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Towards Improved Prediction of Ship Performance: A Comparative Analysis on In-service Ship Monitoring Data for Modeling the Speed-Power Relation

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  • AP Moller Maersk
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Abstract and Figures

Accurate modeling of ship performance is crucial for the shipping industry to optimize fuel consumption and subsequently reduce emissions. However, predicting the speed-power relation in real-world conditions remains a challenge. In this study, we used in-service monitoring data from multiple vessels with different hull shapes to compare the accuracy of data-driven machine learning (ML) algorithms to traditional methods for assessing ship performance. Our analysis consists of two main parts: (1) a comparison of sea trial curves with calm-water curves fitted on operational data, and (2) a benchmark of multiple added wave resistance theories with an ML-based approach. Our results showed that a simple neural network outperformed established semi-empirical formulas following first principles. The neural network only required operational data as input, while the traditional methods required extensive ship particulars that are often unavailable. These findings suggest that data-driven algorithms may be more effective for predicting ship performance in practical applications.
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arXiv Preprint December 27, 2022
Towards Improved Prediction of Ship Performance:
A Comparative Analysis on In-service Ship
Monitoring Data for Modeling the Speed–Power
Relation
Simon DeKeyser,Casimir Morob´
e, Malte Mittendorf
Toqua Ghent, Belgium
Technical University of Denmark Kgs. Lyngby, Denmark
Abstract—Accurate modeling of ship performance is crucial for the shipping industry to optimize fuel consumption and
subsequently reduce emissions. However, predicting the speed-power relation in real-world conditions remains a challenge. In
this study, we used in-service monitoring data from multiple vessels with different hull shapes to compare the accuracy of
data-driven machine learning (ML) algorithms to traditional methods for assessing ship performance. Our analysis consists of
two main parts: (1) a comparison of sea trial curves with calm-water curves fitted on operational data, and (2) a benchmark of
multiple added wave resistance theories with an ML-based approach. Our results showed that a simple neural network
outperformed established semi-empirical formulas following first principles. The neural network only required operational data
as input, while the traditional methods required extensive ship particulars that are often unavailable. These findings suggest that
data-driven algorithms may be more effective for predicting ship performance in practical applications.
Index Terms—Ship Performance, Speed-Power, Added Wave Resistance, Resistance model, Neural Networks
F
1 INTRODUCTION
A
CCURATE ship performance modeling is a
powerful decision-making instrument for
the shipping industry, offering the ability to save
fuel and reduce emissions (Morob
´
e, 2022), e.g.,
through optimized hull maintenance and voyage
optimization. In the last decades, a spectrum
of strategies has been developed, ranging from
simplified empirical formulas derived from
model tests to advanced full-scale 3-D numerical
simulations (Tezdogan et al., 2015). However,
considering both accuracy and computational
effort, Liu and Papanikolaou (2020) argue that
for practical applications where only a limited
amount of ship parameters is available, a semi-
empirical formula seems to be the most efficient
method that captures the underlying physics of
the problem. Unfortunately, the development of
such methods leads to a loss of accuracy, which
we classify into three categories.
(1)
Approximations are made to generalize
the actual hull shape to an efficient number of
parameters, i.e., main particulars. The hull form
and its displacement distribution significantly
impact the magnitude of the wave resistance in
calm water. Nevertheless, incorporating the full
hull geometry into a semi-empirical model would
reduce its practical advantage over full-scale
numerical simulations.
(2)
Assumptions about the range of validity
need to be made to enforce regimes where certain
physical relations hold. Extrapolating results
outside particular validity ranges may result in
poor accuracy and non-physical transitions.
(3)
The theoretical expressions are fitted on
experimental and numerical data obtained from
controlled environments, which are far from
Corresponding authors: simon@toqua.ai,casimir@toqua.ai 1
arXiv:2212.13061v1 [cs.LG] 26 Dec 2022
reality. Most of the experiments are performed
on scaled models in towing tanks, causing an
increase in errors when scaling the measurements
back to full size. The problem lies in a lack of
Reynolds number similarity between the ship and
the model as only Froude number similarity is
obtained. There, further assumptions are made on,
e.g., hull smoothness and weather conditions, as
imitating every possible ship/environment would
be infeasible. Hence, we face data scarcity when
assessing the in-service ship performance.
The overarching goal of this article is to
show that machine learning (ML) approaches
may complement or possibly surpass traditional
methods in the assessment of in-service ship
performance. ML is a subfield of artificial
intelligence that involves the use of algorithms
and statistical models to enable a system to learn
from data and improve its performance on a
specific task. It consists of a set of computational
techniques to discover patterns and relationships
in data and make predictions or decisions
based on those patterns. ML algorithms can be
thought of as an extension of statistical regression
analysis, which is a technique used to model
the relationship between a dependent variable
(the outcome or response of interest) and one or
more independent variables (the predictors or
explanatory variables).
Whereas regression analysis typically involves
fitting a fixed model to data and then using the
model to make predictions or inferences about the
data, ML algorithms can be trained on data and
adapt as new data becomes available, allowing the
system to learn and improve over time without
being explicitly programmed for each task.
In the past, naval engineers have pursued
finding universal semi-empirical formulas that
rely on ship-specific inputs by performing fits
(regression analysis) on experimental data to
model ship performance. Instead, we propose
training ship-specific ML models on real-world
data while ensuring physical relevance without
needing ship particulars. Nowadays, a lot of
in-service sensor data is acquired, allowing us
to leverage the scalability of ML and accurately
model at once all of the ship-specific intricacies
that semi-empirical formulas cannot.
ML is a natural progression for the shipping
industry’s practical purposes, as it allows us to
predict unobserved outcomes in high-dimensional
parameter spaces. In fact, several studies have
examined the use of ML in the maritime
industry, such as fuel consumption prediction
and optimization for Diesel engines (Parlak et al.,
2006), condition-based maintenance of naval
propulsion plants (Coraddu, Oneto, Ghio, et al.,
2016), and modeling marine fouling speed loss
(Coraddu, Oneto, Baldi, et al., 2019; Gupta et al.,
2022).
In this work, the focus lies on predicting the power
delivered by the engine as a function of the ship’s
speed. This so-called speed-power relationship
forms the basis of ship performance modeling as
it is the gateway to crucial economic parameters
such as fuel consumption. An assessment of
the current state of ship performance modeling
research is made by comparing the accuracy of
older, simpler theories with state-of-the-art (semi-
)empirical formulas on in-service monitoring data.
Herein, engine power sensor measurements will
form our ground truth. We follow this approach
to enable the extension to a supervised learning
problem, where a learning algorithm is trained on
observed data (ship speed, weather, and loading
conditions) to predict specific targets (engine
power) in unobserved situations.
2 METHODOLOGY
In the following sections, we use the ship resis-
tance model according to the ISO 19030 (2019)
industry standard to model the speed-power rela-
tionship. This model splits the resistance experi-
enced by a ship into different parts:
R=Rcalm +RAA +RAW +RAH +Rothers (1)
where
Rcalm
is the calm-water resistance,
RAA
and
RAW
respectively are the added resistances due
to wind and waves,
RAH
is the added resistance
due to changes in hull condition such as the
2
accumulation of marine growth on hull and
propeller, and
Rothers
combines the effect of all
other contributions such as steering and shallow
water resistance.
Several (semi-)empirical techniques are elaborated
on and critically examined for the different
resistance contributions in section 3. To simplify
our analysis, we will discuss the calculation of the
first three resistance terms, which constitute the
most considerable contribution to
R
(Dalheim and
Steen, 2020). The other terms require knowledge
of ship-specific and environmental parameters,
which are often unavailable in real-world
situations.
In this study, we select neural networks (NNs) to
prove the value of ML algorithms in predicting
ship performance. We chose NNs for their
simplicity and effectiveness in demonstrating
the potential of ML without adding unnecessary
complexity to the comparison. In section 4, we
provide a brief overview of the NN architecture,
pre-processing techniques tailored for in-service
vessel data, and the training and evaluation
process.
Section 5 discusses a benchmark of different
RAW
estimation procedures evaluated on in-service
ship monitoring data to highlight the uncertainties
of using experimentally fitted formulas for
practical purposes. The benchmark, focusing on
the resistance due to calm water, wind, and waves,
will provide a clear and functional assessment of
the accuracy of the (semi-)empirical techniques
discussed in section 3.
The goodness of fit between the actual
Pi
and
predicted
ˆ
Pi
engine powers (for the observations
i= 1, ..., N
) is assessed with several metrics. We
use the well-known mean absolute error (MAE):
MAE =1
N
N
X
i=1 Piˆ
Pi(2)
, the mean absolute percentage error (MAPE):
MAPE =1
N
N
X
i=1
Piˆ
Pi
ˆ
Pi
(3)
TABLE 1: Input and target variables of the in-service dataset.
Category Variables
Draft aft, Draft forward, Displacement
Wind direction, Wind speed
Input Wave direction, Significant wave height, Wave period
Current speed, Current direction
Speed-through-water, Ship heading
Target Brake power
, the mean bias error (MBE):
MBE =1
N
N
X
i=1 ˆ
PiPi(4)
, and the R-squared score (R2):
R2 = 1
N
X
i=1
(Piˆ
Pi)2
(Pi¯
P)2(5)
with ¯
P=1
NPN
i=1 Pithe mean observed power.
The in-service dataset combines high-frequency
sensor measurements of ship speed and main
engine brake power
a
with weather data and
loading conditions. Table 1 summarizes the
variables into input and target categories. The
data is filtered to steady state voyages in full sea
conditions, where ship accelerations, shallow
water effects, and steering maneuvers are
eliminated.
To further isolate our analysis to the
Rcalm
,
RAA
and
RAW
resistance terms, measurements are
taken from periods after cleaning events, such that
RAH
can be disregarded. Note that the relative
contribution of
RAH
to
R
will increase over time
and for practical use cases, accurately modeling
RAH
is crucial. Although it remains a challenging
task to solve using empirical models (Demirel
et al., 2017; Guo et al., 2022), recent research has
shown that digital twin ML-based methods may
bring solace (Coraddu, Oneto, Baldi, et al., 2019;
Gupta et al., 2022).
Finally, a comparison is made with a simple
ship-specific NN model, which incorporates all
a.
The brake power of a ship’s engine is the power output
of the engine measured at the engine’s crankshaft, before any
transmission losses. It can be calculated by measuring the torque
and angular speed of the crankshaft.
3
of the resistance contributions while requiring
only operational data as input. Here, we also
consider some limitations of using ML for ship
performance modeling. Section 6 will summarize
our findings and use them to propose ML as the
natural next step towards improved prediction of
ship performance.
3 SHIP RESISTANCE MODEL
The brake power
PB
of the ship’s main engine is
calculated according to:
PB=RVS
ηDηM
(6)
with
VS
the ship’s speed-through-water (STW),
ηD
the propulsive efficiency,
ηM
the mechanical (shaft
+ gearbox) efficiency, and
R
the total resistance.
As with
RAH
, empirically modeling the propulsive
and mechanical efficiencies with limited informa-
tion remains difficult (Shigunov, 2017).
In particular,
ηD
depends on the wake fraction,
thrust deduction, propeller diameter, and even the
total resistance experienced by the hull. All these
factors but the propeller diameter are speed and
seaway dependent. Several curves and empirical
models have been proposed to estimate
ηD
(Breslin
and Andersen, 1995; Holtrop and Mennen, 1982;
Kristensen and L
¨
utzen, 2012). However, these are
only valid for calm water, and there is a lack of
data in waves. Therefore, we use the default values
recommended by ISO19030.
3.1 Calm-Water Resistance
A direct method for calculating a ship’s calm-
water resistance is to conduct a sea trial, during
which the speed-power relation is measured
under calm weather conditions. Usually, these
measurements are performed around the design
speed of the vessel. In recent years, however,
operational velocities have been lowered to reduce
emissions (Psaraftis and Kontovas, 2014), meaning
that operating speeds do not always match the
design speeds anymore.
It is widely accepted that the correlation between
speed and power follows a relation
PVc
with
constant exponent
c3
, which holds around
the design speed (MAN Energy Solutions, 2018).
However, previous studies have found that the
cubic law underestimates power at speeds below
the design speed (Adland et al., 2020; Taskar and
Andersen, 2020; Tillig et al., 2018). These findings
suggest that simply extrapolating sea trial curves
to operational speeds lower than the design speed
can lead to underestimations of power.
In this work, we will omit three of the main issues
associated with the use of sea trial curves to
estimate Rcalm:
Sea trial curve availability
Operational speed 6=design speed
Time-dependent performance loss (e.g.,
fouling)
by using the data-driven method proposed by
Berthelsen and Nielsen (2021). This method fits
a calm-water curve using in-service data. Their
model starts from a simple least-squares fit of
(x1, x2) to a log-linearized power law:
P=x1Vx2(7)
ln(P) = ln(x1) + x2ln(V)(8)
They extend the regression model to incorporate
the ship’s draft T:
ln(P) = ln(x1) + x2ln(V) + x3T+x4ln(V)T(9)
and make the exponent speed dependent by
introducing breakpoints that separate different
speed intervals where the exponent remains con-
stant. With one breakpoint
Bp
, the regression is
performed on:
ln(P) = ln(x1) + x2ln(V) + x3T+x4ln(V)T
+x5(ln(V)ln(Bp))Vd
(10)
In their paper,
Vd
is a Heaviside function cen-
tered at
Bp
. We, however, propose a differentiable
dummy index to make the speed-power relation
smooth:
Vd=1
21 + tanh VBp
δ (11)
with δthe smoothing factor.
In practice, the breakpoints are detected with a
binary segmentation algorithm, as implemented
in the
ruptures
Python library (Truong et al.,
4
2020). The algorithm is applied to the measured
power of the speed-sorted data, and it finds
change points in the signal where the slope of
speed and power alters. At last, the regression
is performed on the power data, which is first
corrected from weather conditions (see next
sections) to ensure that we fit calm-water data. As
we do not correct the fouling power loss, a part of
Rfouling
will be included in the fitted calm-water
curves.
3.2 Added Resistance due to Wind
The industry standard for calculating
RAA
is: (ISO,
2015)
RAA =1
2ρAAXV CAA (θrel )V2
wrel 1
2ρAAXV CAA (0)V2
G
(12)
with
ρA
the air density,
Vwrel
the relative wind
speed calculated according to ISO 15016,
VG
the
ship’s measured speed over ground,
AXV
the
transverse projected area of the ship above the
water line,
CAA
the wind resistance coefficient
and
θrel
the relative wind direction. The negative
term is the air resistance due to the ship moving
forward with a headwind caused by
VG
(
θrel = 0
),
which is already included in Rcalm.
CAA
is estimated with the regression formula
based on wind tunnel tests developed by Fujiwara
(2006):
CAA(θrel ) = CLF cos(θrel)+
CXLI sin(θrel)1
2sin(θrel) cos3(θrel)+
CALF sin(θrel) cos3(θrel)
(13)
where the coefficients
CLF
,
CXLI
, and
CALF
consist
of different regression expressions for
θrel <90
and
θrel >90
. Unfortunately, the latter coefficients
depend on several detailed ship geometry-related
parameters, such as the bridge height and longitu-
dinal projected area of superstructures, which are
usually unknown.
However, Kitamura et al. (2017) developed ship-
type-specific regression formulas that estimate the
input parameters
P
of Fujiwara’s formula and
AXV
from the ship’s overall length
LOA
and beam
B:
P
P/LOA
P/B
P/L2
OA
P/(LOAB)
P/B2
=
aB +bLOA +c
aB +c
bLOA +c
(14)
with (
a
,
b
,
c
) the regression coefficients and where
the left-hand-side and right-hand-side expressions
were carefully chosen for each specific parameter
to maximize the accuracy. In the following, this
approach is used to calculate the inputs for
Fujiwara’s expression, and while it introduces
additional approximations, it is necessary to keep
the required parameters at a feasible level.
Fig. 1 shows
RAA
calculated according to
this method for a bulk carrier (
LOA = 190
m,
B= 32
m) with
Vwrel = 8
m/s and
VG=VS= 13
kn, where a distinction is made between laden
and ballast loading conditions. The absolute value
of
RAA
is higher for ballast conditions, which
is expected as the area above water is larger.
Also, the added resistance is more considerable
for headwinds and drops to negative values
for following winds, meaning that the ship is
being pushed forward. The reader may find it
counter-intuitive that
RAA = 0
kN occurs at angles
of only
±40
, but it is stressed that we plot the
added resistance relative to the ship’s headwind
(Eq. 12).
3.3 Added Resistance due to Waves
Modeling a ship’s added resistance due to waves
is a highly complex, non-linear problem, and
most methods rely on simplified assumptions. In
practice,
RAW
is defined as “the unsteady longitu-
dinal force a ship experiences apart from the calm
water, and wind resistances in a realistic seaway”
(Mittendorf, Nielsen, and Bingham, 2022).
The force is a second-order quantity, which de-
pends on the incident wave’s amplitude and speed
(Liu, Papanikolaou, and Zaraphonitis, 2011). One
5
30°
60°
90°
120°
150°
180°
30
20
10
0
RAA
[kN]
Laden
Ballast
Fig. 1:
Polar plot of the added resistance due to wind as a function
of the relative wind direction, calculated with Fujiwara’s and
Kitamura’s regression formulas for different ship load conditions.
of the oldest and simplest modeling approaches is
that of Kreitner, used in ITTC2005 (2005):
RAW = 0.64gH2
SCBρw
B2
Lpp
(15)
with
g
the gravitational constant,
HS
the significant
wave height,
CB
the block coefficient,
ρw
the water
density, and
Lpp
the length between perpendicu-
lars. As this expression only holds for head waves,
a cosine law can be used to account for waves with
an arbitrary heading (Hansen et al., 2011):
RAW = 0.64gH2
SCBρw
B2
3LOA
(2 + cos(αrel)) (16)
with αrel the relative wave direction.b
Another simple, but more recent standard of ITTC
2014 (ITTC, 2014), is the empirical STAwave-1
formula:
RAW =1
16gH 2
SρwBrB
LB
|αrel| 45(17)
b.
We use
αrel = 0
for head waves, and
αrel = 180
for
following waves.
where
LB
is the length of the bow at the waterline.
More advanced (semi-)empirical approaches split
the added wave resistance into two contributions:
RAW =RAW M +RAW R (18)
where the motion induced
RAW M
and wave re-
flection
RAW R
resistances are a function of the
wave frequency
ω
. To accurately model the ship’s
resistance in irregular sea conditions with waves
of varying frequencies, integration is performed
over a wave spectrum
S(ω)
to calculate the mean
added resistance:
RAW = 2 Z
0
S(ω)RAW (ω)
ζ2
a
(19)
with
ζa
the wave amplitude. Hereby, we assume
that the superposition principle holds and that the
calculation is valid for long-crested waves. Often,
a Pierson-Moskowitz type spectrum is used in
fully developed sea states:
S(ω) = 5
16H2
S
ω4
p
ω5exp 5
4ωp
ω4(20)
where
ωp= 2π/Tp
and
Tp
is the wave peak period.
Using this frequency response framework,
ITTC2014 (2014) and ISO 15016 (2015) recommend
using the semi-empirical method called STAwave-
2, which again holds for |αrel| 45.
From 2016 to 2020, Liu and Papanikolaou
(2016;2020) derived an improved formula with
regression analysis on model test data, which
holds for arbitrary wave headings. In 2022,
Mittendorf et al. (2022) enhanced their formula
by performing multivariate regression on the
parameter vector with model test data, thereby
also including the 90 % prediction interval of
RAW
. We note that a discontinuity is present in
the Liu/Mittendorf formulas, which we found
to be caused by the non-linear form factor in the
expression for RAW R:
0.87
CB(1+4F r)f(αrel )
(21)
with F r the Froude number and:
f(αrel) = (cos(αrel)πE1αr el π
0αrel < π E1
(22)
6
30°
60°
90°
120°
150°
180°
0
100
200
300
400
RAW
[kN]
Kreitner*
STAwave-1
STAwave-2
Papanikolaou
Mittendorf
Fig. 2:
Polar plot of the added resistance due to waves as a function
of the relative wave direction, calculated with different wave
resistance theories in laden condition.
where
E1
is an angle that defines where the ship’s
bow ends. The discontinuity can be solved by
shifting the cosine of
f(αrel)
to 0 when
αrel =
πE1, while still forcing f(αrel)=1for αrel =π:
f(αrel) = (cos(αrel)+1
cos(πE1)+1 + 1 πE1αrel π
0αrel < π E1
(23)
We will use this new form of
f(α)
in the
RAW R
expression of the Liu and Mittendorf methods, as
discontinuities will not be present in the measured
data.
To give the reader a visual understanding
of the different wave resistance formulas, Fig. 2
plots
RAW
calculated with the aforementioned
methods for a bulk carrier (
LOA = 190
m,
B= 32
m,
CB= 0.7
) with
HS= 4
m,
Tp= 10
s and
VG=VS= 13
kn in laden condition. Given
Fig. 2, it is stated that while the theories’
RAW
predictions are in the same order of magnitude,
their shapes vary differently as a function of
αrel
.
Although the discontinuity has been resolved,
the semi-empirical Liu and Mittendorf methods
still exhibit non-differentiable irregularities where
the resistance increases or decreases. The reason
is that their formula consists of expressions for
the bow and stern of the ship, including form
factors to incorporate non-linear behavior. Of
course, in natural seas, the waves will come
from multiple directions at once, and Eq. 19
can be extended to a double integration over
an angular wave spectrum. Accounting for
short-crested waves will smooth the
RAW (αrel)
curves to behave as one would expect physically.
Unfortunately, performing a two-dimensional
numerical integration comes at the cost of
computational time.
Another observation is that the more advanced
methods predict a larger wave resistance for
oblique waves compared to head waves. The latter
effect is confirmed in model tests, where oblique
waves cause more pitch motions dissipating
energy (Valanto and Hong, 2015). The magnitude
of the wave angle-dependent Kreitner’s formula
remains similar for the different wave headings
and lies somewhere between the STAwave and
the Mittendorf methods.
Lastly, to highlight the importance of the
peak wave period on the frequency response-type
methods, Fig. 3 shows a surface plot of
RAW
as a function of
Tp
and the wave heading
αrel
calculated with the Mittendorf method. The same
values were used as in Fig. 2 for
B
,
HS
, etc. and
VG
=
VS
= 0 kn. The ship experiences the highest
added resistance in sea states with short waves
that arrive obliquely to the ship’s heading.
4 ANML-BASED APPROACH
Neural networks (NNs) are ML models inspired
by the structure and function of the human brain.
They are composed of interconnected processing
nodes, called neurons, which work together to
recognize patterns in data and make predictions.
Here, we will use one of the most straightforward
NN architectures, the feedforward NN (FNN).
This section will briefly explain the architecture of
the FNN, the data pre-processing steps, and how
7
Tp
[s]
510 15 20 25
Wave Heading [°]
0
50
100
150
RAW
[kN]
150
100
50
0
50
100
150
Fig. 3:
Surface plot of
RAW
as a function of
Tp
and the wave
heading
αrel
calculated with the Mittendorf method in laden
condition.
a standard training procedure is performed and
validated.
4.1 The Feedforward Architecture
An FNN is organized into three main parts: the
input layer, the hidden layers, and the output
layer. The input layer receives input data, the
hidden layers process that data and the output
layer yields the final output of the network. Each
layer consists of several neurons, which receive
input from the previous layer and produce the
output that is conveyed to the next layer.
Mathematically, a neuron can be represented as a
function that takes in a set of inputs,
x1, x2, ..., xn
,
and produces a single output,
y
. The inputs
are multiplied by a set of weights,
w1, w2, ..., wn
,
and summed with a bias term,
b
, to produce
an intermediate value,
z
, which is then passed
through a non-linear activation function,
f
, to
produce the output:
z=
n
X
i=1
wixi+b(24)
y=f(z)(25)
A commonly used activation function is the recti-
fied linear unit (ReLU):
f(z) = max(0, z)(26)
which has the disadvantage of being non-
differentiable at
z= 0
. In this work, we use
a smooth approximation of the ReLU function,
namely the softplus activation function, defined
as:
f(z) = log(1 + ez)(27)
The main premise of activation functions is to
replicate the behavior of brain neurons, which
fire when the electrical impulses coming from
connections with other neurons reach a certain
threshold (at z= 0 here).
4.2 Feature Engineering
The input features are the variables presented
to the network’s input layer. The process of
selecting and transforming the input features to
improve the performance of a NN is called feature
engineering. It involves identifying the most
relevant features in the dataset and pre-processing
them in a way that makes them more suitable for
the NN to learn from.
In this work, we deliberately reduce the
NN’s complexity by using as few features as
possible without losing predictive power by
combining different input variables from table 1.
Table 2 summarizes the engineered features,
which mimic theoretical formulas of wave and
wind resistance. E.g., the Wind Product feature
resembles the form of Eq. 12, while the Wave
Power mimics the power per unit wave crest in
deep water:
P=ρwg
64πH2
STe(28)
with Tethe wave energy period.
Although power is not a vector quantity, we
separate all directional variables into their
longitudinal (long.) and transversal (trans.)
direction to the vessel’s heading. Doing so results
in a correct angular dependence of model w.r.t.
the wind direction
θrel
and wave direction
αrel
relative to the vessel’s heading.
8
The final FNN architecture consists of an
input layer with six features, four fully-connected
hidden layers (64, 32, 16, and 8 neurons,
respectively), and an output layer that predicts
the brake power.
4.3 Training and Validation
During training, the weights and biases of
the neurons are adjusted to minimize a loss
function that measures the difference between the
predicted output and the true output (e.g., with
the MAE (Eq. 2) or mean squared error (MSE)).
The optimization is generally conducted with
algorithms like stochastic gradient descent, which
calculates the gradient of the loss function with
respect to the weights and biases and updates the
weights and biases in the direction that reduces
the loss.
In this study, we employed ADAM (Kingma
and Ba, 2014), a frequently used variation of
gradient descent that incorporates momentum
(Rumelhart et al., 1985) to accelerate convergence
to the minimum of the loss function in the
high-dimensional space of weights and biases.
To assess the performance of the NN, we
use k-fold cross-validation. Here, the data is
divided into k folds, and the model is trained on
k-1 of those folds while leaving the remaining
fold as a validation set. This process is repeated
k times, with a different fold being used as the
validation set each time. The final performance
of the model is then calculated as the average
performance across all k iterations. This approach
ensures that the NN is tested on completely
unseen data, giving us a reliable representation
of how the model would perform in unobserved
real-world conditions.
TABLE 2: Features used as input of the NN.
Feature Formula
STW
Draft Average (Taft +Tfw d)/2
Wind Product (long.) V2
wrel cos(θrel )
Wind Product (trans.) V2
wrel sin(θrel )
Wave Power (long.) H2
STpcos(αrel)
Wave Power (trans.) H2
STpsin(αrel)
5 DISCUSSION
In the following, we will use in-service data
of different vessels to compare the performance
of the resistance models described above and
extend the analysis to include a simple NN. Before
proceeding, we will first justify the use of a speed-
dependent exponent in the calm-water speed-
power relation.
5.1 Calm-Water Resistance
Sea trial curves of a tanker were extrapolated to
lower speeds, and weather corrections
RAA
,
RAW
were calculated with Fujiwara’s and Mittendorf’s
semi-empirical formulas, respectively. The main
engine power is then obtained with Eq. 6, ignoring
RAH
and
Rothers
. Fig. 4 shows the predicted and
measured power as a function of STW, along
with the extrapolated laden and ballast sea
trial curves. Predicted points
ˆ
P
are colored
based on their absolute percentage error (APE)
=
|ˆ
PP|/P
with respect to the measured values
P
.
The scatter plot in Fig. 4 supports our
earlier hypothesis that extrapolating sea trial
curves below the design speed leads to an
underestimation of the power, indicating that
the cubic law does not hold in this range.
Additionally, time-dependent performance losses
such as fouling can also cause an underestimation
of the power, which increases over time.
To address the limitations of the extrapolated sea
trial curves, we perform a least-squares fit of Eq.
10 on calm-water data according to the method
by Berthelsen and Nielsen. The calm-water data
is obtained by correcting the
HS1
m in-service
power data for wind and waves with Fujiwara’s
and Mittendorf’s formulas.
c
First, the binary
segmentation algorithm was used to find the
breakpoint at
VS= 11.53
kn. Fig. 5 shows the
results in the same way as before, but with
the fitted calm-water curves. The speed-power
exponent changes (smoothly (
δ= 0.5
)) at the
breakpoint from
1.8
to
2.8
for the laden curve.
Table 3 compares the accuracy metrics obtained
c. A histogram of HScan be found in Fig. 10.
9
Fig. 4:
Scatter plot of the measured and predicted main engine
power (normalized w.r.t maximum) as a function of STW, together
with the extrapolated laden and ballast sea trial curves. (Log-scaled
color map.)
with the sea trial curves and the fitted calm-water
model. Our analysis shows that the data-driven
calm-water method significantly improves the
accuracy compared to extrapolated sea trial
curves in terms of R2, MAPE, and MBE. This
suggests that the data-driven calm-water model
is a more reliable approach for predicting power
requirements in realistic operating conditions.
However, as shown in Fig. 5, the fitted calm-water
model still tends to underestimate the power
(negative MBE) at low STW. This error is likely
due to the presence of bands in the speed-power
data, where the vessel encounters heavy weather
while operating in constant power or constant
shaft rpm autopilot, resulting in large power
values at low STW.
5.2 Added Wave Resistance Benchmark
Up to this point, we have only used Mittendorf’s
formula to calculate the power corrections due
to waves. In the following analysis, we perform
TABLE 3:
Accuracy metrics evaluated on the predictions from
different calm-water models in combination with Mittendorf’s and
Fujiwara’s resistance corrections.
Calm-Water Model R2 MAPE [%] MBE [kW]
Extrapolated Sea Trial Curve 0.72 15 -1522
Fitted Calm-Water Curve 0.89 7.1 -443
Fig. 5:
Scatter plot of the measured and predicted main engine
power (normalized w.r.t maximum) as a function of STW, together
with the fitted calm-water laden and ballast curves. (Log-scaled
color map.)
a benchmark of different added wave resistance
theories to evaluate their performance, along with
sea trials and the fitted calm-water regression
model, using data from both an oil tanker and
a dry cargo carrier. The method is similar to
that used to produce Figs. 4 and 5, except that
different wave theories are used in combination
with Fujiwara’s
RAA
. Fig. 6 shows the results,
including the MAPE scores for each vessel and
wave resistance theory, along with a comparison
of the fitted calm-water model to sea trials.
Interestingly, we find that the accuracy of
all the added resistance theories is very similar,
despite significant differences in their complexity.
One possible explanation for this finding is that
the more advanced approaches are too complex
for noisy data. When examining Fig. 2, we see
that the Liu theory exhibits abrupt changes (non-
differentiable points) in added wave resistance,
which are not physically realistic. Based on these
results, one might even consider using the wave
angle-dependent Kreitner’s formula, which has
the advantage of requiring fewer ship-specific
parameters and can be evaluated approximately
ten times faster than Mittendorf’s method (
0.1
ms versus 1ms).
10
Oil tanker Dry cargo carrier
0
5
10
15
20
25
30
MAPE [%]
13.83 14.53
8.46 8.13
9.69 9.40
9.73 8.62
8.45 8.60
7.40
10.39
Sea trial curve:
No corrections
Kreitner*
STAwave-1
STAwave-2
Liu & Papanikolaou
Mittendorf et al.
Fitted calm-water curve:
No corrections
Kreitner*
STAwave-1
STAwave-2
Liu & Papanikolaou
Mittendorf et al.
Fig. 6:
MAPE obtained with the different combinations of wave resistance theories and calm-water models evaluated on multiple vessels.
5.3 A Simple NN
In this section, we evaluate the performance of
the simple NN by comparing it with the fitted
calm-water model using different added wave
resistance theories on the in-service data of a
chemical tanker.
d
In contrast to the previous
vessels, including sea trial curves in the analysis is
not possible as they are unavailable. This further
highlights the benefit of using the data-driven
calm-water model by Berthelsen and Nielsen.
Fig. 7 compares the MAPE score achieved
with the fitted calm-water curve using different
wave resistance theories and the simple NN.
The Kreitner and STAwave-2 theories perform
worse than the model without weather corrections
due to multiple data points falling outside the
valid ranges for these empirical formulas. The
simple NN achieves a MAPE of 7.4 %, which
represents a roughly 3 % improvement over
the best (semi-)empirical formula. This is a
remarkable result considering the simplicity of
our NN architecture. It is expected that further
increasing the complexity and effort in developing
the NN architecture would significantly improve
the accuracy.
To further analyze the performance of these
d.
The FNN was trained using mini-batches of 32 samples with
a learning rate of 0.015 for five epochs on each k-fold.
models, we can perform an error analysis using
binned scatter plots. In these plots, the data points
are grouped into bins according to a variable of
interest. Then, the MAPE is calculated for each
bin, which allows us to visualize the relative
accuracy of the model as a function of the variable
of interest, providing insight into the factors that
may influence the model’s performance.
Figs. 8 and 9 show a binned scatter plot
with respect to the significant wave height and
wind speed, respectively. Here, we compare
the fitted calm-water model using Mittendorf’s
Chemical tanker
0
5
10
15
20
25
MAPE [%]
16.36 16.87
10.13
26.58
12.73
10.34
Simple NN
Fitted calm-water curve:
No corrections
Kreitner*
STAwave-1
STAwave-2
Liu & Papanikolaou
Mittendorf et al.
Fig. 7:
MAPE obtained with the different combinations of wave
resistance theories and the fitted calm-water model. A comparison
is made with a simple NN.
11
added wave resistance corrections with the simple
NN from Fig. 7. The histograms show the data
spread in the variable of interest, and the error
bars denote the standard deviation of the binned
MAPE. From both plots, we conclude that the NN
achieves a better accuracy globally, and thus it
generalizes better than the theoretical model.
Taking the significant wave height as the
variable of interest, the theoretical model performs
particularly poorly in bad weather conditions,
with a binned MAPE reaching over 70 %. This
effect is not as noticeable when evaluating the
overall MAPE because the data spread gives
increased weight to low wave heights. Similarly,
using the relative wind speed as the variable of
interest yields similar results for the NN, while
the theoretical model performs worse for a wide
range of wind speeds with a binned MAPE of up
to 45 %.
It is notable that, despite having less data
available for high wave heights and wind speeds,
the NN’s accuracy remains approximately in a
similar range. This finding is remarkable because
NNs generally require considerable amounts of
01234
Wave Height [m]
0
25
50
75
100
Binned MAPE [%]
Simple NN
Fitted calm-water + Mittendorf
101
102
Count
Fig. 8:
MAPE binned w.r.t. the wave height, evaluated on a chemical
tanker. A comparison is made between the fitted calm-water model
in combination with Mittendorf’s wave resistance and a simple
NN. The histogram shows the data spread of the wave height.
training data to learn and generalize effectively.
The fact that the NN can maintain a similar level
of accuracy with fewer data suggests that our
choices of feature engineering were effective.
5.4 Machine Learning as the Next Step
Our findings above demonstrate the potential of
ML methods to improve the prediction of ship
performance in various operating conditions
compared to traditional methods. However, ML
has previously been proposed as a valuable tool
for shipping companies. Alexiou et al. (2022) have
compared a number of the latest data-driven
models used in papers to model the speed-power
relation and concluded that data-driven models
were more suitable for shipping companies,
provided sufficient representative historical data
was available.
The key aspect of this work lies in the direct
comparison of the data-driven methods and
multiple theoretical models on several vessels
with different design types. However, before we
propose ML as the natural next step for ship
performance modeling, we must carefully weigh
0 5 10 15 20
Wind Speed [m/s]
0
20
40
60
Binned MAPE [%]
Simple NN
Fitted calm-water + Mittendorf
101
102
Count
Fig. 9:
MAPE binned w.r.t. the wind speed, evaluated on a chemical
tanker. A comparison is made between the fitted calm-water
model using Mittendorf’s wave resistance and a simple NN. The
histogram shows the data spread of the wind speed.
12
the benefits against the limitations.
One of the main benefits of ML models is
that they can be trained on existing data from a
ship’s performance, which means that they can be
customized to a specific vessel and its operating
conditions. This can provide more accurate and
relevant results than using traditional methods.
Another advantage of ML is that it can automate
some of the more tedious and time-consuming
aspects of ship performance modeling, freeing
engineers to focus on other tasks. Machine
learning algorithms can also learn from data in
real-time, which means that they can adapt to
changes in a ship’s operating conditions and
provide updated predictions on an ongoing basis.
In contrast, one of the main constraints is
the need for substantial amounts of high-quality
data to train the models. This may be challenging
for some ships or operating conditions where data
may be limited or difficult to collect (e.g., only
noon report data available or sensor drift).
Additionally, ML models can be difficult to
interpret and explain, making it challenging
for engineers to apply the results to their work.
This issue can be addressed through the use of
explainable artificial intelligence (XAI) techniques
(Gunning et al., 2019) and physics-informed
ML (PIML) (Karniadakis et al., 2021). XAI helps
to make ML algorithms more transparent and
interpretable, while PIML allows ML methods to
follow physical relations.
Furthermore, ML algorithms can be subject
to bias if the training data is not representative
of the full range of conditions that a ship
may encounter. This can lead to inaccurate or
misleading results, which can be challenging
to detect and correct. However, careful feature
engineering and the use of PIML can help to
ensure that the ML algorithm follows physical
relationships beyond the range of the training
data and is not biased by the data used to train
it. This can help to improve the robustness and
generalizability of the ML model (Vanackere et al.,
2022).
Overall, while the use of ML in ship performance
modeling offers many potential benefits, there are
also significant limitations that one must consider.
However, as the availability of data increases
and the expertise in ML continues to grow, these
restrictions are becoming easier to overcome.
6 CONCLUSION
We have outlined several (semi-)empirical
approaches developed in the last decades of
ship hydrodynamics. Most of these methods use
theoretical approximations, after which they are
fitted to experimental data. Researchers have
been using increasingly advanced regression
algorithms to enhance the experimental fits of
their semi-empirical formulas.
Although their findings are crucial to obtaining
a better understanding of ship performance in a
research-based way, the developed methods might
not be the best choice in practical applications as
they require extensive ship particulars that are
often unavailable.
Our benchmark of different added wave
resistance theories, performed on in-service
data from multiple vessels, supports this idea.
Results showed that simple theories had similar
accuracy to more complex methods, and that
a data-driven calm-water resistance regression
method proposed by Berthelsen and Nielsen
(2021) was more useful than extrapolated sea trial
curves for performance monitoring.
A natural next step to further improve accuracy
and reduce the need for empirical methodologies
and ship particulars, may be to leverage the data
scalability of ML techniques. By applying ML
techniques to ship-specific sensor data, the whole
ship and its environment can be learned directly
from real-world data.
Our results showed that a simple NN outperforms
all of the (semi-)empirical added wave resistance
methods, which were used in combination with
the data-driven calm-water model. This suggests
that ML techniques may be a valuable tool for
improving ship performance modeling in practical
applications. Even a small increase in accuracy
13
could have significant economic and ecological
benefits for the shipping industry. It requires a
tiny change of perspective transitioning from
theoretical regression analysis to a more advanced
approach potentially yielding significant
benefits.
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APPENDIX
Fig. 10: Histogram of wave height.
15
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