30-07-2012, 12:50 PM
Extreme Value Analysis of Wave Energy Converters
Analysis of Wave Energy.pdf (Size: 389.93 KB / Downloads: 139)
ABSTRACT
This paper presents a statistical Extreme Value Analysis (EVA)
methodology to evaluate the design survivability condition of a Wave
Energy Converter (WEC). The technique is applied to the Oyster®
WEC design which is being developed by Aquamarine Power Ltd.
(APL) but can be easily modified to investigate other technologies that
have different operational philosophies. The approach presented
considers the extreme statistics of both the incident wave climate at the
device location and the complex dynamic response of the WEC in such
conditions. This give a more robust evaluation of the WECs
survivability condition than a more traditional single design wave
approach.
INTRODUCTION
An Extreme Value Analysis (EVA) technique is presented in this paper
to evaluate the loading survivability condition of a Wave Energy
Converter WEC. Determining the maximum expected load experienced
by a WEC during its operational lifetime is of paramount importance to
the design process, in particular to the design of the foundation and/or
mooring system. Thorough and consistent EVA can reduce/remove the
implementation of large safety factors to a WEC design load condition
which can have significant cost benefits. Extreme loads experienced by
a WEC are not only dependent on the nature of the wave climate within
which it is operating but also on the phase relationship between each
incident wave and the WECs dynamic response. These unique features
make it necessary to develop a custom EVA technique with specific
application to WECs. A possible hazard of developing an EVA
methodology is that it relies heavily on novel analysis techniques.
Despite its mathematical elegance, industry confidence in such a
technique is often low because of the novel approaches employed.
EXTREME VALUE METHODS
In recent years the significance of extreme value analysis has become
ever more important to a wide variety of disciplines. This has lead to a
large development of eloquent and novel statistical techniques, which
unfortunately have been accompanied by a gross misuse in their
application. Other simpler approaches such as inferring extreme values
from an empirically-approximated parent distribution also leads to
large uncertainties due to the fact that the extreme values are very often
driven by a different physical process than the parent population on
which the approximate distribution is based. In addition to this, basic
empirical fitting is subject to interpretation as very often there is no
basis for choosing one type of distribution over another.
The Peak-Over-Threshold Method
The POT approach generates a subset of data points from a parent set
by only considering those events (data peaks) above a defined
threshold. By only considering peaks above a threshold the data is more
than likely to be from the same distribution. This intrinsically assists
with obtaining an identically distributed data set. In addition to this,
provided the data peaks can be considered statistically independent,
thus the i.i.d. condition is satisfied, the distribution of the peak events
should have a Generalised Pareto distribution.
DETERMINATION OF EXTREME SEA STATES
This section describes the methodology used to derive the extreme sea
states that a WEC is likely to encounter in its lifetime. In particular the
study focuses on the Oyster® WEC located at the EMEC site in
Orkney.
The ultimate scope of this entire piece of work is to evaluate Oysters
survivability load conditions, it is expected that wave height will have
the dominant effect on the extreme loads, potentially modified by the
wave period due to the dynamics of the device and the effect of wave
steepness on wave breaking and water particle acceleration. Thus, the
significant wave height (Hs) and spectral mean wave period (Tm =
m0/m1) are analysed for specification of the extreme sea-states. Also at
this stage of the analysis the POT method is selected over the BM
method to conduct the EVA on the wave climate conditions as seasonal
variations in significant wave height are only accounted for by
selecting block sizes of 1 year. This has severe restrictions on the
amount of data utilised in the BM method which would significantly
increase the uncertainty of the estimate of the extreme values.
Significant Wave Height POT Data Set
The selection of a suitable Hs threshold level is key in achieving a
robust data set. This is done by inspecting the behaviour of the fitted
Generalised Pareto distribution parameters and mean of the excess as
outline in the Peak-Over-Threshold section.
Although the POT method intrinsically assists with selecting an
identically distributed data set, it is good practice to actively ensure that
this condition is satisfied. Different methods have been proposed to
impose statistical independence of events. (Morton and Bowers 1996)
proposed the assumption that meteorological events are independent
over a period of 30 hours. As the wave climate is closely correlated to
meteorological events this assumption was supported by analysis off
wave heights of off the coast of Lowestoft. The methodology adopted
here however defines the temporal extent of each event to last until the
significant wave height has dropped to 1.0 metres less than the
threshold, thereby ensuring that there is a period of relative calm
between events. It is suggested that these periods of calm provide
delimiters to meteorological events to ensure that they are statistically
independent. The statistical independence of the peak events selected
using this method is tested by calculating the correlation coefficient
given in Eq. 1. Analysis of the current data set results in a correlation
coefficient of -0.18 which suggests that the peak events are indeed
independent.
Joint Probability Distribution Function
The joint probability density function of significant wave height and
mean period is required in order to calculate the return period of the
extreme sea states. The joint probability density function is essentially a
theoretical reconstruction of the wave resource scatter table. The joint
probability density function can be calculated in a variety of ways
depending on how the marginal and conditional density functions are
constructed. In this case it is simply a multiplication of the marginal
probability density function of the significant wave height with the
conditional probability density function of the wave period. However,
because the marginal and conditional probability distributions
calculated refer to the probability of a peak event occurring
consideration must be given to large non-peak events as outlined
below.
EXTREME FOUNDATION LOAD
Extensive experimental testing of the Oyster 2 device was conducted in
the wave tank facilities at Queens University, Belfast. In particular,
Oyster was tested in the 1, 50 and 10,000 year return period extreme
sea states defined in the previous section and all foundation loads
experienced by the device recorded. As stated earlier a single event in a
wave resource scatter table is the average over a 3 hour period. Thus
each sea state tested in the wave tank is run for the equivalent of over
hours assuming a Bretschneider wave spectrum. However, because the
short term statistics of the foundation loads are strongly dependent on
wave-device interactions, it is important to capture as many different
wave-device phasing events as possible. Thus all sea states were
repeated (with different wave phasing induced each time) so that at
least 4000 wave cycles were achieved. This also has the added
advantage of increasing the size of the data set which is beneficial for
any statistical analysis. However, there is a trade off between quality or
robustness of the data set and the time taken to execute the
experimental tests.
At this stage of the analysis there is no evidence to suggest that one
extreme value method should be preferred over the other, thus both the
POT and BM methods are employed. To demonstrate each EVA
method the surge foundation load is used as a representative metric for
Oysters survivability condition.
CONCLUSIONS
An Extreme Value Analysis technique has been developed to assess the
extreme foundation loads of a Wave Energy Converter. In particular the
Oyster® device was examined in a wave climate condition typical of
the EMEC test site in Orkney. However the technique could be applied
to any other type of Wave Energy Converter. The long term extreme
wave statistics were evaluated using a Peak-Over-Threshold method to
determine the sea states which Oyster® may encounter at this site. The
short term extreme load statistics induced by wave-device interactions
were explored using both a Peak-Over-Threshold method and a Block
Maxima method. Although the results from both methods were in
excellent agreement, the Peak-Over-Threshold method is favoured as it
is less wasteful of the available data and its implementation is assisted
by some attractive mathematical features.