1989), the isopone method (Klompstra,

1992), dynamic programming method (De Wit, 1990),

DIRECT algorithm (Simonsen et a l. 2014), etc.

Efficiency of a Voluntary Speed Reduction Algorithm

for a Ship’s Great Circle Sailing

Wang, W. Mao & L. Eriksson

Chalmers University of Technology, Gothenburg, Sweden

ABSTRACT: The great-circle is the shortest distance between two points on the surface of the earth. When

planning a ship’s sailing route (waypoints and forward speeds) for a specific voyage, the great circle route is

commonly considered as a reference route, especially for ocean-crossing seaborne transport. During the

planning process, the upcoming sea weather condition is one of the most important factors affecting the ship’s

route optimization/planning results. To avoid encountering harsh conditions, conventional routing

optimization algorithms, such as Isochrone method and Dynamic Programming method, have been

developed/implemented to schedule a ship’s optimal routes by selecting waypoints around the great circle

reference route based on the ship’s operational performances at sea. Due to large uncertainties in sea weather

forecast that used as inputs of these optimization algorithms, the optimized routes may have worse

performances than the traditional great circle sailing. In addition, some shipping companies are still sailing in

or making charting contracts based on the great circle routes. Therefore, in this study, a new optimization

algorithm is proposed to consider the voluntary speed reduction with optimal speed configuration along the

great circle course. The efficiency of this method is investigated by comparing these two methods for optimal

route planning with respect to ETA and minimum fuel consumption. A container ship sailing in the North

Atlantic with full-scale performance measurements are employed as the case study vessels for the comparison.

http://www.transnav.eu

the

International Journal

on Marine Navigation

and Safety of Sea Transportation

Volume 14

Numbe

r 2

June 2020

DOI:

10.12716/1001.14.02.04

302

Most of the optimization algorithms are two-

dimensional and optimization process is performed

by changing a ship’s course based on a reference

route, e.g., the great circle course. The variables that

are optimized are the ship’s sailing waypoints/course

(in a more specific way, ship heading). The optimal

routes obtained from these algorithms can differ

significantly from the great circle courses to avoid

harsh storms, implying the sacrifice of the shortest

travelling distance. Nevertheless, to avoid

encountering harsh storms can be also achieved by

proper management of the ship speed, but still

keeping the shortest sailing distance along the great

circle course.

In the voyage optimization process, sea weather

forecast plays an extremely important role as it is the

main dynamic input for the optimization algorithm.

The reliability of the optimized route highly depends

on the accuracy of the weather forecast. However, the

accuracy of weather forecast often starts to be

questioned after 3-5 days, while even large

discrepancy happens after 5-7 days. An example of

the comparison of significant wave height between

forecast data and hindcast data is presented in Fig. 1.

The observation point is selected in the North Atlantic

Ocean. The figure indicates that the accuracy of

forecast data starts to decrease after 5 days. As the

consequence, even the route given by the

optimization algorithms is optimal based on forecast

data, the actual voyage may end up with

encountering very harsh weather conditions.

Figure 1. Comparison of significant wave heights obtained

from forecast data and hindcast data

In this paper, a new voyage optimization method,

which considers voluntary speed reduction in the

great circle course is proposed. Then, a

comprehensive comparison between the proposed

method and conventional two-dimensional Dijkstra’s

algorithm is conducted to investigate the reliability

and efficiency of the proposed optimization method.

The investigation also includes the influence of the

uncertainty of weather forecast on the optimization

results.

2 DESCRIPTION OF THE PROPOSED METHOD

The great circle course is divided by N waypoints and

each waypoint is associated with a passing time. A

ship’s state variable represents the location and the

associated passing time of the i-th waypoint as,

[ ]

i i ii

xyt=s

(1)

where

i ii

xyt

represent longitude, latitude and

time of the i-th waypoint. All the waypoints along the

route is described by

[ ]

, , ,

= …

S ss s

Normally, for a conventional voyage optimization

algorithm, a ship’s speed and heading at each

waypoint are selected as control variables that needs

to be optimized during the voyage,

( ) ( ) ( ) ( )

( ) ( ) ( )

11 2 2

, , ,

ii ii ii



= …







US us u s u s

us s s

(2)

where

and

are vectors of ship speed and

heading of all waypoints such as

[ ]

,,,

vv v= … v

In this study, we fixed the heading to the great

circle course while keep speed varying. The weather

conditions encountered during a voyage depends on

the state variable vector

, denoted by

( )

. The

sea conditions contain information of wave height,

wave direction, wave period, wind and current, etc.

In addition, a ship’s sailing is often limited under

some operational constraints. The constraints function

is a Boolean function that returns if the ship is feasible

to sail under a state S and a control variable U of a

waypoint. Here, the constraints include the land-

crossing avoid, maximum speed/engine power

constraints.

The objective of the voyage optimization is to

minimize the fuel consumption during the voyage.

The total fuel consumption

is estimated by,

( ) ( )

( )

C f dt=

∫

US WS

(3)

where

( ) (

)

( )

,f US WS

is the instantaneous cost

function (fuel cost between two waypoints) for a

series of ship state

under control

( )

;

are the departure time and arrival time, respectively.

303

Figure 2. Scheme of the proposed method

The proposed method is to optimize the speed

profile, i.e., the speed vector

, along great circle

waypoints

. The workflow of the method is shown

as Fig.2.

2.1 Speed/passing time initiation along the route

The first step is to generate waypoints on the great

circle course between the departure and the

destination. The waypoints contain the information of

longitude and latitude. Then, it will add a time set to

each waypoint. A time set is an array, which contains

a series of feasible arrival times to a waypoint. Figure

3 gives a simple example on how the time set is added

on a waypoint. The precedure is done as follows:

1 Generate a set of discrete time with even space for

each waypoint. The space can be chosen from

several hours to days.

2 Add constraints to the time set. Here we consider

both the sailing speed and ETA as constraints.

3 Select feasible times from each waypoint shown as

round dots in the figure.

Figure 3. Illustration of a “time set” along the route

2.2 Cost function for edge generation

In this study, a ship energy performance model is

used as the cost function through the optimization

and the ship’s speed is considered as the input for the

optimization.

Figure 4. Ship energy performance models

For the optimization, the ship’s engine power

needed to push the ship forward at the input speed

should be estimated as the cost based on encountered

sea conditions, the ship’s characteristics, and

operational profiles etc. The workflow for the ship

speed-power performance modelling is presented in

Fig.4.

2.3 Graph generation

A great circle course is geometrically divided into a

number of waypoints and each waypoint contains a

set of feasible arrival time. With this information a

graph

( )

,=G SA

can be formed, where S is a set of

waypoints (also considered as ship state variables)

and A is a set of edges composed of pairs of nodes. A

waypoint

contains a number of ship states,

,1 ,1

i i ii

im i i im

xyt

  

  

= =

  

  

  

(4)

where

is the

th waypoint,

is the number of

feasible arrival time of the

th waypoint.

Edges are also required to be added for proceeding

the optimization. Adding edges is to pair two

nodes/ship states from two adjacent waypoints as the

following equation shows.

1, 1 1 1,

ij i i ij

i jk

ik i i ik

xyt

+ + ++

  

= =

  

  

(5)

where

is the ith waypoint and

,jk

are the jth

and kth of the time corresponding to the ith and i+1th

waypoints. Constraints must be considered to

eliminate the infeasible edges/connections between

nodes/ship states when

,,i jk

is going to be added

to A:

1 if

1,ik

,ij

;

2 the corresponding shaft power between

,ij

and

1,ik+

doesn’t exceed the maximum engine power.

304

For the great circle course based optimization,

each edge

,,i jk

is assigned with a cost/weight

,,ijk

which is computed by the cost function, where

( ) (

)

( )

, , , , 1, ,

ijk ij ij i k ij

f tt

= −C Us Ws

(6)

The generation of the graph

is complete when

all the feasible nodes



are generated and all the

edges

are added to

2.4 Implementation of Dijkstra’s algorithm

Dijkstra’s algorithm works in a graph by visiting

edges starting from the source node to the target

node. In this study, the source node and the target

node refer to the starting ship state and the end ship

states. It is an algorithm for finding the shortest path

between two graph nodes in a graph. In this study,

instead of using Dijkstra’s algorithm to find the

shortest path, it is used to find the path with

minimum cost.

Before implementing Dijkstra’s algorithm, a vector

of total cost from the starting ship state

0,0

to every

node/ship state C should be constructed. This means

(

)

,ij

should represent the cost from

0,0

to any

node/ship state

0,0

in S. Then a dictionary

representing all unvisited nodes within

generated.

The implementation of Dijkstra’s algorithm is

performed as follows. For every

∈

1 Initialize the cost set

( )

,ij

to infinity. An

infinite cost in

for a given node/ship state

means no path has been found from the start node

0,0

2 Add

,ij

, indicating

is unvisited.

3 Set

( )

0,0

C s

to 0.

4 If

is not empty, select node

,ij

with the

smallest

-value from

5 Remove

,ij

from

. For every adjacent node

,ij

( ) ( )

( )

ij ij

C weight C+<s su u

then

( )

(

)

( )

ij ij

C C weight

= +u s su

. Go back

to step 4, until

gets empty.

In this study, it should be noted that the

weight

function above refers to Eq. (6). We have a number of

end ship states at waypoint N as shown in Fig. 3. This

means a number of optimal speed configuration along

the route can be generated for different ETA.

3 CASE STUDY SHIP FOR INVESTIAGTION

In order to demonstrate the advantage and

disadvantage of the proposed great circle based

voyage optimization method, a case study ship

equipped with full-scale measurement devices is

chosen here. In addition, the influence of the

uncertainty of the weather forecast data on the

optimization results is also investigated. Thus, both

the weather forecast data and hindcast data are used

in this study.

Moreover, the interpolation method used in

interpolating sea weather conditions for the ship’s

waypoints can also influence the optimization results.

In this study, trilinear interpolation (Weiser et al.

1988) is used. The influence of different interpolation

methods is not considered here.

3.1 Details of the case study

In this case study, a 2800TEU container ship sailing in

the North Atlantic is taken as a case study ship (Mao

2014). The main particulars of the ship are listed in

Table 1. For this case study ship, the full-scale

measurements of the ship’s performance at sailing in

the North Atlantic during the year 2008 are available

in this study.

In the practical voyage planning, the voyage

optimization process is conducted before the voyage

starts and the weather forecast data is usually used as

the preliminary data source for optimizing the

voyage. In this study, the optimization is conducted

using the weather forecast information that can be

accessed before the voyage. While the hindcast data is

used to estimate the weather and sea conditions

which the ship actually encountered. The forecast

data is obtained from ECMWF Mars operational

archive server (https://apps.ecmwf.int/mars-

catalogue/?class=od) and the hindcast data is

extracted from ECMWF ERA5 hindcast dataset

(https://cds.climate.copernicus.eu). It should be noted

that the forecast data is updated every 12 hours and

the time resolution of the forecast data is 3 hours. As

the ship’s cost variation is mainly caused by

encountered significant wave height (H

s), Hs is chosen

to represent sea conditions.

Table 1. Main particulars of the case study ship

_______________________________________________

Length Loa 235.0 m

Length L

pp 230.4 m

Beam B 32.2 m

Prop. Diam. 6.9 m

Displacement 55566 m

Wetted surface 10396 m

Length waterline 230.4 m

Draft T 10.78 m

_______________________________________________

In this study, the actual voyages of the case study

ship sailing along the North Atlantic during 2008 are

used as the reference for implementing the proposed

method and comparing the method with other

methods. As the storms in North Atlantic always

moves from the east to the west, two westbound

voyages are selected because it is more important for

captains to plan their voyages for westbound voyages

to ensure cargo safety and save fuel, in particular

during the winter season. Furthermore, to

demonstrate the capability of the proposed method,

all optimizations are conducted in the open sea area

in North Atlantic. The analysis of the strategy of slow

steaming is conducted in the last case study.

3.2 Ship routing algorithms for the case study

This investigation compares three algorithms, i.e., 1)

2D Dijkstra’s algorithm with fixed ship speed, 2)

Great circle course with speed optimized (proposed

method), and 3) Great circle course with fixed ship

speed often for chartering contract.

305

The 2D Dijkstra’s algorithm (2DDA) is one of the

conventional voyage optimization algorithms derived

from 2D Dynamic programming method (De Wit,

1990). It uses the Dijkstra’s algorithm instead of the

traditional dynamic programming approach to

conduct the optimization. The grid system is

generated based on the great circle reference course,

which covers the accessible sailing area of the voyage

shown in Fig. 5. The 2DDA is a two-dimensional

optimization method. It is used to find optimal

waypoints for a certain voyage with certain demands

such as minimum fuel consumption, maximum ship

safety.

The proposed method is used to optimize the ship

speed profile during the voyage instead of the

location of the waypoints for a voyage. The third

method is to estimate the speed and passing time

along the great circle course. It should be noted that

constraints such as maximum-engine-rate constraint

are applied to all of these methods in the optimization

process. Thus, there is involuntary speed reduction

occurring when encountering harsh sea conditions.

Figure 5. Example of the grid system for the 2DDA

4 RESULTS OF INVESTIGATING THE PROPOSED

VOYAGE OPTIMIZATION

An actual voyage during Feb. 2008 is selected to

demonstrate the capability of the proposed method.

Two operation scenarios are investigated. One is for

the ship sailing with her service speed using 85%

MCR of the ship engine. Considering the slow

steaming strategy is quite well adopted in today’s

shipping market, the other scenario is to investigate

the capability of the proposed method for the ship to

sail with a 25% speed reduction as her service speed.

4.1 A westbound voyage sailing in service speed

The trajectory of the optimized routes by the 2DDA

method is shown in Fig.6, as well as the great circle

course. The trajectory got from the 2DDA method

implies that the encountered sea weather conditions

were more severe in high latitudes. Therefore, the

2DDA method could try to find alternative waypoints

(rather than the shortest distance) to avoid these harsh

sea conditions.

Figure 6. Route trajectories by 2DDA and the shortest

distance (the great circle course).

Furthermore, the ETA, fuel consumption and

sailing distance along the routes optimized by the

three methods based on forecast sea weather

conditions are listed in Table 1. It also gives the ship’s

overall operation performance from the post-voyage

analysis. The post voyage analysis is performed by

extracting the “true” weather information from

hindcast data for the same waypoints (i.e., location

and passing time) as the optimized routes. Then the

ship’s new fuel consumption and ETA are re-

estimated based on the hindcast weather information.

If the ship state is invalid in the actual sea condition

(for example, the shaft power estimated by the actual

sea condition exceed the maximum power rate), the

current ship state as well as the following ship state

will be recalculated.

For the voyage optimization based on the weahter

forecast data, Table 1 shows that the proposed

method can save approximate 1% of fuel

consumption in comparison with the conventional

fixed speed great circle sailing method (as reference

here). The optimized route from the proposed method

can also keep the same ETA as the reference method.

While, the ship routing generated from the 2DDA

method gives the largest fuel saving, because this

method can choose routes shifting away from the

shortest distance to encounter calm sea environments.

Table 2. Optimization results for service speed sailing based on forecast and hindcast weather data

__________________________________________________________________________________________________

Method Fuel cost [ton] ETA [hour] Distance [km]

Forecast Hindcast Forecast Hindcast

__________________________________________________________________________________________________

(Method 1) 2DDA 228.0 234.8 100.4 100.8 3324

(Method 2) GC optimal speed 242.4 244.3 102.6 103.3 3121

(Method 3) GC fixed speed 244.4 247.9 102.7 103.3 3121

__________________________________________________________________________________________________

Table 3. Optimization results for slow steaming sailing based on forecast and hindcast weather data

__________________________________________________________________________________________________

Method Fuel cost [ton] ETA [hour] Distance [km]

Forecast Hindcast Forecast Hindcast

__________________________________________________________________________________________________

(Method 1) 2DDA 170.3 160.0 122.8 122.8 3380

(Method 2) GC optimal speed 195.4 188.4 117.6 117.6 3121

(Method 3) GC fixed speed 204.7 190.9 117.7 117.9 3121

__________________________________________________________________________________________________

306

Furthermore, Fig.7 presents the comparison of

significant wave height (H

s) between obtained from

weather forecast in the optimization and extracted

from hindcast data after the voyage, as well as the

comparison between optimized ship speeds (based on

weather forecast) and actual ship speeds (estimated

from hindcast data for post voyage analysis). It shows

that for such a short passage, the weather forecast

data is very reliable at the beginning of the voyage.

However, after approximately 2-3 days sailing, the

forecast data begins to deviate from hindcast data

(but not very much). The post voyage analysis of

speeds in the right column shows the same trend, i.e.,

ship speeds optimized from various method begin to

differ in the late stage of the voyage.

Figure 7. Comparison of Hs and sailing speeds from weather

forecast for route optimization and hindcast data for post

voyage analysis.

In comparison with the post voyage analysis and

the voyage optimization based on forecast data, there

is very little difference of fuel cost and ETA for the

two great circle based routes, the different between

forecast and hindcast based. This is because 1) the

total sailing time is about 4 days and the weather

forecast is quite reliable, 2) there is not so big margin

to change too much of the ship’s speed in the great

circle course to avoid harsh storm and keep ETA at

the same time. If there are too large and long

voluntary speed reduction, the ship will not be able to

increase her speed too much than the service speed to

catch up with the same ETA.

However, on the other hand, even though the

weather forecast is quite reliable for the 4 days’ route,

there are still large variation (3% in fuel cost) between

the forecast and hindcast based optimization by the

2DDA method. It means that the 2DDA method is the

most affected by the uncertainty of the weather

forecast. For even long sailing voyages, this weather

uncertainty may lead to that the 2DDA method will

generate worse route than the simple shortest route.

4.2 A westbound voyage with slow steaming

The slow steaming strategy for the same voyage is

used here to study the capability of the proposed

method. The trajectory of the generated courses by

different method is presented in Fig.8. It shows that

the 2DDA method for the slow steaming have even

large freedom to choose the waypoints different

significantly from the great circle course.

The corresponding ship performance (fuel cost,

ETA and distance) planned by these methods are

listed in Table 3. For the slow steaming, the fuel

consumption for all routes is reduced significantly,

but the reduction is simple proportional (not

exponential) to the speed reduction. The consequence

is obvious that it takes more sailing time. It gives

large margin for the proposed method to optimize the

ship speed configuration along the great circle course.

The result shows that it increases the fuel saving up to

around 5% instead of 1% in the service speed sailing.

For the comparison of fuel consumption between

optimized routes by the forecast data and post voyage

analysis by the hindcast data, the difference can be

more than 7% due to long sailing time, which

increased weather uncertainties. Furthermore, the

proposed method performs much better than the

simple fixed speed great circle route.

Figure 8. Trajectories generated by different methods

4.3 Cons and pros of the proposed method

For the proposed method to consider the voluntary

speed reduction to avoid harsh environment along the

great circle course, the fuel consumption of the

generated route is a bit higher than the 2DDA

method. One possible reason might be that the storms

often appear at least for 12 hours, but the ship speed

cannot be voluntarily reduced too much and last for

too long time in order to keep the ETA.

Figure 9 shows the waypoint number and its

corresponding weight graph with the square of

significant wave height. The weight graph is

generated based on the hindcast data. The solid line is

the optimized result given by the proposed method.

The dash lines can be considered as the boundary

which assume that the ship sails with certain speed. It

can be implied that without delaying the ETA, it is

hard to avoid the bad weather conditions. It is also

shown that the ship can slow down at the beginning

and catch up later to avoid the heavy shadow area

shown like the arrow line. However, to catch up the

ETA, the ship has to accelerate in the shadow area

which would cost more fuel that the optimized one.

Thus, it is possible to avoid the storm by adjusting the

ship’s speed. However, when considering the item of

westboun

307

fuel saving, the speed acceleration to a very high level

may lead to much higher fuel cost, in particular,

sometimes, the maximum speed is limited by the

ship’s engine size.

Figure 9. Waypoints and the weight graph

In addition to the voluntary speed reduction, the

proposed method can deliver a Pareto front for end-

users to select the best combination of ETA and fuel

consumption to meet their interests. Figure 10 shows

the Pareto front obtained from the proposed method

based on the weather forecast data. The dots are a

small part of the results extracted from the proposed

method. The feasible ETA can be selected from

approximately 80h to 140h with 0.3h interval.

Figure 10. Pareto front for the Great Circle route given by

the proposed method

5 CONCLUSION

Large uncertainty of weather forecast will affect a

ship’s actual performance for sailing along the routes

from voyage optimizations. In this study, the fixed

great circle course considering voluntary speed

reduction is compared with the conventional fixed

speed shortest route planning and fixed speed

varying course planned by the 2-dimensional Dijkstra

algorithm (2DDA). Therefore, a new method is

proposed for optimizing a ship’s voluntary speed

reduction along the great circle course. Through the

comparison between the three methods based on an

actual voyage collected by a containership, the

proposed method shows better optimization results

with respect to fuel saving than the simple fixed

speed shortest route. Even higher energy saving can

be expected if the slow steaming strategy is used. For

the current case study, the 2DDA method can

generate ship routes with lowest fuel consumption.

However, the performance of the 2DDA method

can be easily affected by the uncertainty from the

weather forecast. For the case study voyage lasting for

about 4 days, the forecast data can be quite reliable.

But still the variation of weather uncertainty can lead

to more than 7% difference of fuel saving between

optimization based on weather forecast and the post

voyage analysis from hindcast data. It can be expected

for even long-time-range voyages such as across

Pacific Ocean, optimization methods such as 2DDA

may give the bad optimization results in comparison

with the proposed method, due to the deficiency of

weather prediction by forecast. Furthermore, the

proposed method can also generate Pareto Fronts for

end-users to select the best combination with the

respect of ETA and fuel consumption. It also has more

potential for fuel saving in slow steaming scenario

and long-distance sailing as it has more space to

voluntary speed reduction during the voyage.

ACKNOWLEDGEMENTS

The research is supported by the EU Horizon2020 project

EONav project (GA no. 687537). We are also grateful to the

DNV, the crews, and ship owners for providing

measurement data. The authors also would like to thank the

financial support from STINT (CH2016-6673), and National

Science Foundation of China (NSFC-51779202).

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