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Uncertainty Analysis in Ship Performance Monitoring 2014
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Uncertainty Analysis in Ship Performance Monitoring
L. Aldousa, T. Smithb, R. Bucknallc and P. Thompsond
a,b Energy Institute, University College London, 14 Upper Woburn
Place, London, WC1H 0NN, UK c Department of Mechanical Engineering,
University College London, Torrington Place, London, WC1E 7JE, UK d
BMT Group Ltd, Goodrich House, 1 Waldegrave Road, Teddington, TW11
8LZ, UK Corresponding author: L. Aldous, email [email protected],
phone number +44 (0)7887 441349 Keywords: Uncertainty, Noon report,
Ship performance, Measurement, Monte Carlo, Continuous monitoring 1
Abstract
There are increasing economic and environmental incentives for
ship owners and operators to develop tools to optimise operational
decisions, particularly with the aim of reducing fuel consumption
and/or maximising profit. Examples include real time operational
optimisation (e.g. ship speed and route choice), maintenance
triggers and evaluating technological interventions. Performance
monitoring is also relevant to fault analysis, charter party
analysis, vessel benchmarking and to better inform policy decision
making. The ship onboard systems (propulsion, power generation
etc.) and the systems in which it operates are complex and its
common for data modelling and analysis techniques to be employed to
help extract trends. All datasets and modelling procedures have an
inherent uncertainty and to aid the decision maker, the uncertainty
can be quantified in order to fully understand the economic risk of
a decision, if this risk is deemed unacceptable then it makes sense
to re-evaluate investment in data quality and data analysis
techniques. This paper details and categorises the relevant sources
of uncertainty in performance measurement data, and presents a
method to quantify the overall uncertainty in a ship performance
indicator based on the framework of the Guide to Uncertainty in
Measurement using Monte Carlo Methods. The method involves a
simulation of a ships operational profile and performance and is
validated using 4 datasets of measurements of performance collected
from onboard in-service ships. A sensitivity analysis conducted on
the sources of uncertainty highlight the relative importance of
each. The two major data acquisition strategies, continuous
monitoring (CM) and noon reported (NR), are compared.
Abbreviations CM: Continuous Monitoring NR: Noon Reports
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1. Introduction
Ship operational performance is a complex subject, not least
because of the various systems and their interactions in which a
ship operates, the major factors are presented in Figure 1. At the
ship level the ship design, machinery configurations and their
efficiencies determine the onboard mechanical, thermal and
electrical energy flows which, despite automation built in to the
configuration mode settings at the ship design phase, there is
still an appreciable level of human interaction during day to day
operations. The environmental conditions (sea state, wind speed,
sea/air temperature etc.) are dynamic, unpredictable and
complicated to quantify, due in part to the characteristics of the
turbulent flow fields by which they are determined. These
environmental conditions exert an influence on the ships resistance
and therefore the ship power requirements in differing relative
quantities. The rate of deterioration in ship performance (engine,
hull and propeller) is dependent on a vast array of variables;
including the quality and type of hull coating and the frequency of
hull and propeller cleaning which are also dependent on the ocean
currents, temperature and salinity in which the ship operates.
Further, the shipping industry operates in an economic sphere in
which the global consumption of goods and global energy demand, and
conditions in the various shipping markets determine operating
profiles, costs and prices (see for example Lindstad (2013) which
also explores environmental effects). In addition, technological
investment, fuel efficiency and savings are complicated by the
interactions between ship owner-charterer-manager [Agnolucci
(2014)].
Figure 1: Ship performance influential factors
Data collection, either through daily noon reporting procedures
or high frequency, automatic data acquisition systems, and data
processing techniques such as filtering and/or modelling have so
far proven to be useful tools in capturing and quantifying some of
the intricacies and nuances of these interactions to better
understand the consequences of operational decisions. These
datasets and modelling outputs are applied in areas such as real
time operational optimisation including trim adjustments,
maintenance triggers, predicting and evaluating the performance of
new technologies or interventions,
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Uncertainty Analysis in Ship Performance Monitoring 2014
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particularly for cost benefit analysis, fault analysis, charter
party analysis, vessel benchmarking and to better inform policy
decision making.
The need to conduct uncertainty analysis is linked to the
amplitude of the noise or scatter in the data relative to the
underlying, longer term trends that are to be extracted. The ship
system interactions give rise to scatter in the data, not only from
inherent sensor precision but also from unobservable and/or
unmeasurable variables. According to the central limit theorem
(assuming independent, identical distributions), over time the
scatter will tend to a normal distribution with zero mean. This
time period is dependent on the data acquisition and processing
strategy; the temporal resolution of sensors and data collection
frequency, the sensor precisions and human interactions in the
collection process and the modelling or filtering methods all play
a part. There are also uncertainties in the data that will
introduce a potentially significant bias in the results and this
too needs to be understood and evaluated. The magnitude of the
underlying trends to be identified are a function of the modelling
application; for evaluating the performance of new technologies the
signal delta, i.e. the improvement in ship performance, may be a
step change of the order of 1-3% (as in the case of propeller boss
cap fins) or up to 10-15% as in the case of hull cleaning or new
coating applications where analysis of trends in the time domain is
also necessary. Therefore, the required measurement uncertainty
depends on the application and this drives the desired acquisition
strategy which is of course constrained by costs; economic, time
and resources.
Acquisition strategies are broadly separated into two dominant
dataset types. Noon report (NR) datasets are coarse but cheap to
compile and readily available since they are currently in
widespread use across the global fleet. The frequency of recording
is once every 24 hours (time zone changes allowing) and the fields
reported are limited, generally included as a minimum are ship
speed and position, fuel consumption, shaft rotational speed, wind
speed derived Beaufort number, date/time and draught. Given the
economic and regulatory climate there has been a shift towards more
complete, automatic measurement systems referred to in this paper
as continuous monitoring (CM) systems. The uptake of these has been
limited by installation costs in service while improved data
accuracy, speed of acquisition, high sampling frequency (5minutes)
and repeatability are cited as the key drivers. All datasets and
modelling procedures have an inherent uncertainty associated and as
a prerequisite the uncertainty must be quantified in order to fully
understand the economic risk of the decision. The benefit of
reducing uncertainty is a function of the uncertainty of the data
relative to the change in ship performance, the former depends on
the data acquisition strategy (noon reports vs continuous
monitoring, for example) and the latter depends on the application
(the technological / operational decision being made) and the
cost-benefit of both determine the overall economic risk of the
decision. If the economic risk is deemed unacceptable then it makes
sense to re-evaluate investment in data quality and data analysis
techniques.
The uncertainty is also important because of the risks and costs
that are associated with the decision derived from the measured
ships performance. The desire to quantify these (in other
industries) has led to the field of research into methods for risk
based decision making. The application of these methods to the
shipping industry is also important, for example, measurement and
verification is cited as a key barrier to market uptake in fuel
efficient technologies and retrofit. In order to secure capital,
investment projects must be
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expected to yield a return in excess of some pre-defined minimum
[Stulgis (2014)]. Weighing the economic risk of capital investment
against the certainty of the effectiveness of a fuel efficient
technology is therefore key. A study of the sensitivities of the
uncertainty in the ship performance measurement is pertinent to
inform where resources can be invested most effectively in order to
reduce the overall uncertainty to the desired level; is the cost of
obtaining additional information outweighed by the value of the
improvement in the model from which the performance estimate is
derived? [Loucks (2005)]. It is of course not just financial but
legislative drivers that are significant in the uptake of fuel
efficient technologies and modelling ship performance in this case
is also important in order to establish if new policies have been
effective either from a fleet wide or total global emissions
perspective.
An overview of uncertainty analysis methods and their
application to ship performance measurement uncertainty is
described in section 2. This paper is based on a similar framework
but also employs an underlying time domain algorithm to simulate
the ships operational profile and performance trends in order to
propagate the errors through the model by Monte Carlo simulation.
Section 3 presents a brief overview of ship performance methods and
introduces the ship performance indicator used in this study, the
sources of uncertainty in this measurement are then detailed and
quantified in section 5. This method is validated using data from 4
ships; 1 continuous monitoring dataset and 3 noon report datasets,
the validation results are presented in section 5.4. A sensitivity
analysis is employed in section 7 to examine the effect of sensor
characteristics and data acquisition sampling frequencies on the
performance indicator uncertainty given the length of the
performance evaluation period. Different acquisition strategies
(based broadly on noon reporting and continuous monitoring
acquisition strategies) are then compared. The type of data
processing has also been considered and while this paper focuses on
a basic ship model using filtered data, there is ongoing work that
explores how the uncertainty may be reduced by using a ship
performance model (normalising approach). Sections 8 and 9 present
the results, discussion and conclusions.
2. Uncertainty Analysis Methodology
The aim of an uncertainty analysis is to describe the range of
potential outputs of the system at some probability level, or to
estimate the probability that the output will exceed a specific
thresholds or performance measure target value [Loucks (2005)]. The
main aim in the uncertainty analysis deployed in the quantification
of performance trends is to estimate the parameters of the output
distribution and to conduct a sensitivity analysis to estimate the
relative impact of input uncertainties.
Uncertainty analysis methods have evolved in various ways
depending on the specific nuances of the field in which they are
applied. However, a key document in the area of uncertainty
evaluation is the Guide to the expression of uncertainty in
measurement (GUM) [JCGM100:2008 (2008)] which provides a procedure
adopted by many bodies [Cox (2006)]. The methods of each adaptation
essentially distil to that of the original GUM framework;
1. Identify each uncertainty source and classify
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2. Assign probability distributions and their parameters 3.
Propagate the errors through the linearised model (also known as
the data reduction
equations (DRE) or transfer function) 4. Formulate the output
distribution of the result and report overall uncertainty
The GUM framework is itself derived in part from the work of
Coleman (1990). The nomenclature and definitions of Coleman and
Steele are consistent with those of the ANSI/ASME standard on
measurement uncertainty; precision error is the random component of
the total error, sometimes called the repeatability error, it will
have a different value for each measurement, it may arise from
unpredictable or stochastic temporal and spatial variations of
influence quantities, these being due to limitations in the
repeatability of the measurement system and to facility (equipment
/ laboratory) and environmental effects. The bias error does not
contribute to scatter in the data but is the fixed, systematic or
constant component of the total error; in a deterministic study it
is the same for each measurement. Both types of evaluation are
based on probability distributions.
The GUM specifies three methods of propagation of
distributions:
a. The GUM uncertainty framework, constituting the application
of the law of propagation of uncertainty
b. Monte Carlo (MC) methods c. Analytical methods
The analytical method gives accurate results involving no
approximations however it is only valid in the simplest of cases
while methods a. and b. involve approximations. The GUM framework
is valid if the model is linearised and the input probability
distribution functions (pdfs) are Gaussian, this is the framework
followed by the AIAA guidelines [AIAA (1999)] and the ITTC guide to
uncertainty in hydrodynamic experiments (ITTC 2008) of which
relevant examples include applications to water jet propulsion
tests (ITTC 2011) and resistance experiments (ITTC 2008) and (Longo
2005). If the assumptions of model linearity and Gaussian input
pdfs are violated or if these conditions are questionable then the
MC method can generally be expected to lead to a valid uncertainty
statement that is relatively straightforward to find. The Monte
Carlo method was applied in the shipping industry by Insel (2008)
in sea trial uncertainty analysis. A further advantage of the MC
method is that a more insightful numerical representation of the
output is obtained which is not restricted to a Gaussian pdf.
Taking into account the different methods that can be applied in
the case of ship performance analysis, the nature of the data
available and the uncertainties associated with that data (both
aleatory and epistemic), this paper follows the GUM approach and
the errors are propagated through the model using the MCM.
Instrument precision is evaluated from repeated random sampling of
a representative pdf and instrument bias / drift is evaluated by
elementary interval analysis with deterministic bounds. An output
distribution is formed at each time step and the overall precision
is based on the distribution of the coefficient of a linear
regression of the performance indicator trend over repeated
simulations. Further details of the method can be found in section
4.
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3. Ship Performance Methods
One aim of ship performance monitoring is to quantify the
principal speed/power/fuel losses that result from the in-service
deterioration (hull, propeller and engine). A performance indicator
may be defined to identify these performance trends; performance is
the useful output per unit of input. For a ship the most relevant
unit of input is the fuel consumption. Sometimes, its useful to
isolate the performance of the hull and propeller from that of the
engine, in which case it is the shaft power that becomes the input
to the performance indicator estimation. The most aggregate unit of
output is transport supply, which for constant deadweight capacity
and utilisation, ultimately comes down to the ships speed. One of
the most basic methods to extract this information is to control
all other influential variables (weather, draught, trim, water
depth etc.) by filtering the dataset. An alternative method is to
normalise each influential variable to a baseline by employing a
model that quantifies the ships power/fuel for all environmental /
operating conditions.
The main problem of normalising is that the model used for the
corrections may lead to uncertainties that arise from incorrect
model functional form (or model parameters, depending on the
training / calibration dataset integrity) due to either omitted
variables or unknown effects. Instrument uncertainty also becomes
important for all the input variables to the correction algorithm.
On the other hand, from a richer dataset trends may be derived that
have a more comprehensive physical significance and a larger
dataset also reduces the uncertainty making the results applicable
to both short and long term analysis. The filtering approach is
easy to implement and interpret however filtering data removes many
observations and the analysis must be done over a longer time
period in order to collect adequate volumes to infer statistically
significant results and to reduce the uncertainty to a level
appropriate to the application. There is therefore a trade-off
between uncertainty introduced due to the model form and the then
required additional instrument uncertainty and the uncertainty
arising from a reduced sample size, this is the focus of ongoing
work. In this study the influential effects are controlled for by
applying a filtering algorithm and the correction model in the
simulation is therefore a function of ship speed and draft. A cubic
speed-power relationship is assumed (ideally this would be sought
from sea trials) and the admiralty formula is used to correct for
draught, a simple linear relationship between displacement and
draught is assumed.
4. Method
The simulation of the performance indicator is based on the
difference between the expected power and the measured power. The
measured power is equal to the expected power plus some linearly
incremented power increase over time to simulate the effect of ship
degradation (hull / propeller), the simulation also allows for the
inclusion of some instrument error (precision, drift and bias) and
model error. A sampling algorithm is able to simulate the effect of
sample averaging frequency and the change in the number of
observations. Further details regarding the nature and magnitude of
these data acquisition variables are provided in sections 5, 6 and
7.
The method is summarised by the following steps (Figure 2):
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1. Identify sources of uncertainty (instrument / model) and
assign pdfs 2. Actual Power, Ptrue,i: Define underlying ship
performance (the truth) at the
maximum temporal resolution; a. Operating profile: The ships
loading condition is assumed to be 50% loaded
and 50% ballast, the ships voyage speed variability is
represented by a Weibull distribution.
b. Environmental effects: The effect of small changes in the
weather (within the 0 < BF < 4 filtering criteria) is allowed
for by the inclusion of some daily speed variability, assumed to be
normally distributed. This also includes other small fluctuations
that may not be filtered for (acceleration for example)
c. Time dependent degradation: Assumed to be linear with time
(power increases ~5% per year) and independent of ship speed and
draught, the ships speed decreases to simulate constant power
operation
3. Average actual power / speed / draught according to averaging
frequency, fave 4. Measured Power, Pmeas,i: Add instrument
uncertainties to Ptrue as random samples
from pdfs assigned in step 1. 5. Expected Power, Pexp,i: Reverse
calculate from measured power using the same
model as in step 2 (assuming no degradation), add model
uncertainty as random samples from pdfs assigned in step 1
6. Percent change in Power, %Pi : 100(Pmeas,i - Pexp,i)/ Pexp,i
7. Repeat 4 to 6 n1 times at each time step to define the
parameters, mean and
standard error (!,si), of the pdf for %Pi at each time step 8.
Randomly sample from each %Pi for the evaluation period length,
teval and
according to the number of observations, N, and calculate
overall %P by linearly regressing %P on time and finding %P at time
= teval,end
9. Repeat step 8 n2 times to find the distribution parameters of
the result, precision is twice the standard error (95% CI), the
effect of bias is indicated by a change in the mean.
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Figure 2: Diagrammatic presentation of the method
5. Sources of Uncertainty in Ship Performance Monitoring
There are many different classification structures for the
sources of uncertainty proposed in the literature, different both
between industries to incorporate field specific nuances field and
also within industries according to the specific application.
Figure 3 outlines the key sources of uncertainty relating to ship
performance monitoring.
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Figure 3: Source of uncertainty in ship performance
monitoring
The sources of uncertainty presented in Figure 3 are discussed
in detail in the sections that follow. Table 1 presents the
relevant data acquisition variables (pdf and sampling algorithm
parameters) input to the MC simulation and which are categorised as
two different data acquisition strategies; noon reports (NR) and
continuous monitoring (CM).
DAQ decision variable Baseline input, NR
Baseline input, CM
Evaluation Period teval, days 90 270 90 270 Number of
observations, N 24 72 2160 6480 Shaft power / FC sensor precision
(1), % 5.00 0.51 Ship speed (STW) sensor precision (1), % 1.00 1.00
Draught sensor precision (1), m 1.00 1 Averaging frequency, fave
samples/day 1.00 96 Daily speed variability, % 1.74 1.74 Model
precision error, % 9.00 9 Table 1: Continuous monitoring and noon
report data inputs
5.1 Sampling
Sampling error arises from the fact that a finite sample is
taken from an infinite temporal population and because variable
factors are averaged between samples. Both of these effects are
investigated through a sampling algorithm that samples from the
underlying ship performance simulation which is based on a maximum
temporal resolution of 4 samples per hour.
The number of observations in Table 1 are based on a sample rate
which is, realistically, less than the sample averaging frequency
because of periods when the ship is at anchor,
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manoeuvring, waiting or in port and because of the filtering
criteria. The days spent at sea depends on ship type and size,
figures from the latest IMO GHG study (Smith 2014) show that for
2012 the average number of days at sea for all sizes of tankers,
bulk carriers, containers and general cargo ships was 53%. This is
similar to the results found from the 397 day CM dataset analysed
in this study (51.5% at sea days), the results of which are
presented in Table 2.
Number of Observations % of All All data, 397 days 38112 100.0
At sea data 19618 51.5 Wind < BF 4 12799 33.6 Inliers 12798 33.6
Water depth > min* 9570 25.1 Table 2: Effect of filtering on the
number of observations of a CM dataset. *See equation [1]
The proportion of the results that are removed due to filtering
depends on the environmental / operational conditions. From the CM
dataset used in this study, after filtering, 25% of the
observations for use in the ship performance quantification
remained. The NR datasets that were used in this study showed a
high variation in the data filtered for environmental conditions
with 50% being the average, meaning that of the possible number of
samples 26.5% were used in the ship performance quantification
(generally NR data is only recorded while at sea).
The averaging frequency is either equal to the underlying
temporal resolution of the true ship performance parameters
(96/day) as in the CM baseline, or 96 samples are averaged for a
daily frequency (NR baseline). To simulate the effect of daily
averaging of noon reported variables, a daily speed variability is
included at a level of 1.74% of the ships speed. This was the
outcome (median) of a study of the CM dataset covering a 396 day
time period after filtering for outliers and wind speed > BF 4
(not sea water depth since this data is not generally recorded in
the NR data) . The variability is due to the effects of ship
accelerations, rudder angle alterations, wind speed fluctuations
within 0 < BF < 4 range, and crew behaviour patterns (slow
steaming at night for example), which arent included in the ship
performance model or cant be filtered out in the NR data set. The
variability is introduced into the simulation through the addition
of normally distributed noise to the underlying true ship speed.
The effect of averaging daily draught variability (i.e. due to
intentional changes in trim or due to influences on trim due to
fuel consumption) is not included but assumed to be negligible.
The number of observations is also affected by the evaluation
period length and the sensitivities of the overall uncertainty to
this are investigated by evaluating both 90 day and 270 day
periods. There is seen in some datasets the effect of seasonality,
whereby weather fluctuations (even within the BF filtering
criteria) cause discontinuities in the measured power and therefore
a minimum of one years data collection is required. The presence of
this is dependent on the cyclic nature and the global location of
the ships operations and is difficult to generalise, it is
therefore not included in this analysis.
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5.2 Instrument Uncertainty
Instrument uncertainties arise from variations attributable to
the basic properties of the measurement system, these properties
widely recognised among practitioners are repeatability,
reproducibility, linearity, bias, stability (absence of drift),
consistency, and resolution (ASTM 2011). Since the truth is unknown
then bias, and by extension drift (change in bias over time), are
epistemic uncertainties, and are assumed to be zero for all
measurements in the baseline. Their effect on the overall
uncertainty is investigated in the sensitivity analysis by an
elementary interval analysis.
Some manufacturers quote sensor precisions and/or accuracy which
are assumed to reflect manufacturing tolerances, however these are
not well documented or publicly available and also they are more
likely to quote sensor repeatability calculated under laboratory
conditions (same measurement system, operator, equipment) or
reproducibility (different operator) and therefore likely to under
estimate the actual precision in service conditions where different
operators, equipment/facility and measurement system may increase
the overall precision. Quoted shaft power sensor accuracies range
from 0.25% (Seatechnik, http://www.seatechnik.com/) to 0.1% Datum
Electronics (http://www.datum-electronics.co.uk/). The IMO
resolution Performance Standards for Devices to Indicate Speed and
Distance (IMO 1995) stipulate that errors in the indicated speed,
when the ship is operating free from shallow water effect and from
the effects of wind, current and tide, should not exceed 2% of the
speed of the ship, or 0.2knots, whichever is greater. The shaft
power in the CM is the addition by quadrature of an RPM sensor
error of 0.1% (1) and a torque sensor error of 0.5% (1).
Discussions with industry experts and experience of ship
performance datasets suggest that the instrument precisions quoted
in Table 1 are appropriate, however they are estimates and the
sensitivity of the performance indicator to changes in these is
explored in section 7. All sensors are assumed to be consistent
during the simulation (no change in repeatability over time) and
linear (absence of change in bias over the operating range of the
measurement instrument). Sensor resolution is assumed to be
reflected in the quoted sensor precision and not a restricting
factor in the uncertainty of ship performance measurement.
In the NR dataset, fuel consumption is generally recorded rather
than shaft power; fuel flow meter accuracy ranges between 0.05 per
cent and 3 per cent depending on the type, the manufacturer, the
flow characteristics and the installation. Fuel readings by tank
soundings are estimated to have an accuracy of 2-5% [Faber (2013)].
This uncertainty estimate does not include however additional
uncertainties associated with the fuel consumption, for example
fuel oil quality variation (grade and calorific value), pulsations
in the circulation system, back flushing filters, the effect of
cat-fine, waste sludge from onboard processing and inaccuracies in
ullage measurements due to weather, fuel temperature etc. The use
of fuel consumption as a proxy for shaft power is included by
assuming a power sensor precision of 5% for the NR strategy.
The draught uncertainty associated with draught gauges is of the
order 0.1m, in noon report entries the draught marks at the
perpendiculars may be read by eye and, depending on sea conditions,
a reading error of 2 cm can be assumed (Insel 2008). However, noon
report entries are often not altered during the voyage to record
variability due to trim
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and/or the reduction in fuel consumption and therefore a greater
1.0 m uncertainty is assumed. Often the draught field in CM
datasets is manually input and therefore the same uncertainty
applies.
5.3 Model Uncertainty
The model parameter uncertainty depends on whether a filtered or
normalised (corrected by modelling interactions between variables)
dataset is used to calculate the performance indicator.
For a normalised dataset, model parameters and model form may be
based on theoretical or statistical models, both of which have
associated uncertainties. For example, the speed-power relationship
is often approximated to a cubic however in reality the speed
exponent may for example be between 3.2, for low speed ships such
as tankers and bulk carriers, and 4.0 for high speed ships such as
container vessels (MAN 2011). In a statistical ship performance
model, the model parameter uncertainty is a result of the sampling
error that occurs during the calibration/reference period. From
this period a training dataset captures the ship performance in
terms of a larger array of influential environmental/operational
factors and the correction model is defined. Therefore some
sampling error exists because the training dataset is a finite
sample taken from an infinite temporal population, the instrument
uncertainty discussed in the previous section will also be present
in the training dataset. Model form uncertainty may arise because
the variation in the data during the reference period is
insufficient to capture the interaction between variables in the
baseline performance, for example, if the ship is operated at only
one draught then there will be no way to define the general
relationship between draught, shaft power and other variables.
For the filtered approach applied in this paper, the model
parameter uncertainty is from the sea trial dataset. This has been
investigated by Insel (2008) who found that from sea trial data of
12 sister ships that represented a wide variety of sea trial
conditions a precision limit of about 7-9% of shaft power can be
achieved. This uncertainty represents both model uncertainties (due
to corrections applied to trial measurements) and instrument
uncertainties for the measurement of torque, shaft rpm, speed,
draught and environmental measurements etc. The author identifies
the Beaufort scale estimation error as the key measurement error
affecting the overall sea trial data uncertainty. The magnitude of
the uncertainty induced in this method is assumed to be the same
for both CM and NR simulations because the sea trial data is from
the same source in both cases. It is notable however that because
reliable methods do not currently exist, the influence of currents,
steering and drift are not corrected for, for more details see
Insel (2008).
Model form uncertainty is notoriously difficult to quantify, in
this paper the expected power and the measured power are based on
the same model therefore the model form is assumed to be correct
and no model form error is accounted for. Model form error in
reality may arise from, for example, unobservable or unmeasurable
variables that are not filtered for, the effect of these are
exacerbated in NR datasets because fewer variables are recorded
(such as acceleration and water depth).
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5.4 Human Error
Human error (which is often categorised as instrument
uncertainty) may occur in any measurements when operating, reading
or recording sensor values if the report completion is not
automated. For example the noon data entry may not occur at exactly
the same time each day, the recording of time spent steaming may
not be adjusted to compensate for crossing time zones and it is
possible that different sensors are used to populate the same
field, for example, some crew may report speed from the propeller
RPM and others report speed through water. The measurement of wind
speed through crew observations may also be subject to
uncertainties. Human error is not included in this current
analysis.
6. Experimental Precision Comparison
The uncertainty of the performance indicator as calculated from
the propagation of elemental uncertainties via the simulation
method (NR baseline) was validated against the precision calculated
from experimentally collected data from the same number of
observations over the same length evaluation period. The overall
uncertainty in both methods include uncertainties from the
measurement sensor repeatabilitys, the sampling uncertainty
(frequency and averaging effects) and model parameter uncertainty
while only the experimental method also includes uncertainty from
human error, model form, missing model parameters not filtered out
(particularly in the NR dataset) and any possible measurement
sensor non-linearity, bias or drift, the magnitudes of which are
unknown. The data is filtered according to the following criteria
where data is available, (outliers are removed manually):
Mean draught is between ballast and design True wind speed is
between BF 0 and BF 4 Ship speed is greater than 3 knots Water
depth is greater than the larger of the values obtained from the
two formulae: = 3 ! = 2.75 !!
(1)
where h: water depth [m] B: ship breadth [m] TM: draught at
midship or mean draught [m] Vs: ship speed [m/s] g: gravitational
acceleration (9.81 m/s2)
Table 3 shows the comparison of the methods, the simulation
method through error propagation of elemental uncertainties tends
to over-estimate the uncertainty.
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Evaluation period length
(days)
Number of observations
Error propagation of elemental
uncertainties, %
Statistical uncertainty
calculation*, % CM 1 200 1392 2.16 1.4 NR 1 265 70 9.42 8.78 NR
2 377 81 8.67 6.00 NR 3 373 62 10.22 4.95
Table 3: Statistical uncertainty calculation vs elemental
uncertainty propagation for method 2, error reported as the
confidence interval of P (95% level) as a percentage of mean shaft
power (CM) or mean fuel consumption (NR)
Figure 4 shows the experimental precision from the continuous
monitoring dataset and the simulation results. The y-axis is the
change in the number of samples for the same evaluation time
period. The performance indicator uncertainties (plotted as a
percentage of mean shaft power over the evaluation period) are
similar, although again, slightly over estimated in the simulation.
However the difference is again minimal and the comparison provides
evidence in support of the use of the simulation and propagation of
elemental uncertainty in this way.
Figure 4: Simulation and experimental precision comparison for
varying evaluation periods and sampling frequencies
-100
-90
-80
-70
-60
-50
-40
-30
-20
-10
0 0 2 4 6 8 10
% cha
nge in nr o
f sam
ples (from baseline)
Condence Interval, % of mean power
Change samples per day, experimental precision
Change samples per day, simulaAon
Simula>on and Experimental Precision Comparison for Varying
Sampling Sizes
-
Uncertainty Analysis in Ship Performance Monitoring 2014
15
7. Sensitivity Analysis
The sensitivity indices relate to the precision of the
uncertainty in the overall performance indicator. The bias in the
result is investigated and graphically represented also.
7.1 Relative Effects of Data Acquisition Factors
A local sensitivity analysis was conducted by computing partial
derivatives of the output function (y) with respect to the input
factors (xi). The output function, y is the precision of the
measurand; twice the standard error of the P at time = teval (see
section 4). The input factor is one of the data acquisition
parameters of Table 4. Each factor is altered one-factor-at-a-time
(OAT) and a dimensionless expression of sensitivity, the
sensitivity index, SIi is used to measure changes relative to the
baseline.
! = ! . ! The input parameter interval bounds reflect results
from studies of actual data where possible or, in the absence of
data, estimates of realistic values are made given the source of
the uncertainties as detailed in section 5. Sensor precisions are
induced by the addition of a normally distributed error, and in the
baseline this is assumed to have a mean of zero and a standard
deviation according to Table 4. Sensor bias is introduced by a
deterministic offset to the mean of the normal distribution of the
error.
DAQ decision variable Baseline input SA input Evaluation Period
(days), teval 90 270 90 270 Number of observations, N 24 72 45
135
Power sensor precision (1) % 1 5 Power sensor bias % 0 3
Draught sensor precision (1) m 0.1 1.5 Draught sensor bias m 0
1
Ship speed sensor precision (1) % 1 5 Ship speed sensor bias % 0
3
Ship speed sensor drift %/270 days (linear increase) 0 3
Averaging frequency, fave, sample/day 1 96 Daily speed
variability, % 1.74 3.76
Model precision error, (1) % 1.5 9 Table 4: Adjustments to the
baseline for the sensitivity analysis (SA)
The increase in the sample size from 25.0% or 26.5% to 50%
(presented in the table by the increase in number of samples, N) is
to investigate the effect of increasing the sample size to that of
a dataset that is normalised rather than filtered by increasing the
number of observations to the total observations before filtering.
The daily speed variability SA input represents the upper
interquartile of the results of an investigation conducted on a CM
dataset of time period length of 396 days. Model precision error
represents the results of
-
Uncertainty Analysis in Ship Performance Monitoring 2014
16
the study by (Insel 2008) who found 7-9% of shaft power to be
the precision error in the sea trial data of 12 sister ships
studied. If the ship reporting is speed over ground (SOG) rather
than speed through the water (STW) then this will increase the
uncertainty; a study of the difference between STW and SOG for 20
ships over 3 to 5 years indicates a standard deviation of
0.95knots, (Smith 2014). The effect of using SOG as a proxy for STW
is therefore investigated by increasing the speed sensor precision
to 5%. Likewise, power sensor precision is to reflect the effect of
fuel consumption as a proxy for shaft power.
7.2 Comparison of Data Acquisition Strategies
The second study is to compare the uncertainty for the data
acquisition parameters of a standard CM dataset with the data
acquisition parameters of a standard NR dataset. The measurand is
the uncertainty (SE at 95% confidence level) reported as a
percentage of the parameter of interest, for the purposes of this
investigation, the parameter of interest is the change in ship
performance over time, i.e. increase in shaft power for a given
speed and draught (e.g. due to hull fouling), this was set to
linearly increase by 18% over 6months. This is a practical metric
since it puts the uncertainty in a useful context and is of greater
interest when comparing the data acquisition strategies however it
is worth highlighting that the uncertainty measured in this way is
not only representative of the impact of all the elemental
uncertainties but it is also a function of the magnitude of the
performance indicator itself and consequently of the time period of
evaluation. For example, if the rate of deterioration was actually
10%/year and not 5% or if the evaluation time period was set to
increase from 90 days to 270 days then in both cases the
uncertainty would naturally decrease even if the elemental
uncertainties were unchanged
This was carried out for each of the two baselines (CM and NR,
Table 1) and for two time periods of evaluation; 90 days and 270
days. This experiment is repeated once with an increase in number
of observations (N) for both, and once with a reduction in model
precision error Table 5.
DAQ decision variable Alternative 1 NR
Alternative 1 CM
Alternative 2 NR
Alternative 2 CM
Evaluation Period days 90 270 90 270 90 270 90 270 Number of
observations
N 48 144 4320 12960 24 72 2160 6480
Shaft power sensor precision (1) % 5 0.51 5 0.51
Ship speed sensor precision (1) % 1 1 1 1
Ship draught sensor precision (1) m 1 1 1 1
Averaging frequency /day 1 96 1 96
Model precision error (1) % 9 9 1.5 1.5
Table 5: Input parameters for comparison of data acquisition
strategies
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Uncertainty Analysis in Ship Performance Monitoring 2014
17
8. Results and Discussion
8.1 Relative Effects of Data Acquisition Factors: Precision
This section looks at sensitivities of the performance indicator
uncertainty relative to the variations in the data acquisition
decision parameter uncertainties (see section 7.1). The sensitivity
index indicates the change in the parameter of interest (the
precision of overall uncertainty quantified by 2) relative to the
percent change in the input quantity examined (the data acquisition
variable) and then weights it according to the relative size of
each. For example, an SI of unity, similar to the case of the speed
sensor precision, SI = 0.96 (Figure 5), indicates that the change
in overall uncertainty due to a change in the precision of the
speed sensor is almost exactly offset by the magnitude of the ratio
of its absolute precision (5%) to the resultant overall
uncertainty.
Figure 5: Sensitivities of the performance indicator uncertainty
to model, instrument and sampling input uncertainties (evaluation
time period: 90days)
The exponent in the relationship between speed and power means
that the performance indicator uncertainty is more sensitive to the
speed sensor precision than either of the draught or shaft power
sensor precisions. This highlights the importance of investment in
high precision speed sensor, and the criticality of using speed
through the water sensor rather than speed over ground (which may
cause a precision of 5%, see section 7.1) which will have dramatic
consequences for obtaining meaningful information from the data. CM
systems that augment core data acquisition with additional
measurement of GPS, tides and currents would provide a means to
independently calculate and verify the speed through water
measurement from speed over ground. The results emphasise the
importance of a precise draught measurement which may be manually
input, even in continuous monitoring systems, the overall
uncertainty would benefit from investment in a draught gauge which
also records potentially significant variations during the voyage
(including in trim) rather than recording the draught at the last
port call. If fuel consumption is used as a proxy for shaft power,
which may cause an inaccuracy of 5% (due to uncertainties in the
fuel
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Speed precision Model precision
Draught precision Sample frequency Power precision Daily
averaging
Draught bias Power bias
Daily speed variability Speed bias Speed driL
Sensi>vity Index 90 days 270 days
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Uncertainty Analysis in Ship Performance Monitoring 2014
18
consumption measurement) then there could be a significant
effect also on the overall uncertainty. The effect of sensor
precision reduces over a longer evaluation period (270 days
relative to 90 days) because over time the variability cancels and
stabilises towards the mean. Sensor bias has a lesser relative
effect on the overall precision, the order of significance is the
same as for precision; speed, draught then power. For the speed
sensor, the bias and drift cause the precision in the overall PI to
increase because the increased Vmeas is compounded by the cubic
relationship which causes P to drift (see next section, Figure 6)
and therefore decreases the precision in the linear trend line of
the PI. The drift in P means that the effect of bias and drift in
Vmeas increases for a longer evaluation period. It is worth
highlighting however that the presented magnitude of the effect is
influenced by the input parameters of the ships operational profile
and the rate of speed reduction over the time period. There is a
similar but reduced effect occurring due to the draught and power
bias also causing P to drift as the evaluation period increases but
this is to a much lesser extent and small relative to the
counteracting effect of increasing sample size which reduces the
overall precision as the evaluation period increase. The
uncertainty in the performance indicator is second most sensitive
to changes in the model precision error, the model is the
calibration or correction factor applied to correct for the ships
speed as established from sea trial data which will have some level
of uncertainty. The effect on overall uncertainty is significant
because of how this translates to Pmeas through a cubic
relationship. Instead of sea trial data, a continuous monitoring
calibration/reference dataset representing a short period of
operation may be used to derive the ship speed/power performance
curve (the ship performance should be as close to stationary as
practically possible during the time period). The advantage of the
latter method is the increased number of samples in the dataset,
even after filtering for environmental / operational conditions,
which reduces the uncertainty. The CM dataset will also include the
effect of technological enhancements made since the sea trial
dataset was compiled; the effects of these more recent
interventions may not vary linearly with the model input parameters
and may therefore be incorrectly attributed to ship performance
changes as measured during the evaluation period. The effect of
model precision reduces over a longer evaluation period because
over time the variability cancels and stabilises towards the mean.
Increasing the sampling frequency is also significant because the
overall standard error is inversely proportional to the square root
of the sample size, the absolute values, when increased on a
samples per day basis cause the effect over 270days to be more
significant relative to 90 days. An increased sample size may be
achieved by addressing root causes of outliers or missing data in
the datasets (i.e. due to stuck sensors or human reporting errors);
although this is also limited by the ships at-sea days and the days
the ship operates in environmental/operational conditions outside
the bounds of the filtering algorithm. There is therefore a
considered trade-off to be made between the sample size and the
model error which reflects the advantages of filtering over
normalising (see follow up paper). The other sampling effect is
related to the daily averaging frequency, the impact of this on the
overall uncertainty is because the daily environmental fluctuations
are captured; averaging a range of speeds will cause power due to
higher speeds to be mistakenly attributed to deterioration in ship
performance that in reality is not present. The actual
-
Uncertainty Analysis in Ship Performance Monitoring 2014
19
influence on uncertainty is of course a function of the daily
speed variability, and this interaction is not explicitly studied
here however the 1.78% found from the data is realistic and the SA
provides the relative significance under the assumption of
linearity. The daily averaging effect is independent of evaluation
period length because the daily speed variability is assumed to be
constant over time.
8.2 Relative Effects of Data Acquisition Factors: Bias
Generally, the effects of the data acquisition factors studied
do not affect the absolute mean of the performance indicator (the
bias component of the overall uncertainty). Instrument bias however
not only affects the precision component (as demonstrated in the
previous section) but also biases the result; this is observable in
Figure 6.
Figure 6: Performance indicator bias with error bars indicating
the confidence interval (95% level). The baseline performance
indicator is highlighted by the red line
The effect of increasing the STW sensor bias from 0 to 5% is to
increase the average performance indicator value from 790 kW to
1196 kW. The graphic is indicative only since the exact magnitude
of the effect is to some extent dependent on the operational
profile but it is clear that in cases of sensor bias and drift the
actual underlying deterioration trend is difficult to identify.
The error bars in Figure 6 shows how results based on daily
averaging for a short 90 day time period may be inconclusive; if
for example, speed sensor precision is 5% then it might
-2000
-1000
0
1000
2000
3000
4000
Mon
te Carlo Sim
ula>
on: Av
erage Pe
rforman
ce
Indicator, kW
Performance Indicator Bias, 90 day evalua>on period
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Uncertainty Analysis in Ship Performance Monitoring 2014
20
not be possible to conclude (with 95% confidence) if the ships
performance has improved or deteriorated.
8.3 Comparison of Data Acquisition Strategies
This sensitivity index is detailed in section 7.2. Figure 7
shows the results from the MC methods deployment to calculate
uncertainty for a range of changes to the baseline data acquisition
variables. As described previously, the uncertainty measured in
this way is not only representative of the impact of all the
elemental uncertainties but it is also a function of the magnitude
of the performance indicator itself and consequently of the time
period of evaluation.
Figure 7: Simulation uncertainty sensitivity analysis
results
In all cases the simulation based on the CM baseline is
demonstrating a significant improvement in uncertainty compared to
the NR baseline for the same evaluation periods.
For noon report data, a short evaluation period (3months) does
not give useful results since the uncertainty is greater than 100%
of the parameter of interest. This is to be expected since a low
sample frequency, and reduced power sensor precision due to the use
of fuel consumption as a proxy for shaft power, means that a longer
time series is required to achieve the same uncertainty. In fact,
the uncertainty of the 90 day CM baseline is similar to the
uncertainty achievable from a 270 day NR dataset. Both these
findings demonstrate a significant uncertainty benefit of CM data
over NR data; this is of the order of 90% decrease in
uncertainty.
In many instances, shortcomings in the availability or precision
of the measurements can be addressed through the deployment of an
algorithm/model to normalise CM data. This should generally
increase the sample size of data that can be deployed in the
calculation of performance. The effect of increasing the sample
size, as in alternative 1 is to further reduce the baseline
uncertainty in the CM dataset to 2% for 270 days of data, this is
as
235
165
159
36
25
26
22
16
4
3
2
2
0.00 50.00 100.00 150.00 200.00 250.00
Baseline
AlternaAve 1
AlternaAve 2
Uncertainty as a Percentage of the change in Ship
Performance
Chan
ges to Inpu
t Uncertain>e
s
Eect of Input Uncertain>es on CM and NR baselines for Dierent
Evalua>on Periods
CM 270 days CM 90 days NR 270 days NR 90 days
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Uncertainty Analysis in Ship Performance Monitoring 2014
21
effective as reducing the model uncertainty from the sea trial
data from 9% to 1.5% as in alternative 2, but arguably more
achievable given resources.
The extent to which the uncertainty is improved depends on the
quality of the model used; it should be emphasised that every
algorithm/model introduced will also increase the model form
uncertainty as well as the number of data fields with a consequent
possible increase in instrument uncertainty, this needs to be
carefully considered to ensure that the addition of the algorithm
produces an overall reduction in uncertainty. This analysis also
assumes there is no bias and so if the sensors, in particular the
speed sensor, are not calibrated or maintained then the positive
effect of the increased time period on the overall uncertainty will
be reduced (see section 8.1).
There is no uncertainty included to model the influence of human
error, therefore the results for the noon report model is perhaps
optimistic, although the overestimation from the simulation, even
for noon report data, in the experimental precision comparison of
section 6 indicates that this might not always be an issue if
procedures are in place to ensure careful reporting.
9. Conclusions
This paper proposes and describes the development of a rigorous
and robust method for assessing the uncertainty in ship performance
quantifications. The method has been deployed in order to
understand the uncertainty in estimated trends of ship performance
resulting in the use of different types of data (noon report and
continuous monitoring) and different ways in which that data is
collected and processed. This is of high relevance to the shipping
industry, which regularly uses performance quantifications for
operational decision making, and the method and results in this
paper can both inform the decision making process and provide
insight into how lower uncertainty in some of the key decision
variables could be achieved. The desired level should be
appropriate to the particular application (the technological /
operational decision being made); this informs the appropriate data
acquisition strategy as does an analysis of the cost-benefit of
reducing uncertainty which should also be weighed against the
economic, environmental or social risk of an incorrect
decision.
The results indicate the significant uncertainty benefit of CM
data over NR data; this is of the order of 90% decrease in
uncertainty, and is especially relevant to shorter term analysis.
It has been shown in this analysis that the uncertainty of the 90
day CM baseline is similar to the uncertainty achievable from a 270
day NR dataset.
The precision of the speed sensor is of fundamental importance
when it comes to achieving low uncertainty and using SOG rather
than STW is likely to have a dramatic effect on the overall
uncertainty. The sensor precision however only affects the aleatory
uncertainty of the performance indicator and the effect decreases
over longer evaluation periods; the additional confounding factor
in the uncertainty analysis is epistemic uncertainty which may be
introduced through sensor bias and drift. Speed sensor bias and
drift causes significant changes in the performance indicator, it
is difficult to extract the underlying performance trend when bias
or drift is present, after 90 days the effect on the precision
of
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Uncertainty Analysis in Ship Performance Monitoring 2014
22
the overall performance indicator is negligible, however it
becomes significant as the evaluation period increases, this
highlights the importance of proper maintenance and calibration
procedures. Routine sensor checks should be incorporated into the
data acquisition strategy and onboard procedures. CM systems that
augment core data acquisition with additional measurement of GPS,
tides and currents also provide a means to independently calculate
and verify the speed through water measurement from speed over
ground.
The number of observations in the dataset also has a significant
effect, this can be achieved either through data representing a
longer time series (which is less desirable for real time decision
support tools), through a higher frequency of data collection
(highlighting further the positive contribution of continuous
monitoring systems) or through the use of data processing
algorithms rather than filtering techniques whereby applying ship
performance models enables a more comprehensive analysis. In the
latter case, the resultant effect of the increased sample size
(without considering additional model error or instrument
uncertainties) is as effective as reducing the model uncertainty
from the sea trial data from 9% to 1.5%, but arguably more
achievable given resources.
Acknowledgements The research presented here was carried out as
part of a UCL Impact studentship with the industrial partner BMT
group. The authors gratefully acknowledge advice and support from
colleagues within the UCL Mechanical Engineering department and
within UCL Energy Institute, in particular Dr David Shipworth,
Energy & the Built Environment.
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