EWEA Offshore 2015
Systematic quantification of wake model uncertainty
Nicolai Gayle Nygaard
DONG Energy Wind Power
Kraftværksvej 53, 7000 Fredericia, Denmark
[email protected]
SUMMARY
Wake models attempt to describe the complex flow physics through a wind farm mathematically.
Due to the turbulent nature of the flow this is very complicated, and modellers have to balance
the need for accuracy and precision against fast calculation of the wind farm energy output. The
latter is especially relevant in the process of layout optimisation, where calculation of the wind
farm production is required while iterating through a multitude of different layouts. To help strike
the proper balance, and to guide further model developments, wake model validations using
observational data play a crucial role. Here we set up a formal wake model validation procedure
and use it to analyse data from 10 operational offshore wind farms. We develop a systematic
framework for quantification of the wake model uncertainty. Our results show that existing
estimates of wake model uncertainty are conservative.
1.
INTRODUCTION
The flow of the wind through a wind farm is impeded by the turbines themselves. As a turbine is
extracting power from the wind, it leaves a wind speed deficit in its wake. Downstream turbines
in the wake face a reduced wind resource and thus produce less power compared with an
upstream turbine facing the free wind. The wake loss is the difference between the gross
production of the wind farm in the imagined case of no wakes, and the actual production
considering the wake effect, but before applying any other losses. Wake models are designed
to predict this loss based on an approximate mathematical description of the wind farm flow
physics.
Wakes are typically the largest source of losses in an offshore wind farm. Hence it is imperative
for the wind energy industry to understand the bias and uncertainty of wake models when
applying them to predict the annual energy production (AEP) of wind farms. A bias in the wake
model prediction can adversely affect the business case, while the uncertainty represents
financial risk for an investor. Hence the uncertainty associated with wake modelling impacts the
cost of financing for the wind farm developer or the fair value of the asset in a divestment case.
As offshore wind power continues to expand both in cumulative installed capacity and in the
sizes of turbines and wind farms, a mounting body of data is available from operating wind
farms. This contribution emphasises how data analysis can increase confidence of developers
and investors by testing model assumptions and validating model predictions against measured
data.
The benchmarking of wake models against data from operational wind farms is an active area
of research [1-5]. However, much of the work on wake model validation hitherto has been
qualitative and restricted to particular flow cases. In this paper we develop a systematic
framework for validation and uncertainty quantification of wake models, focusing on the ability to
predict the total wind farm energy production. An objective approach to benchmarking facilitates
the rating of different wake models and provides a structured platform for evaluating model
improvements.
The quantification of model uncertainty can follow two distinct paths. The traditional method is
forward uncertainty propagation, where the probability distribution of the model outputs is
assessed from the known or estimated uncertainty of the inputs and model parameters for
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example by Monte Carlo simulation [6] or sensitivity analysis [7]. While rigorous, it relies on
estimates of the constituent uncertainties that may be subjective. The alternative path we follow
here is inverse uncertainty quantification. Based on comparisons between measured data and
model outputs it estimates the distribution of the discrepancy between experimental results and
the model predictions. The width of this model error distribution determines the model
uncertainty; the mean value measures the model bias.
In the next section we define the wake model uncertainty and give examples of its estimated
magnitude characteristic of the current expectations and practices of the wind industry. Section
3 develops the new framework for the quantification of wake model uncertainty . This is based
on a validation procedure coupled with resampling of the validation data to generate the
statistical distribution of the wake model error. The uncertainty quantification framework is
applied to 10 offshore wind farms in Section 4 for an example wake model, producing a
portfolio-spanning estimate of the wake model uncertainty. In the concluding section we discuss
the implications of the results.
2.
WAKE MODEL UNCERTAINTY
A wake model generates a prediction for the wind farm wake loss based on the layout, the
turbine characteristics such as hub height, rotor diameter, power and thrust curves, and the
wind climate. Normally the model is deterministic: given the same inputs, it predicts the same
wake loss. So how can we talk about a wake model uncertainty? What is uncertain is how well
the model prediction reflects reality. On one hand, the inputs have an inherent uncertainty, and
this parametric uncertainty is transferred to the model prediction. On the other hand, the
mathematical model does not correspond perfectly to the physical reality. The wake model may
inadequately capture important aspects of the flow physics such as the influence of atmospheric
stability and turbulence on the wakes. This leads to a structural uncertainty, since the realised
wake loss depends on variables that are not appropriately included in the model .
The existing literature on the benchmarking of wake models for offshore wind farms is extensive
(see [1-4] and references therein). But with few exceptions [5] the emphasis has been on
qualitative comparisons between observations and wake model predictions of power outputs
along a row of turbines. Such validations are typically undertaken for a limited range of inflow
conditions, and they play an important role in verifying the ability of wake models to resolve the
detailed wake structure. However, they are not aligned with the standard use of wake models in
wind resource assessment, which is focussed not on the production of individual turbines for
specific wind directions and wind speeds, but on the power output of the entire wind farm
considering the full wind climate.
In contrast, our interest here is the uncertainty of the predicted annual energy production in a
wind farm energy yield calculation. Specifically, we focus on the component of this uncertainty
arising from the modelling of the internal wind farm wakes. The uncertainties of losses
associated with flow modelling [8], power curves, availabilities, and electrical losses can be
independently quantified. If the losses are independent their uncertainties can subsequently be
combined.
Typically, the probability distribution for each loss factor is assumed to be normal, and the
uncertainty is defined as one standard deviation, equivalent to specifying the 68% confidence
interval around the mean value. Here we do not assume normality of the distribution of the wake
loss. Rather, we define the wake model uncertainty directly as the half-width of the 68%
confidence interval estimated from the sample distribution.
To make results comparable between wind farms with different wake losses we work with the
relative uncertainty. That is the uncertainty normalised with the wake loss. As an example, let
the wake loss be 12% of the gross annual energy production and the (absolute) wake model
uncertainty 6% of the energy yield. The relative uncertainty is then 50% of the predicted wake
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loss and corresponds to the belief that the true wake loss is between 6% and 18% with 68%
confidence.
To get a sense of how wake model uncertainty is currently estimated in the wind energy
industry we list in Figure 1 the relative wake model uncertainty cited by independent consultants
in a number of energy yield reports prepared for DONG Energy for different offshore wind farms.
Note that the wake model used was not the same in all cases. Though unscientific, this
comparison reveals that the estimated uncertainty on wind farm losses ranges from about 25%
to 60% of the modelled wake loss. The average among the reports accessible for this study was
a relative wake model uncertainty of 44%. In the absence of adequate data from operational
offshore wind farms, such estimates naturally contain a certain amount of subjectivity, as
reflected in the large range of values.
Figure 1. Histogram of relative wake model uncertainties as estimated by independent
consultants for DONG Energy.
The remainder of this paper presents a systematic framework for quantifying the wake model
uncertainty based on extensive analysis of data from operational offshore wind farms.
3.
WAKE MODEL VALIATION AND UNCERTAINTY QUANTIFICATION
Our benchmarking observable is the wind farm wake loss 1 ¡ hPneti=hPgross i, defined from the
ratio of the mean net production to the mean gross value. The angle brackets h: : :i denote the
mean value. This is the relevant measure of wake model performance for wind farm planning.
Had we instead used hPnet =Pgross i in the definition, the net power could not be calculated
directly from the wake loss and the gross power. As validation metric we use the relative
difference between the observed wake loss and that predicted by the wake model. The mean
value of this model error is a bias, whereas its variation is a measure of the model uncertainty.
To quantify the relative wake model uncertainty we need to find the distribution of the wake
model error. In the following we describe a wake model validation framework designed to
determine both the bias and uncertainty of a wake model by comparing its predictions with
observed data from a large number of operational wind farms. The uncertainty established
through this systematic framework is traceable, because it derived from a series of well-defined
steps, and it is objective, since it is based not on judgement but on data analys is. In this paper
the framework is applied to determining the bias and uncertainty of the N. O. Jensen wake
model [9,10], but the procedure is generally applicable to any wake model. The only required
input is a power matrix describing the wind farm power production as a function of wind speed
and wind direction with sufficient resolution.
The benchmarking presented here encompasses 10 offshore wind farms located in Denmark
and the United Kingdom, Figure 2. They represent a wide variety of different shapes and sizes,
as indicated in Figure 3 where they are drawn according to their respective proportions. The 10
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wind farms cover a decade of evolution in turbine technology, which is reflected in a 50%
increase in blade length from the oldest to the newest wind farm. The layouts span from regular
arrays to curved and irregular shapes with dense borders.
Figure 2. Map showing the locations of the wind farms included in this validation study. Walney
is included twice in the validation, since the wind farm was built in two stages.
The 10-minute averaged data from the turbine Supervisory Control And Data Access (SCADA)
system are filtered to remove turbine stops and curtailment events that lead to reduced
production, which could be mistaken for wake effects. To increase the data volume we only
require 95% of the turbines to be fully operational for a given timestamp. The set of timestamps
meeting this limit constitute the validation data set that forms the basis of the analysis. Where
neighbouring wind farms have a significant impact on the wake losses, production data from the
period after first power of the neighbour have been disregarded. For each point in the time
series the free stream wind speed and the wind direction are estimated from wind turbine
SCADA data using steps described elsewhere [4]. This defines an effective wind climate for the
validation sample.
We never have access to observations spanning the full lifetime of the wind farm. For practical
reasons the observations are limited to at most a few years. These may coincide with years of
unusually low or unusually high mean wind speeds and could therefore be far from the longterm wind conditions at the site. In addition, the filtering used to eliminate loss of production not
originating from wakes will further distort the distribution of wind speed and wind direction in the
validation sample with respect to the long-term wind climate. Hence it is not possible to directly
validate the long-term AEP of the wind farm or indeed the mean array efficiency it operates at
over its lifetime. Instead, we define the validation wake loss [5]. This is based on the ratio of the
mean value of the produced power to the mean value of the gross power of the wind farm, when
the averages are taken over the validation sample only. The predictive power of the wake model
can be directly compared with this validation wake loss, provided we use wind climate of the
validation sample as input to calculate the modelled net and gross productions.
3.1
Observed wake loss
The wake loss is defined as the wake-induced reduction of the wind farm power production
relative to the gross production in the absence of wake effects:
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Lossobs = 1 ¡
hPnet iobs
hPgross iobs
(1)
Here hPgrossiobs is obtained by calculating the mean power of the free stream turbines at each
time step and multiplying it by the number of turbines to produce a time series of the gross
power. This is subsequently averaged over the validation sample. The mean net power hPnet iobs
is the combined production of the fully operational turbines, scaled to the number of turbines in
the wind farm and averaged over the validation sample. Notice that (1) uses the ratio of the
mean net and gross power, as opposed to the mean of the ratio; hPnet=Pgrossiobs. With the
definition in (1) the wake loss can be used to directly link the net and the gross AEP.
Some timestamps are included, where up to 5% of the turbines may be experiencing stops or
curtailment. The power of these turbines is not counted toward either the gross or the net
production. However, a partially operational turbine does produce wakes on the turbines
downstream of it. Therefore, by allowing some turbines to be shut down or operating in a nonideal state we increase the amount of useable data, but also introduce a small bias in the
observed wake loss, which is slightly overestimated. Numerical experiments have shown that
this effect is small and does not alter the conclusions presented here.
Figure 3. Layouts of the included wind farms, drawn according to their relative sizes and
grouped by the length of the blades. North is toward the top of the page. Note that the Walney 1
turbines have a smaller rotor than those in the neighbouring Walney 2 wind farm to the northwest. Hence it is shaded in a lighter colour (top right corner of the figure).
3.2
Modelled wake loss
The implementation of the N. O. Jensen model used here is equivalent to the PARK model
implemented in WindPRO™. We have used a wake decay parameter of 0.04 for all wind farms ,
which is recommended for offshore applications [1].
The gross power in the modelled results is calculated in each wind speed bin from the power
curve multiplied by the number of turbines. By averaging this over the validation wind climate
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we obtain hPgrossimodel. In the case of Walney 1+2, which consists of two sub-farms with
different turbines, the gross power is calculated separately for each sub-farm and then added.
The average net power hPnetimodel
power production of the wind farm
direction. The entries in this power
validation sample. We can then find
predicted by the wake model is derived by calculating the
as a function of the free stream wind speed and the wind
matrix are then weighted by the wind climate based on the
the modelled wake loss as
Lossmodel = 1 ¡
hPnetimodel
hPgross imodel
(2)
Since the modelled wake loss is based on the validation sample wind climate it will not be
identical to the long-term wake loss for the wind farm. But in this way a direct one to one
comparison with the observed wake loss is possible.
3.3
Wake model error
Our validation metric is the relative wake model error:
"=
Lossobs ¡ Lossmodel
Lossobs
(3)
The significance of the sign is:
 " < 0: The model is overestimating the wake loss. It is AEP conservative
 " > 0: The model underestimates the wake loss. It is AEP optimistic
The validation sample gives us one value for the relative wake model error for each wi nd farm.
To get the distribution of the relative wake model error we need a statistical resampling
approach. This is introduced in the next sub-section.
3.4
Bootstrap uncertainty quantification
The bootstrap is a general and versatile method for inferring arbitrary statistics from sample
data [11]. Starting from a sample of data drawn from an unknown distribution, one draws
random bootstrap samples, by sampling the data with replacement. The distribution of any
sample statistic (mean value, median or something more complicated) over the population can
be estimated by the distribution of the statistic evaluated over the bootstrap samples. From the
bootstrap distribution of the statistic it is thus possible to asses both its mean value and a
confidence interval. In our case the statistic of interest is the relative wake model error (3).
The original bootstrap assumes that the data are independent and identically distributed random
variables. For a time series this is not true, since dependence exists between the observations.
In the case of wind farm data this dependence arises due to the synoptic variations in the wind.
For this reason we use the circular block bootstrap variant of the method [ 12]. It consists of
dividing the time series of SCADA data into overlapping blocks of length k and wrapping the
time series in a circle to connect the beginning and the end. These blocks are then sampled
randomly with replacement, until a synthetic time series of the same length as the original has
been generated. This constitutes a bootstrap sample. The process is repeated to generate B
distinct bootstrap samples, see Figure 4. For each bootstrap sample we calculate the observed
wake loss (1) and the modelled loss (2). For the latter the wind climate specific to the bootstrap
sample is used as input to the calculation.
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Figure 4. Illustration of the bootstrapping procedure. Left: the validation time series of filtered
SCADA data. Each time stamp is a separate colour. The size of the circles represent the wind
farm power production, whereas the direction and length of arrows indic ate the derived wind
direction and free wind speed, respectively. Right: the validation data are sampled with
replacement, generating bootstrap samples of the same length as the validation sample. In the
simplified example here the block length is 3 (30 minutes). The real block length is 2 days.
Each bootstrap sample contains parts of the original sample, but has a unique wind climate and
a distinct distribution of variables such as the turbulence that impact the wake losses.
Consequently, when we evaluate the relative wake model error (3) for all the bootstrap samples,
we get a distribution of values. To be more specific: a wake model may overestimate the wake
loss in some directions and underestimate it in other wind directions. Overall these errors may
tend to cancel each other out, leaving a small model error. But as the wind climate changes
between different bootstrap samples, the degree of error cancellation will be modified. This
results in a range of possible relative wake model error values. On the other hand, while the
wake model may be insensitive to variations in for example the turbulence or the atmospheric
stability, these do affect the observed wake losses in the bootstrap samples. So bootstrap
samples with a large amount of turbulence will tend to have smaller observed wake losses and
therefore smaller values of Lossobs ¡ Lossmodel than bootstrap samples with low turbulence.
Finally, the experimental measurements and the derivation of the free stream wind speed and
wind direction have associated uncertainties of their own, but in a pragmatic and conservative
approach we include those implicitly in the model uncertainty.
Figure 5. Example of the histogram of bootstrap relative wake model error results for a single
wind farm (orange bars). The mean value is indicated by the solid line. The dashed lines are the
th
th
16 and 84 empirical percentiles and are taken to represent the 68% confidence interval.
Figure 5 illustrates a typical histogram of the bootstrapped relative wake model error. The mean
value of the distribution is the wake model bias. This is the value of the relative wake model
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error, when evaluating it for the full validation sample. The width of the distribution gives us the
wake model uncertainty. Specifically, as explained above, we define the uncertainty as the halfth
th
width of the 68% confidence interval. This is contained within the 16 and 84 percentiles of the
distribution. The bootstrap method is non-parametric, since we do not assume normality.
The block length should be long enough to capture the dependence between the observations.
Based on a convergence analysis, where the block length was increased in steps, a block
length of about 2 days was found to be optimal. The number of bootstrap samples
is chosen
large enough to ensure convergence of the bias and the uncertainty.
4.
RESULTS
We have applied the bootstrap methodology outlined in the previous section to quantify the
wake model uncertainty for all 10 wind farms in this study. The resulting distributions of the
relative wake model error are presented together in Figure 6, where the wind farms have been
anonymised and organised in order of increasing bias from top to bottom.
Figure 6. Distribution of relative wake model error for all wind farms. Blue background: negative
model error, equivalent to an AEP conservative model (overestimating the wake loss). Green
background: positive model error, equivalent to an AEP optimistic model (underestimating the
wake loss). The wind farms have been anonymised and organised by increasing bias.
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We remind the reader that a negative wake model error corresponds to an overestimation of the
wake loss by the model, and results in a too pessimistic AEP prediction. Conversely, a positive
wake model error means that the model is underestimating the wake loss, hence being too
optimistic in predicting the AEP. It must be stressed that these are distributions of the relative
wake model error. As an example, if the wake loss is 10%, an uncertainty of 15% corresponds
to a predicted wake loss with a 68% confidence interval of [8.5%,11.5%]. For the “industry
average” uncertainty of 44% from Figure 1 the confidence interval would be [5.6%,14.4%].
Notably the uncertainty of the N. O. Jensen wake model when applied to the different wind
farms is quite small, ranging from 1.4% to 5% of the wake loss. One outlier has a much wider
distribution due to a very small number of data points remaining after filtering. For this wind farm
the wake model uncertainty is 15% of the predicted wake loss. For some wind farms the N. O.
Jensen underestimates the wake loss slightly, for others it leads to a significant overestimation
of the loss.
Understanding the sources of this variation in the wake model bias will be the subject of further
analysis in the future. If the bias of the wake model is systematic, in the sense that it depends
on a set of variables we can characterise, it becomes predictable and then it can be removed.
Equivalently, the wake model can be recalibrated in a closed loop.
However for now we assume that the wake model bias is not predictable. Then the most
pragmatic solution for extrapolating our results to new wind farms is to treat the variation in
observed bias between the wind farms as an uncertainty. We can then combine the bootstrap
results across the different wind farms to generate a cross-portfolio empirical cumulative
probability distribution (eCDF) of the relative wake model error. In essence this means merging
the histograms in Figure 6 into one combined distribution. The result is shown in Figure 7. This
cross-portfolio eCDF of the relative wake model error has a mean value of -2% (a 10% wake
loss prediction corresponds to a real loss of 9.8%), which means that the bias is effectively zero
on average when applying the N. O. Jensen wake model to many different wind farms. The
wake model uncertainty based on the width of the combined distribution of bootstrap results is
15%. We consider this to be a conservative estimate of what the uncertainty would be, if the
wake model was predicting the wake loss of a new wind farm, since widths of the bootstrap
distributions in Figure 6 were almost all well below this level.
Figure 7. Empirical cumulative distribution function based on the totality of the bootstrapped
relative wake model error estimates from all wind farms (blue line). The solid vertical line is the
portfolio mean value of the relative wake model error, while the dashed vertical lines represent
the 68% confidence interval of the distribution. The thin red line is the cumulative distribution
function of a normal distribution with the same mean value and standard deviation as the
combined (portfolio) bootstrap distribution.
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5.
CONCLUSIONS
We have presented a systematic method for quantifying the uncertainty of wake models by
combining the validation of modelled wake losses using observational data with bootstrapping
to generate distributions of the relative wake model error. We have used the N. O. Jensen wake
model as a test case in this study, and find that the uncertainty of this model, when looking
across an entire portfolio of wind farms is 15% of the wake loss. This is substantially below the
uncertainty currently associated with wake modelling by wind energy industry . When focussing
on a single wind farm, the uncertainty is typically even lower. Nonetheless, there is some
conservatism in our wake model uncertainty, since we have not attempted to separate it from
the experimental uncertainty. Hence the latter is included in the wake model uncertainty we find.
The framework for uncertainty quantification is general and can be applied to any wake model.
The only necessary input is a power matrix of the wind farm power output as a function of the
inflow wind speed and wind direction. The framework might t herefore be used to assess
different wake models, and in the future this information might prove useful for building model
ensembles. Later, we will expand the analysis to investigate possible trends in the wake model
bias and uncertainty to enhance the extrapolation of wake model predictions to new wind farms.
ACKNOWLEDGEMENTS
I wish to acknowledge my former colleague Jorge Garza who made important contributions to
the early stages of this research.
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