# Compact Petawatt-Class Laser Wakefield Acceleration with Plasma Telescope

Xuesong Geng  Liangliang Ji<EMAIL_ADDRESS>State Key Laboratory of High
Field Laser Physics and CAS Center for Excellence in Ultra-intense Laser
Science, Shanghai Institute of Optics and Fine Mechanics (SIOM), Chinese
Academy of Sciences (CAS), Shanghai 201800, China  Baifei Shen
<EMAIL_ADDRESS>Shanghai Normal University, Shanghai 200234, China.

###### Abstract

The compactness of laser wakefield acceleration (LWFA) is limited by its long
focal length for high power lasers, e.g., more than 10 meters for 1-peatawatt
(PW) laser pulse and up to hundreds of meters for 10-100 PW lasers. The long
focal length originates from the low damage threshold of the optical off-axial
parabolic (OAP) mirror and consequent large spot size. We propose implementing
an OAP plasma mirror (PM) to form a telescope geometry, reducing the beam size
and hence constraining the focal length to meter-range for LWFA driven by
lasers beyond 1PW. Three-dimensional particle-in-cell simulations are
performed to characterize the reflection of a 1-PW laser by the plasma OAP and
find that optimal condition is achieved within only 1-m optical length. The
new method successfully generates 9GeV electron bunch in the subsequent LWFA
stage with consistent acceleration gradients to that of the 1-PW laser via
ordinary focusing. The proposed geometry provides a solution of compact LWFAs
available for even 100-PW laser systems.

##  I Introduction

Laser wakefield acceleration (LWFA) Tajima and Dawson (1979) is able to
generate high-energy electrons within a short distance, promising the ability
to build compact particle accelerators. The state-of-the-art laser wakefield
accelerator is able to accelerate electrons to the order of 10GeV within tens
of centimeters using 850TW laser Gonsalves  _et al._ (2019). Recent 10-100PW
laser systems worldwide Danson  _et al._ (19ed) have enabled LWFA with higher
laser powers. In general, higher laser power permits longer accelerating
distance and higher electron energy, i.e. $\Delta E_{k}\sim P^{1/3}\ $ Lu  _et
al._ (2007) where $\Delta E_{k}$ is the energy gain and $P$ the laser power.
The 10-100PW laser facilities will raise the energy limit to the level of
100GeV within a single stage, as shown in Fig. 1(a), which can be boosted to
TeV level with the help of multistage LWFA acceleration Luo  _et al._ (2018).
However, LWFA in the blow-out regime Pukhov and Meyer-ter-Vehn (2002); Lu  _et
al._ (2006) requires that the laser field strength should be slightly above
relativistic threshold, and the spot size will be at the order of 100$\rm{\mu
m}$ for 10-100PW laser pulses, which requires focusing mirrors of large
f-numbers, resulting in several hundred-of-meter focal length, as shown by the
dashed line in Fig. 1(b) and 1(c), where the laser spot is assumed to be
focused to a fixed field strength above relativistic threshold. The extremely
long focal length hinders the application of high-power laser systems to
multi-stage LWFA since the actual size of the LWFA system is not only the
acceleration length but also the size of the whole optical system.

Figure 1:  (a) Scaling of electron energy gain in LWFA driven by PW-class
laser systems. The electron energy gain depends on both the laser power and
the plasma density. (b) Scaling law of focal lengths of LWFA for ordinary
focusing system and the proposed plasma telescope. (c) PW-class laser focused
by an OAP mirror with long focal length. (d) PW-class laser focused by plasma
telescope composed of an ordinary OAP mirror with short focal length and a
plasma OAP mirror behind the focal spot. It compresses the size of focusing
system of PW-class lasers from 10-100 meters to a few meters.

It is therefore crucial to shorten the optical length for LWFA if the system
is to remain compact and capable of multi-stage acceleration Gonsalves  _et
al._ (2011); Steinke  _et al._ (2016). Due to the damage threshold of the
amplification media of the laser system, the diameter of the laser spot before
the final off-axial parabolic (OAP) mirror is large for high power lasers,
prohibiting the reduction of the focal length. For 1PW laser, the optimized
acceleration length is about 20cm, while the focal length is at the order of
10m, as shown in Fig. 1(b).

Plasma, as a media free of damage threshold, provides a promising approach to
manipulate high power lasers. Plasma optics provides multiple tools to control
intense laser pulses, e.g., reflection of laser pulses of ultra-high
intensities Thaury  _et al._ (2007); Ta Phuoc  _et al._ (2012) and improving
the beam contrast for laser-solid interactions Thaury  _et al._ (2007) by
plasma mirror (PM), compression intense laser pulses by plasma gratings
Edwards and Michel (2022); Suntsov  _et al._ (2009); Wu  _et al._ (2022) and
plasma lenses for high power lasers Palastro  _et al._ (2015); Zeng  _et al._
(2020).

Here, we propose using plasma telescope to transform a tightly focused pulse
to a quasi-plane-wave beam of spot size suitable for LWFA. By using a
telescope system, the size of the focusing system can be reduced to a few
meters even for 100PW lasers, which is one order of magnitude smaller than the
ordinary focusing system. As shown in Fig. 1(d), plasma telescope is basically
composed of an ordinary OAP mirror of small f-number and an OAP plasma mirror
that reflects intense laser focused by the first OAP. Curved PMs have been
experimentally employed to focus intense laser to higher intensities
Nakatsutsumi  _et al._ (2010); Arikawa  _et al._ (2016); Wilson  _et al._
(2016); Nakatsutsumi  _et al._ (2018); Vincenti (2019); Quéré and Vincenti
(21ed), which is a demonstration of the practicality of the focusing ability
of the curved PMs. By using the telescope system, the jittering of laser spot
on the target, e.g., the plasma channel for LWFA, can be better stabilized
than the hundreds-of-meter-long ordinary focusing system since the distance
between the channel and the OAP-PM is much shorter than the ordinary focusing
system.

In the following, we carry out numerical validation of the reflection of 1PW
laser pulse by an OAP-PM via 3-dimesional (3D) particle-in-cell (PIC)
simulations. The laser wavefront, reflection efficiency, presence of preplasma
and high-order harmonics are investigated in the 3D simulations. The reflected
pulse is then utilized as the driving pulse of the subsequent LWFA stage to
qualify its availability, which is simulated in the quasi-cylindrical
coordinate. The laser pulse reflected by plasma telescope shows consistent
acceleration gradient as compared to the pulse focused by ordinary OAP, where
no significant modification is induced by laser-plasma interaction,
demonstrating 9GeV electron acceleration with 1PW laser power in a 1m optical
length. The scenario can be extended to the 10-100PW lasers since the field
strengths during reflection and the interaction geometry are almost the same.

##  II Reflection by OAP-PM

In the following simulations, we assume the field strength of the circularly
polarized driving pulse at LWFA stage is about $a_{0}=4$, i.e. $a_{0}\approx
2.83$ in y and z directions for laser propagating along x, and peak power of
1PW, where $a_{0}=eE_{0}/mc\omega$ is the normalized field strength with $e$
the electron charge, $E_{0}$ the peak electric field amplitude, $m$ the
electron mass, $c$ the speed of light and $\omega$ the angular frequency of
the laser. In order to achieve short focal lengths, the f-number of the first
ordinary OAP used to focus the incident laser is fixed to $f_{\\#}\approx 6$
which will generate a laser spot of $w_{0}=4\rm{\mu m}$ for 800nm lasers at
the first focus, as shown in Fig. 2(a). The f-number of the first OAP mirror
is flexible by adjusting the subsequent OAP-PM, as long as the combined plasma
telescope transforms the incident laser to desired spot size. In order to get
reflected pulse of $a_{0}=4$, the OAP-PM is placed behind the laser focus at
$z_{0}$, where the field strength decreases to around $a_{0}=4$ and spot size
increases to $w_{0}\approx 43\rm{\mu m}$, corresponding to curvature radius of
the wavefront of $R\approx 680\rm{\mu m}$. To transform the focused laser to a
nearly-plane-wave for later LWFA, the distance between the laser focus and the
OAP-PM satisfies $z_{0}\approx f\approx 675\rm{\mu m}$, where the curvature
radius of the OAP-PM is approximately twice of that of the incident wavefront,
which guarantees the transformation of curved wavefront to flat wavefront. The
chosen off-axial angle is $\theta=0.15$ rad for computational efficiency
consideration, which will be discussed in method section. As a result, the
f-number of the OAP-PM is about $f_{\\#}\approx 10.6$.

Figure 2:  (a) Geometry of reflection of the incident pulse by OAP-PM. (b-d)
Electric fields (red and blue) and PM density (black) in the xy-plane at
$ct=60\rm{\mu m}$, $ct=100\rm{\mu m}$ and $ct=500\rm{\mu m}$.

The reflection and propagation of the pulse is shown in Fig. 2(b-d). During
the reflection, as shown in Fig. 2(b), the field strength is periodically
modulated along the y-direction by the interference between the reflected and
incident pulse due to the extra optical path induced by the curved surface. On
the other hand, no visible distortion of the PM is observed. After the pulse
is reflected and the simulation window starts to move, slight modifications
and noises induced by laser-plasma interaction can be observed in the behind
of the pulse in Fig. 2(c), which become absent after propagation of
$ct=500\rm{\mu m}$ shown in Fig. 2(d). It should be noted that the density
distribution and laser pulse are rotated by the off-axial angle of
$\theta=0.15$ so that the reflected pulse propagates along the x-direction.
The reflected pulse and the transverse profile are shown in Fig. 3(a-b) and
the modification to the pulse is quantified in Fig. 3(c) in terms of
transverse profile of the electric fields and the transverse phase relative to
the pulse center. The relatively flat phase curve indicates that the wavefront
is nearly planar. But the lowered field profile (solid-red/-blue) indicates
that a part of the pulse energy is lost after the reflection, which is 83.6%
of the incident pulse, i.e., 24.0J, due to the absorption and heating of the
PM electrons.

However, in more realistic situations, the pre-pulse or pedestals of the laser
pulse may produce preplasma on the front surface of the PM, which will
influence the energy absorption of PM from the laser pulse. Thus, an
exponentially distributed preplasma of $\exp(-x^{\prime}/l_{\rm{pre}})$ is
added to the PM surface, where $x\textquoteright$ is the coordinate vertical
to the rotated OAP-PM and $l_{\rm{pre}}$ is the scale of the preplasma
Vincenti (2019). In our modelling, we choose $l_{\rm{pre}}=0.1\rm{\mu m}$,
which is a typical situation that can be realized by adjusting the laser pre-
pulse. We notice that the reflectivity can be boosted to 93.4% at the presence
of preplasma of $l_{\rm{pre}}=0.1\rm{\mu m}$, but at the expense of modified
pulse profile, as shown in Fig. 3(d-f), where the squeezed transverse profile
and convex phase indicate that the laser is more focused than that without
preplasma. It is because the presence of preplasma amplifies the denting of
the PM Vincenti  _et al._ (2014) which further shortens the focal length. In
our modelling, larger $l_{\rm{pre}}$ will further amplify the denting effect
and degrade reflectivity. In fact, the formation and evolution should be well
controlled by adjusting the strength and delay of the pre-pulse whenever PM is
utilized Vincenti  _et al._ (2014).

To mitigate the modification, we adjust the focal length of the OAP-PM to
$f=750\rm{\mu m}$ from the designed $f=675\rm{\mu m}$. The transverse profile
and the relative phase get recovered, as shown in Fig. 3(i), where the pulse
profile is closer to the expected Gaussian pulse profile (dashed-gray line)
and the laser phase is as flat as Fig. 3(c), indicating that the wavefront is
almost planar and is suitable for the LWFA stage. As a result, by increasing
the focal length in the presence of preplasma, the OAP-PM is able to reflect
the incident laser pulse with high reflectivity without significant distortion
of the wavefront.

Figure 3:  (a) The reflected pulse at $ct=500\rm{\mu m}$ in xy-plane for
$f=675\rm{\mu m}$ without preplasma. (b) The Ey field in the yz-plane sliced
at the black-dashed lines in (a-c). (c) The profile of Ey at the red-/blue-
dashed lines in (d-e) (red-/blue-solid) and the corresponding phases
(red-/blue-dashed) relative to the pulse center. The gray-dashed lines are the
expected Ey profile without energy loss. (d-f) The results of $f=675\rm{\mu
m}$ with preplasma. (g-i) The results of $f=750\rm{\mu m}$ with preplasma.

When the incident laser is linearly polarized (LP), the reflectivity, however,
drops to around 73% for both s- and p-polarizations in our modelling due to
stronger ponderomotive oscillation of electrons in the LP laser fields than in
the CP laser fields. Therefore, focusing of LP PW-class lasers with the
proposed plasma telescope comes with energy loss of about 30%, which becomes a
trade-off between long focal lengths and lowered laser energies when
reflecting LP lasers. During the laser-plasma interaction, high-order
harmonics could also be generated via the relativistic oscillating mirror
mechanism Teubner and Gibbon (2009); Ghimire and Reis (2019). The normalized
field strengths of the second and third order harmonics are
$a_{0}(2\omega)\approx 0.25$ and $a_{0}(3\omega)\approx 0.15$ in our
modelling, which are negligible compared to the main pulse of
$a_{0}(\omega)\approx 4$ in the context of LWFA. This is because the high-
order harmonics efficiency is suppressed for CP lasers Easter  _et al._
(2013). However, for s-/p-polarized lasers, the strength of the 3rd order
harmonic can reach $a_{0}(3\omega)\approx 1.2$, which could potentially affect
LWFA stage. Thus, the role of HHG in LWFA remains to be investigated for LP
lasers. In the following LWFA stage, only CP laser is considered.

##  III Acceleration by reflected pulse

Plasma channel can guide laser pulse without changing the spot size when the
laser spot size matches the channel Durfee and Milchberg (1993), which is an
effective guiding method for long-distance LWFA Gonsalves  _et al._ (2019);
Leemans  _et al._ (2006). The density profile of the channel is expressed as
$n_{e}(r)=n_{0}+r^{2}/\pi r_{e}w_{m}^{4}$ where $n_{0}$ is the central
electron density, $r_{e}\approx 2.8\times 10^{-15}$m the classical electron
radius and $w_{m}$ the matched spot size.

For demonstration of the acceleration capability of the reflected laser pulse,
we carry out LWFA simulation for both the reflected pulse and the pulse
injected from simulation boundary, i.e., without reflection by PM. The laser
pulses are injected into the plasma channel with central density of
$n_{0}\approx 2.4\times 10^{17}\rm{cm^{-3}}$ and matched spot size of
$w_{m}\approx 43\rm{\mu m}$ for $w_{m}=w_{0}$. The LWFA stage is simulated via
FBPIC Lehe  _et al._ (2016) in r-z coordinates with the reflected pulse as the
initial condition. Simulation details are shown in the method section.

The accelerating field $E_{z}$ and bubble structures are compared in Fig.
4(a-c). One can see that the strengths of the stimulated accelerating fields
$E_{z}$ are almost identical for the reflected and injected pulses, which
remains true in the long-term evolution shown in Fig. 4(d) and 4(e), where the
bubble and injected electron are represented by the variation of the on-axis
acceleration field $E_{z}$ and the on-axis electron density. The yellow
trajectories indicate the electrons of high density, i.e., the tail of the
bubble and the injected electron bunch. It can be inferred from the electron
bunch trajectory and the evolution of $E_{z}$ that the electrons experience
similar acceleration gradients. The consequent electron energy evolution and
final spectrum are shown in Fig. 4(f) and 4(g). It can be seen that the
electrons injected at similar positions are accelerated to similar energies
despite the different bubble evolution. For example, electrons injected at
$ct\approx 60$mm are accelerated to $E_{k}\approx 6$GeV and $ct\approx 40$mm
to $E_{k}\approx 8$GeV in both cases, which can be inferred from the gray
lines in Fig. 4(f) and 4(g).

Figure 4:  (a-c) Accelerating field $E_{z}$ (red and blue) and electron
density (gray) at (a) $z\approx 20$mm, (b) $z\approx 50$mm and (c) $z\approx
80$mm of the pulse reflected by plasma OAP (up) and the pulse injected from
boundary (down). (d-e) Variation of the on-axis accelerating field $E_{z}$ in
the simulation window (red and blue) and the on-axis electron density (yellow)
for (d) the reflected and (e) the injected pulse. (f-g) Evolution of the
electron energy spectrum (gray) and the final spectrum (blue) for (f) the
reflected and (g) the injected pulse.

As for electron bunch emittance, both the reflected pulse and injected pulse
generate well-collimated electrons as indicated by the curves in Fig. 5. The
emittance is calculated by $\varepsilon_{x}=\sqrt{\langle
x^{2}\rangle\langle\theta_{x}^{2}\rangle-\langle x\theta_{x}\rangle^{2}}$,
where $\langle\cdot\rangle$ denotes the standard deviation and
$\theta_{x}=\tan^{-1}(p_{x}/p_{z})$, and $\varepsilon_{y}$ is calculated in
the same way. In other words, for bunch size of 5$\rm{\mu m}$ the angular
divergence is about 0.2mrad when $\varepsilon_{x}\approx\varepsilon_{y}\approx
1\rm{\mu m\cdot mrad}$. In both situations the evolutions of the bunch
emittance are almost identical after the electrons are significantly injected,
i.e., after $ct=50$mm, except that the bunch emittance driven by the reflected
pulse is slightly higher than the injected pulse due to much higher bunch
charge.

Figure 5:  Electron bunch emittance $\varepsilon_{x}$ (red) and
$\varepsilon_{y}$ (blue) driven by the pulse reflected by plasma OAP (solid)
and the pulse injected from boundary (dashed). Electrons below 1GeV are
excluded.

However, since LWFA in the blowout regime is highly nonlinear Pukhov and
Meyer-ter-Vehn (2002); Lu  _et al._ (2006); Esarey  _et al._ (2009) and is
sensitive to initial conditions, any modification of the driving laser could
induce disparate results, especially for long-distance acceleration Pukhov and
Meyer-ter-Vehn (2002); Wang  _et al._ (2013); Kim  _et al._ (2017). The shot-
to-shot instability could be even more significant than the distortion induced
by reflection. In other words, considering the similar acceleration gradient
in our modelling, reflection by OAP-PM will not induce more significant
instability than the shot-to-shot instability of the laser system itself.

Therefore, slight mismatch between the evolution of the bubble structures can
be observed in Fig. 4(a) and 4(b) where the bubble generated by the reflected
pulse is larger than the injected pulse in Fig. 4(a) and smaller in Fig. 4(b).
The inconsistent bubble oscillation during pulse propagation results in
different self-injection moments, as indicated by the start of the yellow
slashes in Fig. 4(d) and 4(e). The reflected pulse induces longer bubble
oscillation time whereas the injected pulse produces continuously expanding
bubble after a few oscillations. Therefore, the reflected pulse induces
discontinuous self-injection, but the electrons driven by the injected pulse
are continuously injected into the end of the bubble. This difference results
in disparate electron bunches as shown by the variation of energy spectrums in
Fig. 4(f-g). The discontinuous injection produces several energy spikes with
significantly higher beam charge but lower cutoff energy at $E_{k}\approx
9$GeV since a few electrons are accelerated over 10GeV by the injected pulse
in Fig. 4(g) as indicated by the gray lines.

##  IV Discussion

The proposed plasma telescope aims to reduce the focal lengths of 1-100PW
laser systems when driving LWFA since higher laser power generates higher
electron energies, which is shown in Fig. 1(a) according to the scaling law in
Lu  _et al._ (2007). When using ordinary OAP focusing geometry, for specific
OAP damage threshold and specific focal intensity, the required spot size onto
the OAP scales with $w_{0}\sim\sqrt{P}$ and the f-number scales with
$f_{\\#}\sim\sqrt{P}$ resulting in linear scaling of focal length with $f\sim
P$, as shown by the black-dashed line in Fig. 1(b). It means generating 100GeV
electron bunch requires hundreds of meters of focal length. By using the
proposed plasma telescope, the f-number of the OAP can be fixed to, for
example, $f_{\\#}\approx 6$ in all scenarios in our modelling, and the scaling
law becomes $f\sim\sqrt{P}$, as shown by the black-solid line in Fig. 1(b),
which reduces the focal length by 2 orders when generating 100GeV electrons.

In terms of laser pointing stability, since the plasma channel can be situated
much closer to the OAP-PM, typically just a few centimeters away, the
positional jittering can be effectively managed. In contrast, conventional
systems place the plasma channel tens to hundreds of meters away from the
focusing mirror. For instance, a 1$\rm{\mu rad}$ angular jittering of the
first OAP mirror results in only about 10$\rm{\mu m}$ of positional jittering
when the plasma channel is 1cm away from the OAP-PM, as illustrated in Fig. 6.
The positional jittering can be further reduced to 1$\rm{\mu m}$ if the plasma
channel can be placed 1mm away from the OAP-PM. However, in conventional
systems, the same angular jittering would cause positional jittering of tens
to hundreds of micrometers at the plasma channel due to the longer focal
length that extends up to hundreds of meters for higher power lasers. The
plasma telescope thus alleviates the stringent requirements on the pointing
stability of the OAP mirror at the plasma channel entrance for high-power
laser systems. It provides a more compact and efficient solution, enhancing
the overall performance and precision of laser systems.

Figure 6:  Sketch of the positional jittering induced by angular jittering of
the first OAP mirror. 1$\rm{\mu rad}$ of angular jittering will result in
about 10$\rm{\mu m}$ of positional jittering at 1cm away from the OAP-PM.

For manufacturing considerations, in the investigated geometry, the
microscopic OAP-PM can be manufactured via 3D-printing technique Gao  _et al._
(2014) or rotating liquid that forms parabolic surface Hickson  _et al._
(1993). On the other hand, the OAP-PM can be replaced by an ellipsoidal PM
(EPM) Nakatsutsumi  _et al._ (2010); Wilson  _et al._ (2016). The foci of the
ellipsoid form a focus-to-focus imaging system where the spot size is
magnified by $\beta/\alpha$ where $\beta$ and $\alpha$ are the distance of the
foci to the reflection point of EPM Stavroudis and Ames (1992). The EPM can be
manufactured in macroscopic scale by tuning $\alpha$, $\beta$ and the
ellipticity of the EPM.

##  V Conclusion

The proposed OAP-PM effectively transforms a 1PW laser pulse focused by a
short-focal-length OAP to a pulse with long Rayleigh length, forming a plasma
telescope. The pulse reflected by OAP-PM successfully generates 9GeV electron
bunch in the subsequent LWFA stage. Though slight modifications are introduced
to the reflected pulse, the acceleration gradient and bunch emittance are
similar to the ordinary focusing system. The proposed method essentially
provides a new option to reduce the focal lengths of 1-100PW laser systems
when large spot size is required like LWFA. Compact LWFA based on PW-class
laser system paves way for multi-stage acceleration towards TeV electrons.

##  VI Methods

Considering the laser-plasma interaction in strong fields, the reflection of
tightly focused laser pulse by OAP-PM is simulated via PIC method in 3D space,
which is carried out using the EPOCH code Arber  _et al._ (2015). Then the
LWFA stage is simulated in quasi-cylindrical coordinate via the FBPIC code
Lehe  _et al._ (2016) based on the reflected pulse from the reflection stage,
which significantly reduces the simulation time and makes it possible to
simulate the 15-centimeter-long acceleration stage.

The 3-D simulations are carried out in the $80\rm{\mu m}\times 200\rm{\mu
m}\times 200\rm{\mu m}$ box with cell size of $0.05\lambda\times
0.2\lambda\times 0.2\lambda$. The OAP-PM is placed at the left boundary of the
simulation box with 8 macro-electrons and 1 macro-proton in each cell with
density of $20n_{c}$ where $n_{c}$ is the critical plasma density. For off-
axial angle $\theta$, the parabolic PM is rotated by $\theta$ along the
z-axis, so that the reflected laser pulse propagates along the x-axis, as
shown in Fig. 1.

Considering the simulation efficiency, the electron density of the OAP-PM is
set to $20n_{c}\approx 3.48\times 10^{22}\rm{cm^{-3}}$ and its off-axial angle
$\theta=0.15$rad. The reflectivity could benefit from higher plasma density in
the modelling, but it will require higher spatial resolution to resolve
smaller plasma wavelength. Larger off-axial angles do not change the
interaction picture but may need high resolution along transverse directions.

The reflected pulse in the 3-D simulation is then transformed into the
cylindrical coordinate via azimuthal Fourier decomposition Lifschitz  _et al._
(2009). First, the electromagnetic field components $F(x,y,z)$ in the
Cartesian grid is converted to $F(z,r,\theta)$. The latter can be expressed by
the summation of the Fourier components

$F(z,r,\theta)=\sum_{m}{F^{(m)}(z,r)e^{im\theta}},$ (1)

where $F^{(m)}(z,r)$ is the Fourier components and m is the mode number. The
Fourier components are calculated via Fourier transformation

$F^{(m)}(z,r)=\frac{1}{2\pi}\int_{0}^{2\pi}F(z,r,\theta)e^{im\theta}d\theta.$
(2)

Such conversion of fields is possible when the symmetry of fields can be
resolved by azimuthal Fourier expansion of $e^{im\theta}$, which is accurate
when the fields are highly symmetric along the $\theta$-axis. In our
simulation, 3 modes is sufficient to model the reflected pulse. The quasi-
cylindrical simulation is carried out in the $4096\times 800$ window with cell
size of $0.05\lambda\times 0.2\lambda$ in z and r directions. Each cell is
filled with $1\times 1\times 12$ macro-particles in z, r and $\theta$
directions. The above calculated Fourier components are loaded into the
simulation window as the initial conditions.

###### Acknowledgements.

The authors acknowledge insightful discussions with Prof. A. Pukhov and Prof.
I. Kostyukov. The work is supported by the China Postdoctoral Science
Foundation (2022M713258), the Shanghai Science and Technology Development
Foundation (22YF1455100), CAS Project for Young Scientists in Basic Research
(YSBR060), National Natural Science Foundation of China (11935008), National
Key Research and Development Program of China (2022YFE0204800), and the
International Partnership Program of Chinese Academy of Sciences
(181231KYSB20200040).

## Data Availability Statement

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

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# Time-Series Analysis Approach for Improving Energy Efficiency of a Fixed-
Route Vessel in Short-Sea Shipping

Mohamed Abuella, Hadi Fanaee, Slawomir Nowaczyk, Simon Johansson, and Ethan
Faghani Manuscript received November 1, 2023; revised November 16, 2023.This
work was supported by the Sweden’s innovation agency (Vinnova).(Corresponding
author: Mohamed Abuella.)
Mohamed Abuella, Hadi Fanaee, and Slawomir Nowaczy are with Center for Applied
Intelligent Systems Research (CAISR), Halmstad University, Kristian IV:s väg
80523, Halmstad, 30118 Sweden (e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>Slawomir.Nowaczyk@hh.se). Simon Johansson and Ethan Faghani
are with CetaSol AB, Gothenburg, 41251, Sweden (e-mail: simon.johansson and
<EMAIL_ADDRESS>
###### Abstract

Several approaches have been developed for improving the ship energy
efficiency, thereby reducing operating costs and ensuring compliance with
climate change mitigation regulations. Many of these approaches will heavily
depend on measured data from onboard IoT devices, including operational and
environmental information, as well as external data sources for additional
navigational data. In this paper, we develop a framework that implements time-
series analysis techniques to optimize the vessel’s speed profile for
improving the vessel’s energy efficiency. We present a case study involving a
real-world data from a passenger vessel that was collected over a span of 15
months in the south of Sweden. The results indicate that the implemented
models exhibit a range of outcomes and adaptability across different
scenarios. The findings highlight the effectiveness of time-series analysis
approach for optimizing vessel voyages within the context of constrained
landscapes, as often seen in short-sea shipping.

###### Index Terms:

Energy efficiency, voyage speed optimization, time-series analysis, short-sea
shipping.

##  I Introduction

The transportation of commercial freight by sea is the most environment-
friendly transportation mode, since it emits less greenhouse gases (GHGs) per
tonne of capacity and kilometer of distance, thus making resulting a smaller
carbon footprint and lower impact on global climate. Short-Sea Shipping (SSS)
is a commercial transportation mode that does not involve intercontinental
cross-ocean. The SSS provides a cost-efficient and environment-friendly
alternative for transportation by utilizing inland and coastal waterways to
transport the commercial freight [1].

On the other hand, the SSS produces some negative effects on the natural
habitats and polluting the air along the coasts of populated cites [2]. As a
response to this, the International Maritime Organization (IMO) have conducted
many studies and recommended standards and imposed policies for the maritime
sector to reduce the carbon dioxide (CO2) to 40% by 2030 and cut 50% of all
GHGs by 2050, based on the emissions in 2008 [3].

Furthermore, COVID-19 pandemic has accelerated the digitalization of the
entire shipping industry globally, and hence attracted a profound
consideration to data collection and preparation stages [4]. The operational
and some environmental conditions can be accessed through an Automatic
Identification System (AIS) messages, which is a service developed by the
International Maritime Organization (IMO) in 2002 to record the sensor
measurements and send the vessel position information for the traffic between
other ships and neighboring shores [5].

In a broader perspective, for improving the vessel’s energy efficiency and
harnessing more fuel savings and less GHG emissions, there are mainly two
procedures. The first strategy is in the ship design stage, where the ship is
built to obtain a body and equipped machinery that work efficiently. The
second strategy is during the ship operation over the water or at the ports.
This latter procedure can be achieved by adopting energy management plans that
optimally enhance the energy efficiency and fuel consumption [6].

This paper proposes a data-driven framework for optimizing vessel speed
profiles to improve energy efficiency in short-sea shipping.
The main contributions of this paper can be summarized as follow:

  * •

Modeling of energy efficiency: Develop a data-driven model for voyage energy
efficiency, including:
A spatiotemporal aggregation of operation and navigation data from onboard and
external sources to capture the impact of both spatial and temporal factors on
voyage energy efficiency.
Introduce an efficiency score that considers both total fuel consumption and
voyage duration to measure the voyage energy efficiency.

  * •

Data clustering: Clustering the data of voyages and sorting them based on
their efficiency scores. This clustering enables the voyage optimization
algorithm to learn more insights for better actions, either by selecting the
best voyages or by eliminating the worst voyages.

  * •

Time-series analysis models and comparative analysis: Four time-series based
models are implemented as algorithms of voyage speed optimization. Then, a
rigorous evaluation of their performance is conducted across different data
clusters and using metrics that account for voyage efficiency.

  * •

Practical implication: Demonstrate the significant effectiveness and
practicality of the proposed approach for fixed-route vessels in short sea
shipping, where the options for obtaining efficient voyages are limited. The
approach also aligns with the guidance of domain experts, adhering to safety
and traffic considerations.

The remainder of the paper is organized as follows: related work is presented
in Section II. The methodology of the proposed framework is covered in Section
III. Section IV introduces the case study, including a description of the data
used and the framework’s implementation. The results and outcomes of the
models are discussed in Section V. Finally, conclusions are addressed in
Section VI.

##  II Related Work

Various techniques can be applied for the voyage optimization problem, as
demonstrated in prior studies such as [7, 8, 9, 10]. Some of these approaches
have been implemented specifically for short-sea shipping, as indicated in the
domain of [11] and [12].

Different modeling and optimization algorithms for ship weather routing have
been investigated thoroughly in [13]. It was found that the effectiveness of
these algorithms strongly depends on specific requirements concerning the
objectives, control variables and constraints as well as the implementation.
The majority of these voyage optimization approaches are applied to ocean-
crossing ships by controlling mainly the vessel speed and its route. Whereas,
in the SSS, especially for fixed-route vessels like ours, there are fewer
options available for voyage optimization. Therefore, the scope of this paper
is mainly focused on short-sea shipping and the pertinent research literature.

In the third IMO GHG study 2014 [14], it was assumed that weather effects
alone would be responsible for 15% of additional power margin on top of the
theoretical propulsion requirements of ocean-going ships, and a 10% additional
power requirement for coastal ships.
In a recent adaption of the Ship Traffic Emission Assessment Model (STEAM),
propelling power is determined by wave height and directions, accounting for
the environmental conditions in a highly detailed manner [5].

Recent research studies [15, 16] have explored energy-efficient routing for an
electric ferry in Western Norway. They rely on operational data from onboard
measurements and environmental conditions from the Norwegian Meteorological
Institute, and proposed a hybrid physics-guided machine learning model for
optimizing the ship route. Based on their findings, the hybrid model was
showing an energy reduction of 3.7% compared to the actual consumption, simply
by applying minor route and speed profile alterations as guided by the
provided weather forecasts.

Researchers from Napa Ltd. in Finland conducted several studies on voyage
optimization. In two of their studies [17] and [18]. These experts stated that
their products of voyage optimization can achieve a fuel cost reduction of
more than 10% with 2% to 4% savings from trim optimization and 6% to 8% from
speed and route optimization. [19].

In the study conducted by Huotari et al. [20], where they used a combined
model with both dynamic programming and convex optimisation to obtain optimal
speed profiles. The fuel savings were around 1.1% and for voyages with
substantial variance in environmental conditions, the fuel savings reached as
high as 3.5%.

In the review paper by Wang et al. [21] numerous studies are explored,
including coastal and inland shipping, also revealing a diverse range of fuel
savings outcomes.
Meanwhile, regarding weather routing, specifically through speed and route
optimization, it has been demonstrated in [22] that the potential savings of
carbon dioxide emissions and fuel costs are in range of 5% to 10%.

In accordance with the main contributions highlighted in Section (I), our
paper distinguishes itself from previous research by tackling the distinctive
operational challenges of SSS, employing time-series analysis models for
optimizing voyage speed, and conducting a comprehensive comparative analysis
to rigorously evaluate the robust performance of these models across diverse
data clusters.

##  III Methodology

The main objective of this study is to improve the vessel voyage by optimizing
its speed to enhance the vessel’s energy efficiency. In other words, reducing
the vessel’s fuel consumption within constrained arrival time.

###  III-A Framework of Voyage Optimization

The framework of the developed approach for improving the vessel voyages is
depicted in Figure 1.

Figure 1: Framework of vessel voyage optimization.

Data Processing and Clustering is a vital component of our framework. The
primary objective is to identify and sort the voyages based on their
efficiency scores. Then, train the models with the sorted data clusters
iteratively, to distill insights from the voyages with different behaviours.
Thus, the trained models will gain valuable insights into the performance and
operational patterns of the vessel.

###  III-B Energy Efficiency Modeling

This part presents a mathematical and visual overview of the fundamental
theoretical background that forms the basis for modeling vessel energy
efficiency—an indispensable element within our comprehensive framework.

To estimate the vessel’s energy efficiency in this framework of voyage
optimization, we employed a previously developed model equipped with
artificial intelligence (XAI) and machine learning techniques. More details
about this energy efficiency modeling approach can be found in [23].

The efficiency score ($\mathrm{Eff_{score}}$) is calculated from the
normalized total fuel and time for every voyage, as following:

$\mathrm{Eff_{Score}}=1-\frac{2\times[Fuel_{Tl_{Nm}}\times
Time_{Tl_{Nm}}]}{[Fuel_{Tl_{Nm}}+Time_{Tl_{Nm}}]}$ (1)

The efficiency score considers the proportional reduction in both fuel
consumption and time, assessing the vessel’s efficient use of resources during
the voyage.

Figure 2 facilitates to visualize the process of aggregation for the vessel’s
voyages. First by illustrating the voyages in terms of space (i.e., latitude
and longitude) as shown in Figure 2(a), and second by representing the
aggregated voyages as points in new dimensions of Efficiency Score versus
total fuel and total time. The aggregated data and its new dimensions are
projected as in Figure 2(b).

(a) Vessel’s route.

(b) Data are projected by Efficiency Score vs. fuel and time.

Figure 2: The vessel voyages and the aggregated data projected by the
Efficiency Score.

The representation of voyages in terms of fuel and time is done by adopting
the concept of the Efficiency Score (Eff-Score). The efficiency scores for all
vessel’s voyages are presented in Figure 2(b). It is evident that the voyage
with lower fuel and shorter time have higher efficiency scores, and vice
versa.

Data: Voyages data of the vessel

Result: Clusters of Voyages

Load the operational and navigational data, including speed, course, fuel,
position, distance, and weather;

Tag the datapoints to its corresponding voyage, $V_{id}$;

foreach  _voyage $V_{i}$ in voyages data_ do

Calculate total fuel consumption and time for $V_{i}$;

Normalize total fuel and time for $V_{i}$ based on their maximum values of all
voyages;

Calculate the Eff-Score as described in Eq. (1), and assign it to all
datapoints of this voyage $V_{i}$;

Initialize four empty lists for each cluster: $Top75Pr$, $Top50Pr$, $Top25Pr$,
$Top10Pr$ (Percentiles of Eff-Scores);

foreach  _data point in all data_ do

Extract the Eff-Score of the data point;

if  _Eff-Score is in the top 75%_ then

Append the data point to $Top75Pr$;

else if  _Eff-Score is in the top 50%_ then

Append the data point to $Top50Pr$;

else if  _Eff-Score is in the top 25%_ then

Append the data point to $Top25Pr$;

else if  _Eff-Score is in the top 10%_ then

Append the data point to $Top10Pr$;

Algorithm 1 Modeling of Energy Efficiency and Clustering of Voyages Data Based
on Their Energy Efficiency

There are four voyage data clusters, namely $Top75Pr$, $Top50Pr$, $Top25Pr$,
and $Top10Pr$, as shown in Figure 3, These clusters are categorized on their
respective Eff-Score percentiles, enabling a structured analysis of voyage
efficiency across various percentile groups.

Figure 3: Four clusters of voyages based on their efficiency.

###  III-C Time-Series Analysis Models for Voyage Optimization

For the purpose of vessel voyage optimization, we need a model to optimize the
vessel’s speed profile for improving the vessel’s energy efficiency. This
ideal model should mainly be able to:

  * •

Model the temporal dependencies in vessel speed profiles.

  * •

Incorporate external variables, such as weather conditions, into the modeling
process.

  * •

Adapt to changing conditions to provide real-time optimized speed profiles.

In order to meet these requirements, we adopt several time-series analysis
models, which are described with some details as follow.

####  III-C1 Long Short-Term Memory (LSTM)

It is a type of recurrent neural network (RNN) architecture designed for
sequential data analysis. It is particularly useful for modeling and
predicting time-series data, making it a valuable tool for improving energy
efficiency in vessel operations [24].
LSTM networks are well-suited for tasks involving time-series sequences of
varying lengths, capturing long-term dependencies, and handling irregular
temporal patterns. The LSTM architecture deals specifically to address the
problem of vanishing gradients that often occurs in other RNN structures [25].

The LSTM’s capability to learn the hidden insights from selected data,
particularly in terms of vessel’s efficiency scores, comes with a trade-off.
The data clustering may result in a smaller dataset, which potentially
limiting the LSTM’s deep learning capabilities. Additionally, data clustering
increases the susceptibility of LSTM to overfitting.
LSTM can be employed to model and predict the vessel’s speed to improve its
fuel efficiency, taking into account various factors such as weather
conditions and operational parameters

####  III-C2 k-Nearest Neighbors (kNN)

The kNN algorithm is a non-parametric model that makes no assumptions and
operates on the principle of determining an unknown observation’s class by
measuring distances to nearby observations, attributing the observation to the
majority class of its nearest neighbors. The parameter k represents the number
of neighboring observations considered when classifying a given observation,
and its value is determined through a search for the optimal choice that
maximizes accuracy in the training set.

The paper by Cover and Hart [26] is a foundational work in pattern recognition
and machine learning, introduced the KNN algorithm for pattern classification
by utilizing the nearest neighbors to classify data points based on majority
vote.

KNN can be employed to analyze and predict energy consumption patterns based
on similar historical data. By considering the nearest neighbors of a given
operational scenario, KNN provides a straightforward approach to making
energy-efficient decisions. The choice of the parameter ’k,’ representing the
number of neighbors, plays a crucial role in the accuracy of KNN-based models.

####  III-C3 1-Nearest Neighbor with Dynamic Time Warping (1NN-DTW)

This model is a distinctive variant of kNN with (k=1) and with a distance-wise
measure that utilizes Dynamic Time Warping (DTW) algorithm for a comparison of
two time-series sequences that may have different lengths, time shifts, and
speed variations [27, 28, 29, 30, 31]. The 1NN-DTW algorithm can be utilized
to measure the similarity between the given speed profile and other measured
vessel speed profiles with higher efficiency scores. 1NN-DTW is capable to
capture the temporal dependency of speed profiles, even in short-sea shipping
where the options of control the ship to improve its energy efficiency are
limited.

Thus, the 1NN-DTW identifies the efficient observed speed profile that most
similar to the given speed profile to be chosen as suggested an optimized
speed profile for the current journey of the vessel that can lead to improved
energy efficiency.

####  III-C4 Hidden Markov Model (HMM)

This model is based on probabilistic modeling and use a variety of techniques
from the statistical modeling, and they are widely used in time-series
analysis and pattern recognition.
The work by Rabiner in 1989 [32] is an essential reference in the application
of HMM. This tutorial delved into the formulation of a statistical method for
representing speech, showcasing a successful HMM system implementation with a
focus on discrete or continuous density parameter distributions.

In the context of improving energy efficiency in constrained environments,
HMMs can be applied to model the underlying patterns and transitions in ship
operational data, allowing for more informed decisions on optimizing energy
consumption.
The HMM estimates three main weather states and dynamically adjusts the
SpeedOverGround (SOG) predictions. During calm weather, it selects the maximum
observed speed in the historical data at that condition. In moderate
conditions, it uses the average speed, while in rough weather, it relies on
the minimum speed profiles. For instance, Eq. 2 indicates that the predicted
speed (SOG${}_{\text{Pred}_{i}}$) by HMM is the minimum value of measured
speed profiles in rough weather state, and likewise for other weather states.

$SOG_{\text{Pred.}_{i}}=\min(SOG_{\text{Meas.}}\,|\,\text{Rough weather})$ (2)

This adaptability to changing weather conditions enhances voyage optimization,
making it a crucial component of the overall framework.

##  IV Case Study

###  IV-A Data Collection

The ship’s onboard data have been received from our industry partner CetaSol
AB in Gothenburg [33]. The data has been gathered over a period of 15 months,
between January 2020 and March 2021. It has a 3Hz frequency and records about
the ship’s position, course direction, and speed. It is also including some of
operational and meteorological data, such as fuel rate, engine speed, torque,
acceleration, wind speed and direction.

Some information about the ship and its voyage can be found on Marine Traffic
website [34].
Other weather variables such as wave height and sea current speed and
direction have been collected from external sources, Copernicus Marine Service
[35] and Stormglass [36] APIs.

TABLE I: The navigational variables and their data sources. Variable | Source | Variable | Source
---|---|---|---
Latitude | Onboard | WindSpeed_cps | Copernicus
Longitude | Onboard | WindDirection_cps | Copernicus
SpeedOverGround | Onboard | WaveHeight | Copernicus
HeadingMagnetic | Onboard | WaveDirection | Copernicus
Pitch | Onboard | WindSpeed_sg | Stormglass
Roll | Onboard | WindDirection_sg | Stormglass
WindSpeed_onb | Onboard | CurrentSpeed | Stormglass
WindDirection_onb | Onboard | CurrentDirection | Stormglass

###  IV-B Data Preparation and Validation

The external weather data are past forecasts (hindcasts), which have
reanalysed to become hourly in temporal resolution and with 0.25 to 0.5 degree
as a spatial resolution. Trilinear interpolation in time and space dimensions
has been applied on external weather data to be more suitable for time and
position frames of the given vessel routing. Therefore, the weather and
onboard data are used in this analysis with a temporal resolution of 1-minute
in average.

The data validation is conducted through the cruising-speeds mode is to reduce
the other vessel effects on the fuel consumption, and thus, producing graphs
that can be then compared with the general ship’s standard performance. The
operational and weather data validation is carried out visually, as shown in
Figure 4.

Figure 4: Vessel’s data analytics and standard performance graph for the case
study [37]

In a prior publication [23], we have addressed the challenges of data
representation and energy modeling in short-sea shipping (SSS), by proposing a
data-driven modeling approach of voyage efficiency, which combines and
aggregates data from multiple sources and seamlessly integrates explainable
artificial intelligence (XAI) to attain clear insights about the energy
efficiency for a vessel in SSS.

###  IV-C Implementation of the Framework

The implementation of our approach of a time-series analysis-based voyage
optimization framework for a fixed-route vessel of our case study is depicted
in Figure 1, and the step-by-step process is described by algorithms 1 and 2.
For more detailed information about setting up the models and their
specifications including various parameters, you may refer to the source
codes, which are developed in Python 3.9.7 to produce the results of this
study. These source codes are available at:
https://github.com/MohamedAbuella/TSA4EESSS.

Data: Refer to Algorithm 1 for data processing and clustering.

Add SOG${}_{\text{Meas.}}$ and Weathers to Inputs.

foreach  _$C_{k}$ in CT (sorted by Eff-Score)_ do

foreach  _model in [LSTM, KNN, 1NN-DTW, HMM]_ do

Train model with voyages in $C_{k}$;

foreach  _Voyage $v_{i}$ in test dataset $\notin C_{k}$ _ do

if  _model is LSTM_ then

Predict SOG${}_{\text{Pred.}_{i}}$ using trained LSTM;

else if  _model is KNN_ then

Predict SOG${}_{\text{Pred.}_{i}}$ using trained kNN;

else if  _model is 1NN-DTW_ then

Predict SOG${}_{\text{Pred.}_{i}}$ based on the most similar voyage in
$C_{k}$;

else if  _model is HMM_ then

Predict SOG${}_{\text{Pred.}_{i}}$ based on weather states (Calm, Moderate,
Rough);

Estimate efficiency for SOG${}_{\text{Meas.}_{i}}$ and
SOG${}_{\text{Pred.}_{i}}$;

Evaluate efficiency gains for test voyages;

Algorithm 2 Speed Optimization Models for Improving Voyage Efficiency

##  V Results and Discussion

The vessel has a fixed-route which starts from the southern port to the
northern port or vice versa. This route can be divided into four segments,
specifically North, Middle, South, and Direct, as depicted in Figure 2(a).
Cruising speeds are more common in North, South, and Direct segments of the
vessel’s route. Meanwhile, in the Middle segment, the vessel typically
operates at maneuvering speeds, due to the presence of two ports located on
west and east sides of the canal.

We have first conducted a statistical analysis on the dataset. Figure 5
illustrates some important statistic for all aggregated voyage, with regard to
the accumulated fuel, time, and distance at different route segments. The
route segments are depicted in Figure 2(a).

Figure 5: Barplots for statistics of Fuel, Time, and Distance in different
route segments

As it can be seen from Table II, the difference of fuel consumption of the and
cruising speeds is 5.47%, so that and also based on the recommendations from
domain experts in compliance with maritime regulations including safety and
traffic considerations, it might be more practical to primarily focus on
optimizing cruising speed.

TABLE II: Statistics of the dataset for different speed modes Variable | Speeds | Difference (%)
---|---|---
All | Cruising
Fuel, total (Liter) | 1329.2 | 1256.62 | 5.47%
Time, total (Hour) | 48.04 | 22.32 | 53.57%
Distance, total (km) | 608.72 | 349.2 | 42.68%
Speed, average (m/s) | 2.67 | 1.67 | 37.5%

After that we have implemented the framework for improving the vessel’s energy
efficiency, as presented in IV-C of Section IV. Then, we evaluate the results,
and for a sake of a fair comparison, we are injected both the actual measured
and optimized speed profiles into the same estimation model of energy
efficiency to predict fuel and time before and after the improving framework
of energy efficiency is being implemented. Once the fuel and time estimated,
we compute the efficiency to determine how much energy has been saved.

One of our main metrics to evaluate the model performance of voyage efficiency
optimization is the gain of efficiency scores, as represented in (3).

$Eff._{Gain}=\frac{Eff.Score_{{Pred.}}-Eff.Score_{Meas.}}{Eff.Score_{Meas.}}\times
100$ (3)

Where $Eff.Score_{{Meas.}}$ and $Eff.Score_{{Pred.}}$ represent the voyage
efficiency obtained with measured and predicted speed profiles, respectively.

TABLE III: Average efficiency gains (Eff. Gains %, see Eq. 3) and counts of improved voyages (IV Count#) out of 162 voyages in the test dataset. Cluster | Efficiency Score | LSTM | KNN | 1NN-DTW | HMM
---|---|---|---|---|---
Top10Pr | Eff. Gains (%) | 2.61 | 2.13 | 3.20 | 6.05
IV Count (#) | 134 | 114 | 127 | 139
Top25Pr | Eff. Gains (%) | 2.38 | 1.58 | 3.23 | 1.30
IV Count (#) | 129 | 107 | 128 | 107
Top50Pr | Eff. Gains (%) | 0.97 | 0.98 | 2.58 | 7.34
IV Count (#) | 100 | 106 | 117 | 140
Top75Pr | Eff. Gains (%) | -0.84 | 0.50 | 2.28 | 9.31
IV Count (#) | 60 | 93 | 119 | 141
Average | Eff. Gains (%) | 1.28 | 1.30 | 2.82 | 6.00
IV Count (#) | 105.75 | 105.00 | 122.75 | 131.75
TABLE IV: Average and deviation of gains (%) of Eff-Scores in three weather states, when models are trained with the four data clusters Model | LSTM | KNN | 1NN-DTW | HMM
---|---|---|---|---
Weather | Avg | Std | Avg | Std | Avg | Std | Avg | Std
Calm | 0.17 | 4.72 | 1.17 | 3.29 | 2.21 | 4.30 | 3.96 | 6.19
Moderate | 1.53 | 4.12 | 1.26 | 4.19 | 3.4 | 4.47 | 5.33 | 6.56
Rough | 1.94 | 3.48 | 1.42 | 3.65 | 2.84 | 4.90 | 8.17 | 9.08
Average | 1.21 | 4.11 | 1.28 | 3.71 | 2.82 | 4.56 | 5.82 | 7.28

As shown in Table III, the HMM model achieves the highest average efficiency
gain of 6.00%, followed by the 1NN-DTW model (2.82%), the KNN model (1.30%),
and the LSTM model (1.28%). In terms of the number improved voyages out of 162
voyages in test dataset, the HMM model also improves the energy efficiency of
the most average number of improved voyages (131.75 out of 162 voyages).

The HMM model achieves its best performance when trained on the Top75Pr
cluster, which includes voyages with lower Eff-Scores and frequently
encountered adverse weather conditions. Such performance underscores the HMM
model’s capability to learn the hidden patterns between the vessel speed and
weather states, ultimately facilitates for developing more efficient speed
profiles.

Table IV indicates that the HMM model yields the highest average efficiency
gain in all three weather states, namely calm, moderate, and rough. The HMM
model also has highest deviation of gains in all three weather states,
indicating its adaptability to produce optimized speed profiles for more
efficient voyages.

(a) LSTM-based model

(b) KNN-based model

(c) 1NN-DTW-based model

(d) HMM-based model

Figure 6: Predicted speed profile for a test voyage. From four time-series
based models incorporate weather data as inputs and are trained by Top10Pr
cluster.

(a) LSTM-based model

(b) KNN-based model

(c) 1NN-DTW-based model

(d) HMM-based model

Figure 7: Sorted Eff-Scores based on their gain, for 162 test voyages. From
four time-series based models incorporate weather data as inputs and are
trained by Top10Pr cluster, $Eff._{\text{Gain}}$ as in Eq. 3.

We also present the results of our analysis through two key visualizations.
The first plot illustrates the predicted speed profiles for a test voyage,
which are generated by four time-series based models that incorporate weather
data as inputs. These models were trained using data from the Top10Pr cluster.
The second plot showcases the sorted Eff-Scores for 162 test voyages, based on
their efficiency gains. These models, similar to the first plot, also
integrate weather data as inputs and were trained by the Top10Pr cluster. The
efficiency gain is quantified using the formula as defined in Eq. 3.

In summary, the HMM-based model is the most effective model for improving
energy efficiency for a vessel voyage in short sea. The HMM model is able to
learn the complex relationships between the input features (e.g., speed and
weather) and the output feature (Eff-Score), even in different weather
conditions.

##  VI Conclusion

We have introduced a data-driven framework designed to optimize vessel speed
profiles and enhance the energy efficiency of voyages in short-sea shipping
(SSS). The framework integrates a data-driven approach, incorporating
explainable artificial intelligence (XAI) for modeling vessel energy, along
with a time-series analysis model for optimizing speed profiles.

Four time-series analysis models, namely LSTM, KNN, 1NN-DTW, and HMM, were
employed to analyze and evaluate the framework’s effectiveness using real-
world data. Our findings underscore the framework’s capability to
significantly improve vessel energy efficiency, especially when faced with
limited options as in SSS.

The results highlight the superiority of the framework with the HMM model over
LSTM, KNN and 1NN-DTW. Particularly, the HMM model demonstrates enhanced
performance when trained with data including voyages with diverse efficiency
levels and frequent exposure to different weather conditions.

In summary, the proposed framework, especially when incorporating the HMM
model, consistently yields efficiency gains, showcasing its adaptability in
generating optimized speed profiles tailored to varying environmental
conditions. This establishes the HMM model as a robust choice among time-
series analysis models, effectively improving energy efficiency in maritime
voyages facing challenges and constrained options, such scenarios are typical
in SSS.

## Acknowledgment

This research project is funded by Sweden’s innovation agency (Vinnova).
The authors wish to thank the diverse group at the Center for Applied
Intelligent Systems Research (CAISR), Halmstad University, for helpful
discussions.

[Supplementary Materials] The source codes that are implemented on Python
3.9.7 to produce the results are available at:
https://github.com/MohamedAbuella/TSA4EESSS The dataset is private and cannot
be shared due to the crucial commercial interests of the startup company
operating the iHelm system.

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|  Mohamed Abuella received his M.S. and PhD degrees in Electrical and
Computer Engineering from Southern Illinois University at Carbondale and
University of North Carolina at Charlotte, in 2012 and 2018 respectively. He
is a postdoctoral researcher at Halmstad University since 2022. His research
interests include energy analytics and AI for sustainability.
---|---
|  Hadi Fanaee-T is an Associate Professor at Halmstad University, Sweden. He
completed his PhD (with distinction) in Computer Science, under supervision of
Professor Joao Gama at the Faculty of Science of University of Porto, Portugal
in November 2015. He was the finalist in ERCIM Cor Baayen Young Researcher
Award 2017. Prior to this position he was a postdoctoral fellow at the
department of biostatistics, University of Oslo, Norway, and also worked as a
postdoctoral researcher in European FP7 Project ”MAESTRA” at INESC TEC
research institute, Portugal. His main research interests are
interdisciplinary applications of tensor decompositions, data fusion,
anomaly/event detection and spatiotemporal data mining. He is the first-author
of several journal and conference papers. He has served as a PC member to over
20 prestigious conferences (e.g. IJCAI, AAAI, ECML-PKDD, ISMIS, ACM SAC, IEEE
DSAA, etc.) and also reviewer to several high-impact journals (e.g. TDKE,
TKDD, DAMI, ML, KAIS, KBS, CSUR.
---|---
|  Slawomir Nowaczyk is a Professor in Machine Learning at the Center for
Applied Intelligent Systems Research, Halmstad University, Sweden. He received
his MSc degree from Poznan University of Technology in 2002 and his PhD from
the Lund University of Technology in 2008. During the last decades, his
research has focused on machine learning, knowledge representation, and self-
organising systems. The majority of his work concerns industrial data streams,
often with predictive maintenance as the goal. Given that accurate and
relevant labels are usually impossible to obtain in such settings, Slawomir’s
contributions primarily take advantage of proxy labels, such as transfer
learning and multi-task learning, or concern semi-supervised and unsupervised
modelling. He is a board member of the Swedish AI Society and a research
leader for the School of Information Technology at Halmstad University.
Slawomir has led multiple research projects on applying Artificial
Intelligence and Machine Learning in different domains, such as transport and
automotive, energy, smart cities, and healthcare. In most cases, this research
was done in collaboration with industry and public administration
organisations – inspired by practical challenges and leading to tangible
results and deployed solutions.
---|---
|  Simon Johansson is a MSc graduate of Chalmers University of Technology’s
program in Engineering Mathematics and Computational Science in 2020,
currently works in Cetasol, a marine company specialising in CO2 reduction and
energy optimisation for vessels. His practical application of computational
methods and dedication to environmental sustainability align with his role,
contributing to global efforts to mitigate climate change. Simon’s commitment
to advancing eco-friendly solutions in the marine industry reflects a seamless
integration of academic excellence and real-world impact.
---|---
|  Ethan Faghani is the CEO and founder of Cetasol. Before Cetasol, Ethan was
Chief Engineer of Automation and AI at Volvo Penta. Ethan has experience
working with cutting-edge technologies in other transportation segments in
both big enterprises and his own founded startup. Ethan obtained his Ph.D. in
mechatronics from UBC and Innovation and Entrepreneurship from Stanford
Business School.
---|---