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Uncertainty Analysis for a Wave Energy Converter: the Monte Carlo Method

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Developing wave energy converter technology requires physical-scale model experiments. To use and compare such experimental data reliably, its quality must be quantified through an uncertainty analysis. To avoid uncertainty analysis problems for wave energy converter models, such as providing partial derivatives for time-varying quantities within numerous data reduction equations, we explored the use of a practical alternative: the Monte Carlo method (MCM). We first set out the principles of uncertainty analysis and the MCM. After, we present our application of the MCM for propagating uncertainties in a generic Oscillating Water Column wave energy converter experiment. Our results show the MCM is a straightforward and accurate method to propagate uncertainties in the experiment; thus, quantifying the quality of experimental data in terms of power performance. The key conclusion of this work is that, given the demonstrated relative ease in performing uncertainty analysis using the MCM, experimental results reported in the future literature of wave energy converter modelling should be accompanied by the uncertainty in those results. More broadly, this study aims to precipitate awareness among the wave energy community of the importance of quantifying the quality of modelling data through an uncertainty analysis. We therefore recommend future guidelines and codes pertinent to uncertainty analysis for wave energy converters, such as those developed by the International Towing Tank Conference (ITTC) and International Electrotechnical Commission (IEC), to incorporate the MCM with a practical example.
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Uncertainty Analysis for a Wave Energy Converter:
the Monte Carlo Method
Jarrah Orphin#1, Irene Penesis#2, Jean-Roch Nader#3
#National Centre for Maritime Engineering & Hydrodynamics, Australian Maritime College, University of Tasmania Locked
bag 1395, Launceston, Tasmania, 7250, Australia
1jarrah.orphin@utas.edu.au
2i.penesis@utas.edu.au
3jeanroch.nader@utas.edu.au
Abstract— Developing wave energy converter technology requires
physical-scale model experiments. To use and compare such
experimental data reliably, its quality must be quantified through
an uncertainty analysis. To avoid uncertainty analysis problems
for wave energy converter models, such as providing partial
derivatives for time-varying quantities within numerous data
reduction equations, we explored the use of a practical alternative:
the Monte Carlo method (MCM). We first set out the principles of
uncertainty analysis and the MCM. After, we present our
application of the MCM for propagating uncertainties in a generic
Oscillating Water Column wave energy converter experiment.
Our results show the MCM is a straightforward and accurate
method to propagate uncertainties in the experiment; thus,
quantifying the quality of experimental data in terms of power
performance. The key conclusion of this work is that, given the
demonstrated relative ease in performing uncertainty analysis
using the MCM, experimental results reported in the future
literature of wave energy converter modelling should be
accompanied by the uncertainty in those results. More broadly,
this study aims to precipitate awareness among the wave energy
community of the importance of quantifying the quality of
modelling data through an uncertainty analysis. We therefore
recommend future guidelines and specifications pertinent to
uncertainty analysis for wave energy converters, such as those
developed by the International Towing Tank Conference (ITTC)
and International Electrotechnical Commission (IEC), to
incorporate the MCM with a practical example.
KeywordsWave Energy Converter, Experimentation,
Uncertainty Analysis, Monte Carlo Method
I. INTRODUCTION
Uncertainty analysis is an indispensable tool in each phase
of an experimental program. Its uses include ensuring an
experiment can indeed answer a proposed question or set of
questions; estimating and using uncertainties in the design,
construction, debugging, execution, and data analysis phases;
and finally to provide a quantitative indication of the quality of
experimental results in the reporting phase [1]. This last use, in
essence, makes data meaningful. Without such an indication,
not only is it impossible to assess the reliability of results, but
these results cannot be compared, either among themselves or
with reference values [2]. It is, therefore, apparently obligatory
to perform an uncertainty analysis for a wave energy converter
(WEC) experiment, if it is to be of scientific or practical value.
Strikingly, one rarely encounters evidence of uncertainty
analysis in the literature on WEC experimentation. Few studies
on this subject have been published [3-5]. Given developing
WEC technology toward commercial readiness is requiring
multiple stages of physical-scale model experiments [6, 7], with
each stage acquiring ostensibly valuable information for due
diligence and future technology and business development, this
lack of uncertainty analysis evidence is alarming. In the context
of the wave energy industry, unknown reliability of
experimental results, at bottom, hinders the ultimate goal of
reducing levelised cost of energy (LCOE).
A key reason why WEC experimentation literature lacks
uncertainty analysis is due to the challenge of modelling a
complex WEC system in a complex environment [6, 8].
Oftentimes it is not straightforward to, first, provide a
mathematical description of the model and, second, evaluate
how variables and their associated uncertainties influence the
model’s behaviour. In the latter case, such an evaluation
requires propagating the uncertainties of each variable through
the data reduction equations (DRE’s) that describe the model.
This task can be difficult or inconvenient when the model
contains many variables with multiple DRE’s common for
most WECs. While the wave energy community has recently
provided general guidance on the principles and use of
uncertainty analysis as it relates to WEC experiments [6, 9],
treatment of other methods for propagating uncertainty is
lacking. In particular, the Joint Committee for Guides in
Metrology (JCGM) provides a supplement to the “Guide to the
expression of uncertainty in measurement” (GUM) [2]
concerned with the propagation of uncertainty using the Monte
Carlo method (MCM) [10]. The MCM is a practical alternative
to the GUM uncertainty framework based on the law of
propagation of uncertainty. It can be applied to overcome
various problems in uncertainty evaluation, for example, when
the probability density function (PDF) for the output quantity
is not a Gaussian distribution; when a model is arbitrarily
complex; or when it is difficult or inconvenient to provide the
partial derivatives of the model, as needed by the law of
propagation of uncertainty.
This paper deals with demonstrating an application of the
MCM for evaluating uncertainty in a WEC experiment. The
rationale for exploring the use of the MCM in this instance was
to overcome difficulty in providing the partial derivatives of an
Oscillating Water Column (OWC) wave energy converter
model. While we focus on uncertainty analysis in the reporting
of results phase, that is, a post-test uncertainty analysis, the
general methodologies presented can be easily adapted for the
use of uncertainty analysis in all other phases of a WEC
experiment. Moreover, this work may also be applied to
uncertainty analysis for numerical models.
The paper is structured as follows. First we provide
methodologies of uncertainty analysis according to the guide to
the expression of uncertainty in measurement (GUM)
uncertainty framework [2], and the MCM [10]. Following this
are descriptions of the mathematical model of the OWC WEC
and experimental setup. Section III presents the results and our
discussion of these results. After, we make a recommendation
for the adoption of the MCM in future versions of guidelines
and codes on uncertainty analysis for WECs
II. METHODOLOGY
There are two primary theoretical considerations of this
paper. The first deals with methodologies for evaluating
measurement uncertainty through an uncertainty analysis. The
second deals with characterising the hydrodynamics and
performance of the OWC wave energy converter.
A. Uncertainty analysis
This section sets out, first, the general procedure for
evaluating measurement data and expressing uncertainty in
measurement [2, 10, 11] and, second, presents our application
of these in evaluating measurement uncertainty in an OWC
wave energy converter experiment, using the Monte Carlo
Method (based on [10] and [1]). For convenience, notation is
take from the International Towing Tank Conference (ITTC)
guideline 7.5-02-07-03.12 Uncertainty Analysis for a Wave
Energy Converter” [12]. Noted, this section is an extension
similar previous work on this subject [5].
The principles of evaluating uncertainty in measurement are
categorised into three groups: (a) standard uncertainty, (b)
combined uncertainty, and (c) expanded uncertainty. Standard
uncertainty is itself grouped into two types: Type A (random
uncertainty) and Type B (systemic uncertainty). Combined
uncertainty is evaluated using the law of propagation of
uncertainty or, as in this paper, through the Monte Carlo
Method. Expanded uncertainty is obtained by multiplying
combined uncertainty by a coverage factor, typically taken
from the Student t-distribution (see [13]).
Evaluating Type A standard uncertainty us-A is based on
statistical analysis. It requires determining the average 𝑞 of a
quantity q and the deviation s(q) of its random variation,
obtained from n independent observations qk under the same
measurement conditions:
𝑞
=
(1)
Type A standard uncertainty us-A is the experimental standard
deviation of the mean s(𝑞), given by
𝑢

=
𝑠
(
𝑞
)
=
1
𝑛
(
𝑞
𝑞
)
𝑛
1
(2)
Type B standard uncertainty us-B is evaluated by scientific
judgement based on available information on the possible
variability of an input quantity Xi. The pool of information may
include calibration data or previous measurement data,
experience, manufacturers’ specifications, uncertainties
assigned to reference data, or any other reliable source of
information. It is also important to identify and evaluate
uncertainties arising in linear models due to nonlinear
phenomena. For example, in this experiment we have assumed
linear damping, but we can see in Fig. this assumption contains
a non-negligible amount of uncertainty, which must be
quantified and propagated through the data reduction equations
of interest. Hence, it is important to evaluate the significance
and quality of each variable of the experiment, where inputs
may be measured, assumed, or calculated.
Type B uncertainty in this experiment was evaluated
primarily using end-to-end instrument calibration data, by
performing a regression analysis to determine the standard error
of the estimate SEE:
𝑆𝐸𝐸
=
𝑢

=
𝑦
𝑦
𝑀
2
(3)
where and 𝑀 is the number of calibration points and 𝑦𝑗𝑦𝑗
is
the difference between the calibrated data point and the fitted
value.
In a similar manner, Type B uncertainty uB of the
hydrodynamic damping coefficient uB(δ), assumed to be linear
with a linear relationship between pressure pc and air volume
flux Q inside the OWC chamber (Fig. 8), was evaluated through
a multivariate normal regression analysis:
𝑦
=
𝑋
𝛽
+
𝑒
,
𝑖
=
1
,
,
𝑛
,
(4)
where yi is the vector of responses, Xi is a design matrix of
predictor variables, 𝛽 is vector of regression coefficients, and
ei is a vector of error terms, with multivariate normal
distribution. The error terms (difference between actual and
predicted values) are calculated for each independent
observation and substituted into Eq. 3 giving a measure of both
the repeatability (Type A) and accuracy in the linear damping
assumption (Type B).
The standard uncertainty us is the combination of Type A and
Type B uncertainties through the root-mean-square:
𝑢
=
𝑢

+
𝑢

(5)
In most cases, the DRE, for example OWC hydrodynamic
power, is not measured directly, but is determined from N other
(measured or assumed) input quantities X1, X2, …, XN through
a functional relationship f:
𝑌
=
𝑓
(
𝑋
,
𝑋
,
,
𝑋
)
,
𝑤𝑖𝑡
𝑒𝑠𝑡𝑖𝑚𝑎𝑡𝑒
𝑜𝑓
:
𝑦
=
𝑓
(
𝑥
,
𝑥
,
,
𝑥
)
(6)
The combined standard uncertainty uc(y) is the positive
square root of the combined variance uc2(y), given by
𝑢
(
𝑦
)
=
𝜕𝑓
𝜕𝑥
𝑢
(
𝑥
)
+
(7)
2
𝜕𝑓
𝜕𝑥
𝜕𝑓
𝜕𝑥
𝑢
𝑥
,
𝑥

Eq. 7 is referred to as the law of propagation of uncertainty.
The partial derivatives 𝜕𝑓/ 𝜕𝑥 (often referred to as sensitivity
coefficients) are equal to 𝜕𝑓/ 𝜕𝑋 evaluated at 𝑋=𝑥; 𝑢(𝑥)
is the standard uncertainty associated with the input estimate xi;
and 𝑢(𝑥,𝑥) is the estimated covariance associated with xi and
xj. For cases where estimated quantities xi are independent and
thus uncorrelated (𝑢(𝑥,𝑥)~=0), the second term in Eq. 7 can
be neglected. For cases where input estimates are correlated,
for instance pc and Q (Fig. 8), the degree of correlation between
xi and xj is estimated using Pearson's correlation coefficient (see
[2]).
Expanded uncertainty U is the combined uncertainty us
multiplied by a coverage factor k to give an overall uncertainty
with a 95% level of confidence. Coverage factor in this
investigation was determined using a student T-Distribution
[13].
𝑈
=
𝑘
𝑢
(8)
Alternatively, the Monte Carlo method may be used to
determine the combined uncertainty of a quantity, or of any
number of DRE’s which are a function of multiple independent
and/or correlated quantities, which can themselves be functions
of multiple quantities. Among other benefits of using MCM for
uncertainty propagation, the method avoids the need to provide
partial derivatives of difficult or inconvenient models, as
required by the law of propagation of uncertainty. This is the
primary reason we have explored its use in this paper.
Drawing on the basic methodology presented in [1], the
MCM flow chart shown in Fig. 1 (noted, this methodology
assumes us-A is calculated for the DRE of interest from multiple
observations using Eq. 2). First, the assumed true or nominal
values Xtrue of each quantity of a DRE are input. The estimates
of the elemental Type B uncertainties us-B for each quantity are
then input. An appropriate probability distribution function
(Gaussian, rectangular, triangular, etc.) is assumed for each
error source βj, with a Gaussian distribution used in this
example. For each quantity Xj, random values from a
(pseudo)random number generator are assigned to its error
source(s). The individual error sources are then summed and
added (or subtracted) to the true values of each quantity to
obtain “measured” values. Using these measured values, the
result of the DRE is calculate.
This process corresponds to running the Monte Carlo
simulation once (M = 1). The sampling process is repeated M
times to obtain a distribution for the possible DRE result values.
From this distribution, mean 𝑞MCM and standard deviation sMCM
statistics can be calculated (sMCM is the combined standard
uncertainty uc of the DRE). The selection of M depends on
when the standard deviation has converged (M = 1000 was used
in this study, see Fig. 3 for convergence study used to determine
M). Typically, convergence of 1-5% is reached after relatively
few iterations (< 500). Once a converged value of uc is obtained,
the expanded uncertainty for the result at a 95% confidence
level is U = 2 uc.
Fig. 1 Schematic of MCM for uncertainty propagation. Directly calculated
random standard uncertainty of the data reduction equation, us-A, is used
(shown as random error, εr).
B. Mathematical model
1) Hydrodynamic power: In line with previous experiments
of a similar nature [9, 11, 12], the air turbine power take-off
(PTO) system was assumed to exhibit essentially linear
pneumatic damping characteristics; that is, a linear relationship
between OWC chamber pressure pc and air volume flux Q
across the turbine, thus representing a Wells turbine [13-16]:
𝛿
=
𝑃
𝑄
(9)
The numerical derivation of volume flux Q (oscillating
airflow caused by change in wave elevation inside the chamber)
is:
𝑄
=
𝜕𝜂
𝜕𝑡
𝑑𝑠
=
𝑣
𝑑𝑠
=
𝑣
𝑆
(10)
where Sc is the area of the free surface, vs is the velocity at
which the free surface oscillates, and 𝑣
is the mean free
surface velocity inside the OWC chamber. The instantaneous
power P of the OWC is:
𝑃
(
𝑡
)
=
𝑃
𝑄
(11)
The mean power Ph is determined by integrating the
instantaneous power over the wave period.
𝑃
=
1
𝑇
𝑃
𝑄
𝑑𝑡
(12)
where T is the phase-averaged wave period. By introducing
Equation (9) into Equation (12), assuming linear wave theory,
the relationship is expressed as:
𝑃
,
=
1
𝑇
𝑃
𝛿
𝑑𝑡
=
1
2
𝐴
𝛿
(13)
where Ap is the amplitude of the chamber pressure measured by
the pressure sensor.
2) Incident wave power: The regular wave power Pi per
device width, assuming intermediate water depth, is calculated
as
𝑃
=
1
𝑇
1
2
𝜂
𝜌𝑔
𝐶
𝐿
(14)
where η0 is incident wave amplitude, ρ is water density, g is
gravitational acceleration, Cg is the group velocity and L is the
wave crest width corresponding to the OWC inlet width.
3) Capture width ratio: Hydrodynamic power Ph is non-
dimensionalised by the incident wave power Pi in order to
quantify its hydrodynamic efficiency in harnessing incident
wave power Pw. This parameter, commonly known as the
capture width ratio (CWR), is given by,
𝑃
=
𝑃
𝑃
(15)
And capture width ratio Pw,δ calculated with OWC power as
function of δ:
𝑃
,
=
𝑃
,
𝑃
(16)
C. Experimental setup
The 2D experiments were performed in the Australian
Maritime College Towing Tank. This 100 m long, 3.55 m wide,
1.5 m deep tank has a single paddle flap type wave generator at
one end and a damping beach at the other. The 1:8 scale,
breakwater integrated bent-duct OWC wave energy converter
(see [12]) was installed about halfway in the tank (Fig. 2). The
OWC model was built into a fully-reflective, flat-faced wall.
Refer to Fig. 2 and Table 1 for other further details about the
experimental program and parameters.
D. Data processing
Data processing and analysis has been performed following
ITTC recommended procedures and guidelines [8, 10, 17, 18],
and the Joint Committee for Guides in Metrology (JCGM) with
their guide to the expression of uncertainty in measurement
(GUM) framework [2, 7]. Time series data were trimmed such
that data used for analysis contained only that which was
considered stationary (see Fig. 5).
Fig. 2 Drawings of 1:8 scale OWC wave energy converter integrated into a flat-faced breakwater showing A side, B top and C front views. This model was
tested in the Australian Maritime College Towing Tank.
TABLE I
EXPERIMENTAL PROGRAM AND PARAMETERS
Parameter Details
Model 1:8 scale breakwater integrated OWC wave
energy converter, with linear PTO damping
simulated using porous mesh cloth (Enviro-
Cloth).
Water depth 1.47 m
Waves Long-crested regular waves:
H = 0.05 m, f = 0.3 – 0.45 Hz.
Measurement
Instruments
Incident wave probe, WPinc
OWC wave probe 1, WPowc1
OWC wave probe 1, WPowc2
OWC pressure sensor, PSowc (Honeywell
Controls TruStability)
Conditions 6 x linear PTO damping values:
δ1 ~= 1237 Nsm-5
δ2 ~= 2390 Nsm-5
δ3 ~= 3482 Nsm-5
δ4 ~= 4607 Nsm-5
δ5 ~= 5412 Nsm-5
δ6 ~= 6497 Nsm-5
E. Monte Carlo method for uncertainty propagation
Here we present a graphical presentation of the Monte Carlo
method simulation setup for propagating uncertainties, applied
to a generic OWC wave energy converter experiment (fig. 4A),
and the MCM simulation output of expanded uncertainty of
quantities and DRE’s fig. 4E). Illustrated in fig. 4A are the
nominal inputs for each quantity (measured or assumed), their
associated standard uncertainties (evaluated or assumed), and
the DREs required to calculate the final result of capture width
ratio Pw. Assumed nominal and uncertainty values have been
taken from standard reference materials. Some quantities (S, g,
h, ρ and L) are wave independent, whereas the remaining
quantities are wave dependent. figs. 4B, C, D show exemplar
results (i = 1, H = 0.05 m, f = 0.34 Hs, δ5 ~= 5412 Nsm-5 ) of
propagating quantities that are not single values, but time series.
For each simulation i, the basic method is that of multiplying
each individual data point of the selected time series with a
random number, taken from a normal distribution N.
The MCM was applied because most quantities are
inherently transient in typical WEC experiments, due to waves.
This leads to difficulty in evaluating and propagating
uncertainties. For example, propagating uncertainties through
integral functions is not practical using the law of propagation
of uncertainty, because one must provide the standard
uncertainty for each measured and influence quantity, and
partial derivatives of each DRE, some of which themselves
become variables in higher-level DRE’s, for example Cg. In
contrast, it can be seen that the MCM provides a
straightforward way in which to propagate uncertainties; only
the DRE and the standard uncertainty associated with each
quantity have to be provided.
To glimpse what the outputs of a MCM uncertainty
propagation looks like, expanded uncertainty functions of
primary DRE’s – incident wave power Pi, OWC power Ph, and
capture width ratio Pw – are given (fig. 4E). Also included are
example results of OWC power Ph and its constituents (Pc, Q),
after M = 1000 simulations. fig. 4F, H, J show the mean value
(black line) and the variation of each data point (grey dots). fig.
4G, I, K show the histogram of peak amplitude values,
including the mean (black line) and expanded uncertainty to a
95% confidence interval with k = 2 (black dashed lines). These
peak amplitude uncertainty values are reported as the final
uncertainty associated with final result statements.
To determine how many iterations of the MCM simulation
are necessary, a convergence study must be conducted (fig. 3).
Convergence is determined by calculating at each iteration the
combined standard deviation sMCM of a DRE of interest, for
example OWC power Ph. Subsequently plotting sMCM shows the
convergence behaviour. From fig. 3 it may be seen that after
only 300 iterations the value has converged to within 2% of the
fully converged value. We repeated this process for all primary
DRE’s, and deemed that after 100 iterations the values were
fully converged.
As one may see, performing a pre-experiment uncertainty
analysis using the MCM would be also be relatively
straightforward, by assuming nominal values and analytical
wave signals with estimated elemental uncertainties
Fig. 3 Convergence of expanded uncertainty U (2.sMCM = U) of OWC power
Ph.
III. RESULTS & DISCUSSION
We report the results and discuss them in three parts. First,
time series data show how data were trimmed in the stationary
region for analysis, as well as provide an indication of the
repeatability of experimental runs. Second, uncertainty of
measured quantities is presented and, third, results for the
propagation of uncertainty using the MCM. Throughout this
section, while we report and discuss the results of dependant
variables in their context, we do so only briefly; the focus is on
reporting and discussing uncertainty results. Uncertainty results
reported are those calculated at the peak amplitude quantities
and data reduction equations, to a 95% confidence interval (CI).
Fig. 4 A graphical representation of the Monte Carlo method for propagating uncertainties in a generic OWC wave energy converter experiment. A, MCM
simulation (M = 1000), with example results of time-variant quantities of OWC power Ph for i = 1, with black profiles being the actual time series and the grey
dots the standard uncertainty multiplied by a random number taken from a normal distribution N. (B, C, D). E, MCM simulation output: expanded uncertainty
U (95% confidence interval) of each quantity and data reduction equation, with example results plotted for i = 1:M (F, H, J). G, I, K, histogram subplots
showing the mean (black line) and expanded uncertainty (dashed lines) of Pc,max, Qmax and Phmax respectively. Example results are H = 0.05 m, f = 0.34 Hs, δ5
~=
5412 Nsm-5.
A. Time series
Selected repeat experimental runs of one wave condition
(H=0.05 m, f=0.45 Hz) indicate minor observed variation in
both magnitude and phase for measured quantities (Fig. 5).
Incident wave profiles in Figs. 5A, B, and C include one run for
each of the six damping conditions (coloured lines) along with
nine dedicated repeat runs (black lines). The standard deviation
of the mean (Type A standard uncertainty uA) for wave height
H and period T, determined from each individual wave defined
by the zero-up crossing, was 0.0008 m and 0.0008 s
respectively. While in general the profiles show sinusoidal
characteristics, there are slight nonlinearities observed in the
first three waves of the trimmed time series profiles. It is
assumed these nonlinearities have a negligible influence on the
primary wave parameters of H and T.
Measurements inside the OWC chamber of pressure pc and
surface elevation ηowc in Figs. 5D, E, and F include nine
dedicated repeat runs. Similarly, these results indicate only
slight variation and, in effect, are highly repeatable
experimental runs for this wave condition. There are minor
nonlinearities observed in the pc profiles around the zero
crossing, which may be attributed to a physical effect in the
hogging and sagging of the damping simulator (porous mesh
material) after ηowc reaches its maximum and begins to fall
causing a positive-negative change in pc. Wave probes inside
the OWC chamber WPowc1 and WPowc2 evidently measured
similar ηowc quantities, indicating the free surface inside the
chamber was essentially level during its cycle. This confirms
the validity in the flat-plate assumption used to numerically
derivate air volume flux Q. Noted, wave frequencies around
resonance exhibited up to 10% variation between ηowc1 and ηowc2,
which hinders the said assumption, however this uncertainty is
accounted for in the standard error of the estimate of the
pneumatic damping coefficient.
B. Uncertainty of measured quantities
To quantify the uncertainty of measured quantities, we first
evaluated standard uncertainty components and then used the
MCM to combine these standard uncertainties, giving the
expanded uncertainty. For each of the measured quantities in
the experiment, expanded uncertainties U were less than 10%
on average, with Type A uS-A and Type B uS-B standard
uncertainties less than three percent on average (Fig. ). All uS-A
results for measured quantities contained outliers, the
maximum of which reached five percent. Looking at uS-B results,
we can see that all the data fall below 2.5% uncertainty. These
results are very similar to those presented in [5] and the
example in [12], with the exception of uS-B(pc) in this
experiment, which had a smaller uncertainty of 2.5%, with 0.5%
arising from calibration data of PSowc, and 2% assumed to
account for for the uncertainty of the instrument used to
calibrate the pressure sensor. In comparison, uB(pc) in [5] was
~ 5%. This reduced uB(pc)is a consequence of testing at a larger
scale of 1:8 (compared with 1:20), such that the calibration
range of PSowc for the 1:8 scale was larger and thus the variation
relatively smaller.
Fig. 5 A-E, Synchronised time series profiles of repeated runs (overlayed) of one wave condition (H=0.05 m, f=0.45 Hz). A, incident wave surface elevation,
ηinc; B, cropped data in the stationary region; C, a crop of one incident wave to aid examination of the repeatability of incident waves; D, surface elevation
inside the OWC chamber, ηowc1,2, measured by wave probes WPowc1,2 (left axis), and pressure inside the OWC chamber, pc (right axis); E, cropped data in the
stationary region, used in the analysis; and F, a crop of one wave showing repeatability and relation of pc and ηowc1,2 inside the OWC chamber.
C. Uncertainty of data reduction equations
Uncertainty results for the data reduction equation’s (DRE’s)
of interest were generally larger compared with measured
quantities’ uncertainty (Fig. ). In terms of standard uncertainty,
uS-A was always smaller than uS-B. The largest uS-B result was
observed in pneumatic damping δ, with approximately four
percent uncertainty on average, and a maximum of 15%. This
result was primarily due to the systematic uncertainty arising
from the linear damping assumption (Eq. 4). Air volume flux Q
showed an approximate three percent uS-B value, with this uS-B
uncertainty component being a combination of the OWC wave
probes’ calibration data (see Eq. 10) and an assumed 2%
uncertainty to account for the piston assumption of water
elevation inside the OWC chamber. There were no uS-B
components estimated for incident wave power Pi, OWC power
Ph,, OWC power Ph,δ as function of δ, capture width ratio Pw, or
capture width ratio Pw,δ as function of δ, as these are functions
of other DRE’s and quantities. Noted, Ph,δ and Pw,δ are the result
of calculating OWC power using Equation 13, with Ph,δ being
a function of Pc and δ, and then using this Ph,δ to calculate
capture width ratio Pw,δ as function of δ. These power
calculations with δ are useful for scaling up power results and
associated uncertainty, as they take into consideration the linear
damping assumption.
Expanded uncertainty U results for DRE’s were less than
~15% on average. However, some values of δ, Ph,δ and Pw,δ
reached 25-30%. Pw,δ showed largest overall expanded
uncertainty, as expected considering it is a function of both
OWC and wave power. Ph,δ and Pw,δ calculations as a function
of δ and Pc (Equation 13) were larger compared with Ph and Pw,
due to δ having a large Type B uncertainty. In the context of
relevant work [5], where an experiment was conducted on a
1:20 scale model version of the 1:8 scale model investigated
here, these expanded uncertainty results are less than half of
those reported in previous work, for reasons discussed above.
To further the analysis of uncertainty in the primary DRE’s
– incident wave power, OWC power and capture width ratio –
we then plotted the results of each experimental condition
across the wave frequency range tested (Fig. ). The following
sections report on these results and discuss their significance.
(1) Incident wave power: Incident wave power Pi decreased
monotonically as wave frequency increased, and the
uncertainty associated with these results was less than 5% on
average (Fig. A). Of this, ~60% was due to the measurement of
the incident wave elevation ηin, with the other four variables of
Pi making up the remaining ~40% (Fig. B). Slight variation of
incident wave power was observed between experimental
conditions (δ1-6) of the same wave condition, evident in
different magnitudes for a particular wave frequency. The
largest difference in magnitude was observed to be 5%.
Interestingly, there was also a systematic difference in
magnitude between experimental runs for the frequency range
tested. This is observed by inspecting, for example, δ4 (square
marker) and δ5 (diamond marker) where Pi for δ5 is always
larger than δ4, by ~2% on average. This illuminates a systematic
error in the experiment; the measurement of ηin is the
predominant source of this systematic error, which we infer is
due to experimental runs being conducted on different days,
and therefore subject to a separate calibration set.
2) OWC hydrodynamic power: The effect of pneumatic
damping on OWC power was observed to be significant around
the resonance frequency of the WEC, both in terms of
magnitude and the broadness of the response spectrum (Fig. C).
At the resonant frequency of f = 0.37 Hz, Ph,δ for the lightest
damping condition δ1 was approximately twice the value
observed for the heaviest damping condition δ6. In general, the
heavier the damping, the lower the Ph,δ, except at ~0.1 Hz either
Fig. 6 Uncertainty analysis results, including A Type A us-A, B Type B us-B, and C expanded uncertainty U. Boxplots represent uncertainty results for each
experimental run of each condition, with normal convention (box = median with 25th and 27th percentiles, whiskers = extreme points not considered outliers,
circles = outliers. B, sensor calibration data constitutes us-B for measured quantities, whereas us-B for δ is the systematic uncertainty in the linear damping
assumption. C, U is to 95% CI, with k varying depending on n of each respective experimental run.
side of the peak, where this trend begins to reverse, due to the
peakier behaviour of light damping. In terms of the power
response spectrum, increasing damping increased the
broadness of the spectrum across the frequency range.
Uncertainty results (bar plots) for Ph,δ showed a trend of
increasing absolute uncertainty with increasing magnitude of
Ph,δ. This trend agrees with findings of a similar study [5]. The
largest observed uncertainty of 17.8% occurred at the resonant
frequency for the δ1 condition. Fig. D shows that of this 17.8%,
~80% of the uncertainty was due to δ, and ~20% due to Pc.
Fig. shows the damping results for each condition, whereby
Pc and Q are plotted against each other, from which the
uncertainty in the linear damping assumption be visualised.
From Eq. 9, theoretical δ is calculated from the ratio of pc and
Q; however, experimentally δ is determined by taking the
gradient of the multivariate linear regression line of pc vs. Q.
This damping coefficient is assumed to be frequency
independent; thus, it represents the damping on the system for
all frequencies tested for that damping condition. Evident in
Fig. , however, is nonlinear behaviour exhibited by pc and Q. The
nonlinearity generally increased with decreasing δ, that is,
fewer number of porous mesh damping simulators. The SEE of
the smallest δ (δ = 1237 Nsm-5), corresponding to one layer of
damping mesh, was on average 8.5%, with maximum 15.0%,
and minimum 5.5%. This variation can be attributed to a
physical effect where the mesh damping simulator does not
exhibit completely rigid characteristics, rather it hogs and sags
slightly due to pc and Q driven by ηowc. This linear damping
uncertainty is considered a Type B uncertainty, and is
propagated as such through the hydrodynamic power Ph DRE
using the MCM.
4) Capture width ratio: The trends and uncertainty results
of pw are similar to those of Ph,δ (Fig. E). This was expected due
to Pi‘s linear decreasing trend. It can be see that Pw,δ reaches up
to a value of three. Put another way, the breakwater integrated
WEC is harnessing three times the incident wave energy, due
to reflection and resonance. The uncertainty results are very
similar to those of Ph,δ, due to Pi uncertainty making up
Fig. 7 Main data reduction equation (DRE) results (left axes) with associated expanded uncertainty (right axes), evaluated using the Monte Carlo metho
d (95%
CI, k = 2). A, Incident wave power Pi, with B showing the proportion of each quantities’ uncertainty within the Pi
DRE; the maximum observed uncertainty of
P
i is selected to show by example (box on bar plot). C, OWC power Ph, with D, proportions of maximum uncertainty. E, Capture width ratio Pw, with F,
proportions of maximum uncertainty.
approximately one third of Pw,δ ‘s total uncertainty. These
results compare favourably to similar work [5]; however, it is
assumed these results are more accurate in describing the
overall uncertainty of the presented DRE’s due to using the
integral of the phase-averaged result rather than simply the
mean of the amplitude. In this way, the uncertainty in nonlinear
behaviour is captured and propagated through the DRE’s.
Fig. 8 Pc vs. Q, showing the damping δ for each condition (δ1-6) and the
associated Type B uncertainty determined through linear regression (Eq. 6).
IV. CONCLUSIONS
To avoid uncertainty propagation problems, such as
providing partial derivatives for complex models with many
time-varying variables and data reduction equations, we
explored using the Monte Carlo method as a practical
alternative for uncertainty analysis. We provided the principles
of uncertainty analysis and the methodology of a MCM
uncertainty analysis, along with a graphical representation and
demonstration of propagating uncertainties of time-varying
quantities.
Standard and overall uncertainty results for measured
quantities in the experiment were presented. Standard
uncertainty results – Type A and Type B – averaged less than
three percent to a 95% confidence interval. These uncertainties
along with other influence quantities were propagated through
the DRE’s related to OWC power performance. Incident wave
power expanded uncertainty across all frequencies testing
averaged less than five percent. The largest expanded
uncertainty for pneumatic damping was 31% for δ1,
corresponding to one layer of porous mesh, with a mean of
8.6%. Hydrodynamic power and capture width ratio showed
similar result trends and expanded uncertainties, with a
maximum of ~18% observed for δ1 condition.
While we focused only on a post-experiment MCM
uncertainty analysis (reporting phase), the general
methodology may be applied to pre- and during-experiment
phases, including planning, design, debugging, construction,
execution, and data analysis. Moreover, the MCM is a suitable
and straightforward way in which to perform an uncertainty
analysis on complex WEC models. Therefore, we recommend
future guidelines and codes pertinent to uncertainty analysis for
WECs, such as those developed by the International Towing
Tank Conference (ITTC) and international Electrotechnical
Commission (IEC), to incorporate the MCM and provide a
simple practical example.
In future work we intend on investigating the use and
usefulness of pre-test uncertainty analysis using the Monte
Carlo method.
ACKNOWLEDGMENT
We thanks the AMC staff, technicians, and students for their
contribution of this work.
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