The latest development of the ship-routing model published in

The new version of the model targets large ocean-going vessels by considering both ocean surface gravity waves and currents. To effectively analyse currents in a graph-search method, new equations are derived and validated against an analytical benchmark.

A case study in the Atlantic Ocean is presented, focussing on a route from the Chesapeake Bay to the Mediterranean Sea and vice versa. Ocean analysis fields from data-assimilative models (for both ocean state and hydrodynamics) are used. The impact of waves and currents on transatlantic crossings is assessed through mapping of the spatial variability in the tracks, an analysis of their kinematics, and their impact on the Energy Efficiency Operational Indicator (EEOI) of the International Maritime Organization. Sailing with or against the main ocean current is distinguished. The seasonal dependence of the EEOI savings is evaluated, and greater savings with a higher intra-monthly variability during winter crossings are indicated in the case study. The total monthly mean savings are between 2 % and 12 %, while the contribution of ocean currents is between 1 % and 4 %.

Several other ocean routes are also considered, providing a pan-Atlantic scenario assessment of the potential gains in energy efficiency from optimal tracks, linking them to regional meteo-oceanographic features.

The strongest water flows are generally observed in western ocean boundary currents, in tidal currents, in the circulation of straits and fjords, in inland waterways, and in the vicinity of river runoffs

The impact of ocean currents on navigation can be examined from several perspectives.

One approach can be based on ship drift (SD) and dead reckoning.
Dead reckoning refers to the computation of a vessel's position by means of establishing its previously known position and advancing it, based on its estimated speed and course over elapsed time. In the study of

In the contexts of robust control and dynamic positioning, currents and other environmental fields, such as gravity waves and winds, are regarded as a disturbance that needs to be compensated for such an objective to be achieved, such as keeping the vessel's position and heading. To achieve this task, numerical schemes typically assume that such disturbance is constant in time

Path following, a specific problem of motion control involving steering a marine vessel or a fleet of vessels along a desired spatial path, can account for the presence of unknown, constant ocean currents in addition to parametric model uncertainty

The impact of ocean currents significantly affects slow-speed vehicles, such as autonomous underwater vehicles (AUVs) or underwater gliders.

A reconstruction of the Kuroshio current by means of drifter data is used by

Currents may also be exploited for optimising navigation between given endpoints with respect to various strategic objectives (e.g. track duration, fuel oil consumption, or

An exact method based on the level-set equation was developed by

Other mathematical techniques are reviewed in the introduction of

In the latest edition of the World Meteorological Organization's

The International Maritime Organization (IMO) recommends avoiding “rough seas and head currents”; this is among the 10 measures within the Ship Energy Efficiency Management Plan, or SEEMP

The above review of the literature shows that the question of the impact of sea or ocean currents on navigation, despite its classical appearance, is still open. The results are difficult to compare because (i) they are not validated against exact solutions, (ii) with some exceptions, they do not declare the computational performance, (iii) generally, their model source codes are not openly accessible, (iv) they are limited to case study analyses on a specific date, without any assessment of seasonal and geographical variability in their quantitative conclusions, and (v) they generally cannot account for both surface gravity waves and ocean currents.

All these considerations have motivated the development of the discoVerIng Safe and effIcient Routes (VISIR) ship-routing model presented in this paper, which is organised into three main sections.

The theoretical framework for inclusion of currents into the model is presented in Sect.

Throughout this paper “track” indicates a set of waypoints joining two given endpoints or harbours, in relation to departure on a given date, and the “route” or “crossing” indicates when there is no reference to a specific departure date; “wave” is a short form for surface gravity wave. The shortcut “w” is for computations accounting for only waves, and “cw” is for both ocean currents and waves.

This section comprises all theoretical and numerical advancements of VISIR-1.b with respect to the previously published version (VISIR-1.a).

The basic hypotheses are described in Sect.

All model features that are not explicitly mentioned in this paper are unchanged from the previous version. A summary of the main changes to the VISIR-1.a code is provided in Table

Some nautical abbreviations used in this paper.

VISIR optimisation corresponds to the top layer in a hierarchical ship motion control system. It determines long-term routing policies that affect the motion of the vessel, viewed as a particle. The related kinematics occur over a long period of time with respect to the timescale of the lower control layer, corresponding to the motion control level, and determine the behaviour of the vessel as a rigid body under the influence of external forces and moments (see Appendix

In terms of the nomenclature used, “vehicle” is here used as a more general term than vessel for the theoretical results that do not refer to any specific ship feature. The term “flow velocity” is used for referring to the velocity resulting from either ocean surface current, tidal current, and non-linear mass transport in surface gravity waves (Stoke's shift) or their composition. Also, when not otherwise specified, the qualification “over ground” is assumed for both speeds and courses.

Assuming that a linear superposition principle holds for vehicle and horizontal flow velocity, the vector speed over ground (SOG) of the vehicle is given by

Equation (

However, we note that the superposition principle in the form of Eq. (

Finally, the aerodynamic drag on vessel superstructure is also neglected in Eq. (

Along the vessel path, course over ground (COG) may need to be constrained for navigational reasons (traffic constraints, fairways, shallow waters, or any other reason for preferring a specific passage), and in the computation of an optimal path, the algorithm (such as a graph-search method) may resort to spatial and directional discretisation, which again is a form of course assignment.

Making reference to Fig.

To keep the course constrained as per Eq. (

VISIR-1.b directional conventions on top of the compass protractor. Shown is the vessel speed through water (

After defining the vector components of the water flow,

Linear superposition Eq. (

Inserting

An equation formally identical to Eq. (

Furthermore, both Eqs. (

Finally, by taking the module of both sides of Eq. (

In this section we report the procedure for ensuring that the graph used by VISIR is safe for navigational purposes.
A note on use of non-regular meshes can be found in Appendix

Due to the non-convexity of the shoreline and the presence of islands, the maritime space domain is not simply connected, and thus not all graph edges correspond to navigable courses. To account for this, the following graph pruning methodology is used. It starts from the observation that in a large ocean domain, most of the edges do not intersect the coastline. Thus, the procedure consists of the following three steps:

Retrieve the indices of edges within a small bounding box around each coastline segment.

Check edges within the bounding box for intersection with the coastline.

Create all edges in the selected domain, pruning just those from the previous step and intersecting the coastline.

Thus, when creating the graph, only the sea and land arcs that do not intersect the shoreline are included in the graph. When the code for track computation is then run, for each of the requested track endpoints (i.e. start and end location of the route), the nearest node on the graph is determined. This can even be a land rather then a sea node. In the subsequent step, the graph arcs are screened for the condition that the under keel clearance (UKC) is UKC

In VISIR-1.a graph nodes were linked only to all other nodes that can be reached via either one or two hops. In this work, a larger number of hops

We refer here to a regular latitude–longitude mesh with

The computational cost of VISIR-1.b graph generation procedure is linear in the total number of edges (from step one of the procedure above) within all the bounding boxes around the shoreline. For a given number of nodes, the computation time for preparing a graph of order

As in VISIR-1.a, edge weights are also computed out of Eq. (

The shortest path algorithm is still derived from that of Dijkstra, which is a deterministic and exact method

In

The performance could be improved to

The VISIR-1.b vessel propulsion and seakeeping model is the same as in VISIR-1.a, but with a minor update. It is reviewed and updated in Sect.

STW together with the ocean current velocity determines SOG (Eq.

That model considered the balance of thrust and resistance at the propeller, neglecting the propeller torque equation

In line with IMO guidance

At present, VISIR includes checks of intact stability related to parametric roll, pure loss of stability, and surf-riding and/or broaching-to at an intermediate level between

All vessel speeds at any location and direction (i.e. on each of the

Verification of VISIR-1.b vs. benchmark solutions. Both least-distance (blue) and least-time (red) trajectories are displayed, and the tracks originate at the black star symbols.

Summary parameters of benchmark case studies (see Fig.

Therefore in terms of vessel stability, the sole update in VISIR-1.b is in the actual values of the vessel parameters and the parametric roll stability check.
The new vessel parameters are suited for modelling a container ship and are listed in Table

In this subsection the impact of track optimisation on voyage energy efficiency is estimated.

Following the Paris Agreement

The third IMO GHG study estimated the share of emissions from international shipping in 2012 to be some 2.2 % of the total anthropogenic

In line with the United Nations Sustainable Development Goal 13 (

The IMO previously introduced the Energy Efficiency Operational Indicator (EEOI) as the ratio of

If a track is plied at a constant

Depending on the subscripts

If vessel stability checks (Sect.

Since currents can be either advantageous or detrimental to SOG (Eq.

VISIR-1.b path kinematics described in Sect.

For the verification, VISIR-1.b includes a verification option to run synthetic fields as the input,
instead of those from data-assimilative geophysical models (as described in Sect.

The remainder of the processing (generation of the graph, evaluation of the edge weights, and computation of the shortest path) is identical for both synthetic and modelled environmental fields. However, as identified in Sect.

The least-time route in the presence of waves is computed using VISIR by assuming that waves affect the speed through water of the vessel (Sect.

Analytical solutions are available for a subclass of these problems, in which STW depends on only one of the spatial coordinates

The cycloidal benchmark was also exploited in

For VISIR-1.b, we compute graphs of higher connectivity (Sect.

The cycloidal solution exploits the fact that a functional of the spatial coordinate is minimised under some necessary conditions provided by the Euler–Lagrange equations

The optimal control formalism provides the framework for computing extremals of a function, not only explicitly depending on spatial coordinates but also on time

Several benchmark trajectories are provided by

Figure

The computational performance (Sect.

Figure

Fit parameters for the data displayed in Fig.

In any two-dimensional regular mesh, the number

Without time interpolation, the optimal path algorithm is about 8 times faster (Fig.

Figure

This is even more apparent in Fig.

To further clarify the memory space requirements of VISIR, we focussed on its shortest path algorithm and collected and analysed additional datasets, as described below. These consist of the following:

time series of RAM allocation of the VISIR MATLAB job

The following shell command is used:

stopwatch timer readings at specific VISIR processing phases

The following MATLAB commands are used:

For each graph's angular resolution (indexed by

In this section, the capacity of VISIR-1.b to deal with both dynamic flows and sea-state fields in realistic settings is demonstrated using the ocean current and wave analysis fields from data-assimilative ocean models.

Section

VISIR-1.b uses both static and dynamic environmental fields obtained from official European and US providers. The static environmental datasets are of the bathymetry and shoreline. The dynamic datasets are of the waves and ocean currents. The specific fields used are described in the following subsections.

The General Bathymetric Chart of the Oceans (GEBCO) 2014 bathymetric database

The Global Self-consistent, Hierarchical, High-resolution Geography Database (GSHHG;

Meteorological fields have not as of yet been used for computing VISIR-1.b tracks. Surface wind fields have only been used in VISIR-1.a for sailboats

Wave analyses are obtained through Copernicus Marine Environment Monitoring Service (CMEMS;

This uses the optimal interpolation of significant wave height from Jason-2 and Jason-3 and SARAL and CryoSat-2 altimeters. The model also takes into account the effect of currents on waves

The spatial resolution is

The wave dataset name is GLOBAL

Ocean currents are obtained through CMEMS from the operational Mercator global ocean analysis and forecast system based on the NEMO v3.1 ocean model

This uses the SAM2 (SEEK Kernel) scheme for assimilating the sea level anomaly, sea surface temperature, mean dynamic topography (CNES-CLS13), and more.
The spatial resolution is

The dataset name is GLOBAL

For the results shown in this section, optimal tracks are computed on a graph with the order of connectivity of

Database of vessel propulsion parameters and principal particulars
used in this work. The values of

Vessel response functions for the parameters given in Table

We first consider a transatlantic crossing in the North Atlantic Ocean, located between Norfolk (USNFK), at the mouth of the Chesapeake Bay (

Geodetic (blue) and optimal tracks (red) for the USNFK–ESALG route in the presence of different environmental forcings and departure dates: panels

First of all, we note that the geodetic (or least-distance) track is bent northwards, as it is to be expected from an arc of GC of the Northern Hemisphere on an equi-rectangular projection. The track is piecewise linear, and its northern edge is flattened due to the finite angular resolution of the graph:

For these tracks, meteo-marine conditions are first introduced (Sect.

The synoptic situation in the North Atlantic during the week following 21 June 2017 (departure date for the eastbound track) was dominated by the Azores High blocking descent of subpolar lows to the middle latitudes. This led to relatively calm ocean conditions (significant wave height

In the week following 16 February 2017 (departure date for the westbound track) a low with storm-force winds that formed near (41

In terms of the currents, we note that the eastern edge of the crossing is N of Cape Hatteras and, thus, N of the GS branch known as the Florida Current

The topological and kinematical features of the optimal tracks of the case study are discussed in this subsection.

Four different solutions for the optimal tracks of the USNFK–ESALG route are given in Fig.

For the eastbound voyage, when only considering waves (w type; Fig.

When the optimal track is computed for the same departure date and direction but also considers ocean currents too (cw type), the solution is significantly modified (Fig.

On the westbound voyage of w type (Fig.

The optimal track for the same departure date and direction but different cw type (Fig.

To gain a deeper insight into the results, in Fig.

Along-route information for both the eastbound

Starting from the eastbound route (Fig.

The geodetic westbound track displays heavy oscillations in SOG, with two deep local minima at

In Fig.

Finally, the angle of attack

Per Eq. (

The stability constraints given in Sect.

In addition, on this specific route and these departure dates, the voluntary speed reduction (Sect.

Furthermore, all time-dependent edge weights along the optimal tracks fulfil the FIFO hypothesis (Sect.

Two simple metrics for summarising the kinematics of a track are proposed here: the optimal track duration

Concerning time gains, it is important to specify whether they refer to the geodetic track (

In this subsection we consider the extent to which the seasonal variability in the ocean state and circulation affects the variability in the optimal track of a given transatlantic crossing.

In order to address it, VISIR-1.b computations are conducted for departure dates spanning the whole calendar year 2017. Departures on six dates (1st, 6th, 11th, 16th, 21st, and 26th) in each month are considered, resulting in 72 dates per year. This is aimed at considering the decorrelation of the ocean current fields after a Lagrangian eddy timescale of about 5 d

To analyse the massive data resulting from these computations, four levels of analysis are considered:
spatial variability in the tracks (Sect.

A direct visualisation of the annual variability in the track topology is shown in Fig.

Route tracks of the same transatlantic crossing of Fig.

Each panel displays a bundle of trajectories relative to the 72 departure dates. The extent of the diversions makes it clear that the case study of Sect.

Some tracks are found to sail quite far inshore towards the Canadian coast, and for this we refer to a related comment in Sect.

The general impact of ocean currents on eastbound tracks is that the bundle of tracks squeezes and shifts S in the vicinity of the GS proper (W of 67

It should be stressed that the computed spatial variability depends heavily on how ship energy loss in waves is parametrised (see Sect.

While the paths of the tracks displayed in Fig.

Evolution lines for the tracks in Fig.

Furthermore, in the presence of currents, the slope can exceed that relative to navigation at SOG equal to the maximum STW. This is due to the speed superposition per Eq. (

Finally, the envelope of the evolution lines along the geodetic tracks is displayed as a grey shaded area. This reveals the kinematical benefit of the optimal tracks, as they can be sailed at an higher SOG (coloured dots are generally left of the grey areas), resulting in shorter voyage durations.

To reduce and better analyse the information contained in Fig.

Distribution of optimal sailing time

Such a plane contains a strictly forbidden region, left of

We first focus on eastbound tracks. The distribution for w-type tracks is given in Fig.

For the westbound tracks (Fig.

These findings are also mirrored in Pearson's correlation coefficient

The dots relative to the tracks selected for the featured analysis of Sect.

Route length

For assessing the benefit of track optimisation in terms of voyage energy efficiency, in Fig.

EEOI relative savings for the tracks in Fig.

In reference to Sect.

With the notation of Eq. (

For the eastbound route,

Furthermore, in Fig.

Thus, the magnitude and location of the GS is critical for voyage energy efficiency along this route in summer. In this respect,

The degree of optimisation of ship tracks that were actually sailed is an open research question. Weather ship-routing systems are used both offshore and on-board for planning, but the final decision is up to the shipmaster

VISIR can be used with either analysis or forecast environmental fields, as it is not constrained by any of the equations of Sect.

The unavailability of forecasts that are long enough can be addressed by either rerouting or using supplementary information.
Rerouting or replanning involves the dynamic updating of the optimal track as new information (forecast) is made available

In a non-operational mode, the unavailability of forecasts is not critical. Analysis fields can then be used, enabling a better reconstruction of the environmental state. A product derived from analyses may be quite useful for scenario assessment, but the uncertainty associated with forecasts

For nine ordered couples of harbours from the list in Table

Database of harbours. Coordinates refer to the graph grid point selected by VISIR. Wherever available, GRT is the annual throughput for the year 2016 from

This exercise demonstrated the generality of the VISIR-1.b code for assessing the potential EEOI savings depending on various wave and ocean circulation patterns. This required that graph, shoreline, bathymetry, and environmental datasets of waves and ocean currents, among other datasets, be made available for wide enough regions of the Atlantic Ocean to account for the spatial variability in the tracks.

By using Table

EEOI savings in the North Atlantic are dominated by waves, with a contribution from currents that is not negligible. At the Equator, currents are the main reason for EEOI saving. In the South Atlantic, the largest savings are computed, and they are mainly due to waves.

Routes mainly affected by ocean currents exhibit a large reduction of the correlation coefficient

The FIFO hypothesis is not satisfied in just a tiny number of edges, which are not used for the optimal tracks. This supports the use of the time-dependent Dijkstra algorithm, as in Sect.

Intentional vessel speed reduction (EOT

Maximum ROT never exceeds 20

Mean relative EEOI savings (%) for several routes in the Atlantic Ocean. The values displayed in the vertical bars refer to the annual average of the mean savings for the return voyages (i.e. mean values along the rows of Fig.

Database of routes.

Route-specific results are discussed in the following paragraphs. In the Supplement of this paper, related figures are published, and the web application for interactive exploration is available at

The geodetic track is bent southwards in the Mercator projection. The (Northern Hemisphere) winter tracks are closer to the geodetic track, while summer tracks exhibit greater diversions. This route is characterised by the highest impact of waves on energy efficiency savings. This can be ascribed to the strength of the Antarctic circumpolar winds, causing large waves in the Southern Ocean

This route does not join any major harbour and is just meant for sampling the equatorial currents. In fact, the w-type optimal tracks are quite close to being an arc of the Equator. Nearly all of the optimal eastbound cw-type tracks instead divert up to 5

This is the route discussed in the featured case study of
Sect.

At their western edge, these optimal tracks tend to sail inshore of Nova Scotia and Newfoundland and in some cases even in the Gulf of Saint Lawrence (Canada), also experiencing the effect of the Labrador Current. This solution may not be viable in practice for two reasons. First, in winter, sea ice can extend several tens of miles off the coastline. Second, coastal Canada is part of the Emission Control Areas (ECAs;

This route spans across both hemispheres. The optimal tracks of w type do not significantly differ from the geodetic track, with the Equator being crossed at about 31

This route connects the Atlantic Ocean to the Mediterranean Sea. In both sailing directions, it is dominated by waves. The tracks of cw type are influenced by both the Atlantic jet past Gibraltar and the Canary Current. They approach the energy-efficient region (Sect.

This route links the major harbour of the Atlantic (Table

The spatial variability in this route is dominated by currents, as waves from subpolar lows are not relevant in the Caribbean region. The bundle shows a waist W of Cuba (

This route is heavily influenced by the Florida current. The northbound tracks tend to align with the ocean flow. The southbound tracks (sailing against the main flow) split into two sub-bundles, W and E of the Florida current. The western sub-bundle is populated by mainly winter tracks. In fact, these tracks sail more inshore, avoiding the rough ocean state and thus reducing the speed loss in waves.

The VISIR ship-routing model and code have been updated to version 1-b. Optimal tracks can now be computed in the presence of both time-dependent ocean currents and waves. Vessel interaction with currents is described in terms of new equations which are validated by means of an analytical benchmark. To represent vessel courses with a higher degree of accuracy, the previous model version has been improved with respect to the capacity of computing graphs of a higher order of connectivity, thus accounting also for the shoreline. The computational cost and memory allocation of the new model version is also assessed, and the inclusion of ocean currents leads to a total CPU time overhead not exceeding 30 % for realistic computations (Fig.

While the code of VISIR-1.a was tested through its operational implementation in the Mediterranean Sea

Several routes are considered, and the variability in the optimal tracks is mapped across a full calendar year (2017). Both spatial and kinematical variabilities in the tracks are accounted for through various types of diagrams. The optimal exploitation of ocean currents may in some cases lead to average speeds greater than the maximum vessel speed in calm water (see Figs.

Furthermore, the analysis of the track dataset is simplified by means of metrics such as the optimal track duration and length, their Pearson's correlation coefficient, and the maximum rate of turn of vessel heading. The correlation coefficient carries a signature of ocean currents, which tend to make optimal track duration and its length less correlated to each other. Furthermore, the approximation of a FIFO network (Sect.

We regard the main computational limitation of VISIR-1.a and VISIR-1.b to be its requirement on computer RAM allocation (Sect.

However, it should be noted that a more realistic representation of the marine state is likely to correspond to a more accurate description of the mechanical interaction between it and the vessel, particularly with reference to speed loss in waves and wind

VISIR-1.b is coded in MATLAB 2016a, which was used on both the workstation (Mac OS 10.11.6 El Capitan; used for the performance analysis of Sect.

The additional figures referred to in Sect.

Animations for each of the panels of Fig. 7 can be found at

The most relevant changes of VISIR-1.b described in this paper are listed in Table

List of main code changes of VISIR-1.b with respect to VISIR-1.a, with indication of their use within this paper.

In order to head as prescribed by the optimal track, the ship has to be manoeuvred (e.g. acting on rudder and/or lateral thrusters;

Motions of the bottom layer (rudder and main engine), as related to electromechanical devices, should occur on a much shorter timescale (probably seconds to a few minutes) than the top-level controls needed for implementing the optimal track (requiring changes of the order of minutes – see ROT

Following

In any case, to ensure navigation safety, the intersection between graph arcs and the shoreline (Sect.

Thus, at this stage we still use a regular grid which enables a relatively quick and easy graph computation at the cost of a longer path computing time. This is not critical, given the non-operational functioning of VISIR for the present exercise. In future model versions, also depending on coding options, domain, and type of application, we may reconsider this choice.

Since the VISIR solution is based on Dijkstra's algorithm, it is not just guaranteed to be exact; its performance (for a given route and vessel departure date) is stable over subsequent runs. This is a difference to evolutionary (EA) and, generally speaking, to heuristic-based algorithms. For that class of algorithm, both the quality and the computational cost of the solution may vary over subsequent runs, as they are driven by random effects. The issue of randomness can be mitigated by statistical averaging over many simulations. However, a more fundamental issue is that, as clearly stated in

The supplement related to this article is available online at:

GM worked on conceptualisation, methodology, software, supervision, validation, visualisation, writing, reviewing, and editing. LC contributed to methodology (Sect.

Links MT worked together with CMCC to run the operational service (

Research results are not to be used for navigation. Neither the authors nor CMCC are liable for any damage or loss to assets or persons deriving from use of tracks computed by VISIR.

We would like to thank Nadia Pinardi (University of Bologna) for her advice on the validation of the model, Fabio Montagna (CMCC) for consultancy on graph indexing, and Florian Aendekerk (Compagnie Maritime Belge) for providing realistic parameters of a container ship.

This research has been supported by the European Commission project AtlantOS (grant no. 633211).

This paper was edited by David Ham and reviewed by two anonymous referees.