1
Wind Power Ramps: Detection and Statistics
Raffi Sevlian1 and Ram Rajagopal2
1. Department of Electrical Engineering
Stanford University, Stanford, CA
2. Department of Civil and Environmental Engineering
Stanford University, Stanford, CA
{raffisev,ramr}@stanford.edu
Abstract—Ramps events are a significant source of uncertainty in wind power generation. Developing statistical models
from historical data for wind power ramps is important for
designing intelligent distribution and market mechanisms for a
future electric grid. This requires robust detection schemes for
identifying wind ramps in data. In this paper, use an optimal
detection technique for identifying wind ramps for large time
series. The technique relies on defining a family of scoring
functions associated with any rule for defining ramps on an
interval of the time series. A dynamic programming recursion is
then used to find all such ramp events. Identified wind ramps are
used to perform an extensive statistical analysis on the process,
characterizing ramping duration and rates as well as other key
features needed for developing future models.
I. I NTRODUCTION
Uncertainty in wind power production is a significant hindrance to its widespread adoption [4], [5]. This uncertainty
can be mitigated by efficient control strategies on the distribution network and electricity market design, [2], [15]
. However, designing such systems requires a deep insight
into wind power production [14], introducing the need for
statistical models for wind power.
Wind power has uncertainty in multiple time scales, each
requiring its own modeling approach [7]. In the range of
seconds to minutes, fast fluctuations of small amounts are
modeled well with autoregressive or persistence techniques.
These models however fail to capture longer term trends [11].
ARIMA techniques, which include an integral step, can model
such processes but do not capture abrupt changes in wind
power generation [12]. On scales higher than a few minutes,
trend modeling is needed.
A typical long term trend with large positive or negative
change in a short period is called a ramp. Ramp event
modeling offers an additional insight beyond Autoregressive
Integrated Moving Average (ARIMA) techniques by modeling
these large deviations observed in wind power time series
[6], [13] [3]. While autoregressive methods model the short
term turbulence in the wind, ramp events can model more
long term fluctuations which have a basis in the physical
aspect of short term atmospheric dynamics. Up ramps are
generally caused by intense low pressure systems, low level
jets, thunderstorms, wind gusts or other longer term weather
phenomena. Likewise, down ramps occur from the reduction
or reversal of these physical processes [6]. These events are
important in wind power management because large swings
978-1-4673-2729-9/12/$31.00 ©2012 IEEE
in power generation must be curtailed to prevent damage to
wind turbines and distribution network. Likewise down ramps
must be met with increasing generating capacity through other
means. Therefore, developing statistical models are useful
since they offer insight into both forecasting of wind ramps
as well as formulating stochastic control strategies for dealing
with ramp events.
There exist two major problems in characterizing wind
power ramps. The first is correctly detecting wind ramps in historical data and the second is modeling these events in a useful
statistical framework. The detection problem requires precise
definitions of wind ramps as well a standardized technique
for segmenting data into positive and negative ramp events
which are robust to fluctuations in wind power in short time
scales. Standardized definitions are difficult to agree upon,
however further work in this field will lead to a convergence
of definitions. Given a set of definitions, detection schemes
rely on heuristic techniques finely tuned to the definitions
and data at hand. Therefore an optimal detection scheme
which can incorporate any set of definitions is quite useful.
The second problem is that of modeling wind ramp events.
Statistical models are useful since they offer insight into both
forecasting of wind ramps as well as integration into power
dispatch formulations. This requires strong physical insight as
well as an abundance of data along with an accurate detection
algorithm for given definitions of a ramp event.
The contributions of this work are two fold. First we
introduce an optimal ramp detection algorithm and discuss
its implementation and approximations. We then use publicly
available data to perform optimal ramp detection and provide
an in depth empirical statistical analysis of key ramp features.
Then we use the gathered statistics to show the implications
of wind power ramps in dispatch operation using the CAISO
as an example.
The paper is organized as follows. Section II reviews
previous and related work on the topic. Section III discusses
the various ramp definitions found in literature. Section IV
outlines the methodology used to detect wind power ramps in
a time series. Section V presents an array of statistics from
available data.
II. P REVIOUS W ORK
The field of wind ramp detection, characterization and modeling is a recent topic in the statistical forecasting community.
2
Ramp End
Power Swing
2.99"
Therefore, the definition of what constitutes wind ramp events
has been arbitrary but has followed a general property of
identifying large positive or negative swings in wind power
output within a short time window. Another characteristic used
in defining ramps has been high rate of power increase. There
is no set definition on the size of the increase nor the rate of
increase needed to be considered a ramp. A result of this has
been that varying these definitions leads to different sets of
detected ramp events.
There is also no clear detection method in the literature
to classify all wind power time series intervals that satisfy a
given set of definitions in a reliable and reproducible way.
They do not offer a clear method of detecting all ramps
of varying durations or rates which satisfy some minimum
threshold value. This problem has been avoided generally by
looking at large time scales (order of hours) and sufficiently
smoothing the signal.
In [9], [8] a technique is proposed for detecting wind
ramp events of a set length of 15, 30, and 60 minutes. Each
time series sample is tested to see if it satisfies the ramp
definition under each duration value. A scheme like this is
adequate for large sample intervals in the time series or data
has been sufficiently smoothed, but runs into a problem of over
counting. However, this work provides a clear set of rules to
test whether an interval of a wind power time series is in fact
a ramp event or not.
Currently, the California Independant System Operator
(CAISO) as well as other operators analyse historical wind
ramp data following the technique used in [9]. An hour
long window is tested to see whether the maximum wind
speed minus the minimum wind speed is greater than some
threshold value. This test is performed every 5 minutes, which
leads to overcounting of ramp events. Currently the CAISO
looks only at the statistics of the ramp swings, looking at the
hourly distributions of ramp swing. This statistic is used to
characterize the capacity reserve by the system. With a more
accurate detection and classification scheme, more thorough
modelling and analysis is possible.
Work has been done on applying statistical methods such as
machine learning to predicting the parameters of wind ramps
events. These have come from a data driven approach without
a heavy focus on model formulation of the ramping process. In
[8], regression and feature selection is performed on a binary
variable indicating whether a particular day will correspond to
a single ramp event or not. The results indicate that additional
information acquired in nearby weather stations can be useful
in predicting ramp events. This however focuses on a time
period of a day, therefore additional work is needed for models
of wind ramps in shorter time horizons. In [17], the authors
use a variety of machine learning tools to predict the the next
ramp rate via previous ramp rates. The definition of a ramp rate
is merely the first derivative of the underlying wind power time
series. Likewise in [16], ramps are grouped in classes and a
support vector machine approach is proposed to predicting and
classifying the next ramp event. These methods in general will
be very good in prediction, but offer less in terms of model
building for future control applications. In [1], a graphical
model framework is proposes a switched ARMA model for
Ramp Rate
Ramp Start
Duration
Fig. 1.
Sample wind power signal. Indicated are items ramp detection
algorithm aims to identify correctly. Ramp start and end times, as well the
power change in interval as well as ramp rate. These variables are basis of
statistical model.
wind.
The model learns the inherent regimes of wind power in the
data represented as the mean and variance. This model does
not incorporate any ramp phenomenon since the underlying
regimes are hidden variables.
III. P ROBLEM S TATEMENT
A ramp represents a large increase or decrease in wind
power within a limited time window. Ramps are parameterized
by the following variables: ramp start, ramp duration and ramp
rate. These uniquely identify such events and are shown in
Figure 1. In addition to ramp rate and duration, specifying
a power swing or ramp end will over parameterize the ramp
event but are useful parameters in analysis. Their detection
and modeling are difficult problems presented in this paper.
The detection problem can now be formulated with the
following elements. We are given an input time series X =
{(t1 , p1 ) , . . . , (tN , pN )} which represents a set of time and
power pairs. Each interval E = (i, j) represented indices
i, j : 1 ≤ i < j ≤ N into the the time series, as well as
a possible ramp starting at i and ending at j. A ramp rule
R : E × X → {0, 1} maps intervals of X to a decision on
whether the interval constitutes a ramp. The optimal ramp
detection problem is to design an algorithm that recovers the
sequence of intervals E1 , . . . Ek each corresponding to a ramp.
In [9] and [6], the following three rules are proposed to
determine whether an interval of the wind power time series
is in fact a wind ramp event.
R0 (i, j) = 1{pj −pi >Pval }
R1 (i, j) = 1{max(pi ,...,pj )−min(pi ,...,pj )>α}
R2 (i, j) = 1 pj −pi >α
(1)
(2)
(3)
tj −ti
Eq. (1) is a power swing threshold test which checks
whether the wind power has increased by a specified amount in
a given time span. The parameter Pval specifies the threshold,
which Δ species the time window for detecting such swings.
3
Eq. (2) checks whether the maximum and minimum observed
power within an interval is greater than a threshold value.
Here, α is used to specify this threshold. Eq. (3) checks
whether the rate of increase for a specified interval is greater
than a specified value α is the power ramp rate threshold used.
We require one additional rule beyond those in Eq. (1),
(2), (3) to exclude intervals which contains sudden drops in
power inside an interval. This rule is helpful in determining
the proper start and end of ramp events:
Rc (i, j) =
j
1{pm >β
max(pi ,...,pm )} ,
β<1
(4)
m=i
A generalized rule in Eq. (5) allows any definition to be
considered.
R (i, j) = Rc (i, j)
M
Rr (i, j)
(5)
r=1
Here Rc (i, j) is given by Eq. (4) and Rr (i, j) is given by
any definition of a wind ramp.
IV. D ETECTION A LGORITHM
i<k≤j
(6)
Ramp Score Function In order for optimal ramp detection
to take place, we require that the score function satisfy the
following super additivity property.
∀i < k < j : R (i, j)
W (i, j)
=
>
1,
W (i, k) + W (k + 1, j) (7)
Since this condition defines an entire family of weight
functions, we use Eq. (8) as a cost function for intervals of
data. Eq. (8) satisfies the condition in Eq. (7).
W (i, j) = (j − i)2 1{R(i,j)=1}
(8)
Trend Fitting We make use of a piecewise linear fitting
preprocessing step. A simple and tractable technique for
generating piecewise linear fits to data is done by solving the
following convex program [10].
min
x̂
1
x − x̂2 + λ Dx̂1
2
The optimal ramp detection scheme and subsequent statistical modeling is applied to a year long data set from
the Bonneville Power Authority. The first data set contains
15,768,000 samples representing measured wind turbine power
output sampled every 2 seconds. The data represents power
generation from January 1, 2005 - December 31, 2006 at
the Klondike I facility. The capacity of the plant is 24 MW.
First, the data is normalized to the nameplate capacity of the
system. Then, negative recorded power values are replaced
with zero power output, since they occur when the power
output is low. Then, the data is interpolated to correct for
uneven sampling timestamps of the original data. Finally, the
data is downsampled by 100 so make the effective sample rate
200 seconds. A linear trend fitting algorithm is applied to the
power time series to remove fluctuations in time scales outside
the desired range of wind power ramps as well as decrease the
number the points needed in the dynamic program. A λ value
of 0.5 was chosen by visual inspection. The result of the trend
fitting is a set of time power pairs used in the sliding window
dynamic program. With a total size of the approximate signal
|X̂P W | = 13217.
B. Ramp Detection
Assuming some scoring function on intervals of the time
series W : E × X → R+ . The objective function J
is maximized, as well as the optimal ramp start and stop
times are recovered by the following dynamic programming
formulation.
J (i, j) = max W (i, k) + J (k + 1, j)
A. Data Description
(9)
The approximate signal can then be used to generate piecwise segments that are then used in the dynamic programming
recursion.
A Ramp detection experiment is performed with the ramp
rules outlined in Section III. The rules in Eq. (1), (3),
(4) have tuning parameters set to Pval = 0.25, α = 0.003,
β = 0.9. The Pval corresponds to 6M W of increased power
for detected ramps while α corresponds to F ILLT HIS
MW/minute minimum ramp rate. The dynamic programming
method as described in section IV is used for ramp detection.
For an input of length 13217 points representing a year
of data, the computation is performed in 3.2 seconds on a
1.67 GHz dual core cpu with 2G of RAM. The software was
implemented in the C programming language. For a year long
dataset, a total of 546 up-ramps were detected as well as 552
down-ramps. A sample of the original power time series and
the detected up and down ramps is shown in Figure 7.
V. E MPIRICAL S TATISTICS OF W IND R AMPS
Analysis of the wind power ramp events is performed on
the BPA data set. The detected ramp are used to perform a
basic statistical survey of various parameters describing the
ramp event process. In Section V-A, overall statistics for
ramp rate and ramp duration are presented as well as power
swing. In Section V-B joint distributions of ramp duration and
slope as well as ramp duration and swing are presented and
discussed. Section V-C presents arrival information, while in
V-D describes ramp parameters conditioning on wind power
quantile.
A. Distribution of Ramp Rate, Duration and Swing
An empirical probability density function (pdf) is constructed for the various ramp parameters. This is done by binning the data and then applying a smoothing spline. Empirical
distributions for ramp duration, rate and swing are shown in
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Figure 2. The results show a high degree of symmetry between
the up ramp and down ramp parameters.
For both ramp types, a mean duration of around 2.6 hours
was observed as well as a minimum duration of 30 minutes.
About 95% of ramps were less than 5.6 hours long for both up
and down ramps. Note the heavy tailed nature of the data: most
ramp durations are centered near the mean value with some
infrequent longer ramp events occurring infrequently. For both
up and down ramp types, a mean ramp rate of 2.5 × 105
KW/hour was observed. The middle 50% of both ramp rates
are around 5.8 to 8.9 MW/Hour while 95% of rates are less
than 13 MW/Hour. The distribution is also very fat tailed. For
ramp swing, a mean swing of around 12 MW was observed
as well a minimum of around 6 MW. The minimum observed
swing corresponds to the maximum power output times the
minimum swing threshold Pval . This is an expected by product
of the ramp detection algorithm. A maximum swing of 24 MW
is also observed which corresponds to the nameplate capacity
of the farm. The middle 50% of up ramp swings are 11 to 16
MW in magnitude while 95% of them are less than 20 MW.
B. Joint Distributions of Ramp Duration with Ramp Rate and
Swing
Joint distributions for ramp duration and rate as well as
ramp duration and swing are shown in Figure 3. The joint
distribution of rate and duration shows a clear inverse relationship. This is expected for the following reason. Ramp
swing
is plotted against duration. Since the
rate defined as duration
variable swing can be bounded: Pval < swing < 24M W , we
Pval
can bound the ramp rate as well. More precisely, duration
<
24M W
rate (duration) < duration .
We see that there is less dependence between ramp swing
and duration as opposed to duration and rate. We also see a
distinct mode of 11M W of ramp swing and 2 Hours of ramp
duration. These correspond to the modes of duration and swing
shown in Figure 2
C. Interarrival Time
Distributions for interarrival times for up ramps and down
ramps as well as combined ramp event interarrivals are shown
in Figure 4. We focus on the combined interarrival distribution
and report important statistics. The mean interarrival time
observed was 0.33 days or about 8 hours. The middle 50%
of combined ramp interarrival times are between 1.8 and 7.2
hours in length while 95% of interarrival times are less than
27 hours long. The minimum observed interarrival time was
30 minutes in length. This an important property in potential
modeling applications, since Poisson arrival models assume
an exponential distribution which can allow for infinitesimally
small interarrival times. Clearly, this is not the case. This is a
natural consequence of the ramp detection algorithm. If two up
ramp events are very close together, the dynamic program will
combine them and report a single larger wind power ramp.
D. Conditioning on Wind Power Quantiles
In this section, we report various statistics for wind ramp
events conditioning on the wind power observed at the start
of the ramp. We condition the variables on the following
quantiles: 25% 50% 75% 80% 90% of observed wind power.
We first use the empirical distribution of the wind power
time series to determine the wind power occurring at these
quantiles. For the BPA dataset, the following power values
were recorded for each quantile: p25% = 0.0008, p50% =
0.1022, p75% = 0.4603, p80% = 0.5831, p90% = 0.8552.
We group each detected ramp event by the power recorded
at the start of the ramp. The total counts for the events are
given in Figure 4. We can clearly see up-ramps occurring
at higher numbers at lower power levels, and down-ramps
occurring at high numbers at higher power levels. This is quite
obvious, since the power is bounded in a finite interval. We
also note that the highest number of up ramps and the highest
number of down ramps do not occur at the smallest and highest
quantiles. This indicates a degree of persistence in the wind
power. That is when the wind power is low, there is a tendency
to stay low. Likewise, when the wind power is high there is a
tendency to stay high.
Figure 6 presents the distributions for ramp duration, rate,
and swing conditioning on the various wind power quantiles.
We can draw some very simple and intuitive conclusions from
this data. We can see that there is a slight dependence on power
ramp duration on the quantile at which the ramp originated.
This is seen more in the up ramp events. There is no clear
reason why up ramp and down ramp events do not follow
the same pattern. Looking at the power swing for both event
types, there is a pronounced difference in distribution shape
depending on the current power quantile.
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indicate the range of normalized power to fall in the range 0 → 0.25 . . . 0.9 → 1. Not all quantiles will be present in every plot. For example since there are
no up ramps detected when current power is greater than 0.8, no ramps were recorded.
7
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E. Seasonality of Ramp Counts
Figure 5 illustrates the two apparent forms of seasonality
in the detected ramps. We see first that ramp events occur
much more often in the spring and summer than in the fall
and winter. In fact, they increase by a factor of two. Second,
we notice that up ramp and down ramp events tend to occurs
at different parts of the day. Up ramps tend to occur more
often during th mid-day and afternoon periods, while down
ramps occurs during the evening and night times. These are
consequences of two different weather cycles. The first being
the yearly cycle corresponding to changes in the seasons. The
second, and more short term is due to daily weather patterns.
Hourly up and down ramp swing distributions are shown in
Figure 5. The figure’s show a dependance of the distribution
on hour of the day. The time dependent mean’s and variances
are usefull for computing the reserve capacity of the power
grid.
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VI. E MPIRICAL M ODELING
Given a ramp detection scheme and sufficient empirical
statistics, we can develop refined models for wind power.
One useful model is marked arrival process model to describe
the dynamics of wind ramp events. In this model, the arrival
rate of a discrete event, either up ramp or down ramp can
be constructed from the available statistics. The rate can be
dependent on various inputs. Figure 7 illustrates the proposed
arrival process. Such a model is useful for wind power
integration studies with parametric models for wind power
ramp parameters.
VII. C ONCLUSION
We have described an optimal method for detecting wind
ramp events for varying lengths under any arbitrary rule set.
A dynamic programming recursion with L1 trend fitting is
used to optimally segment wind power data to ramp events
and non-ramp events. The technique is shown to be optimal,
and is used to provide empirical statistics of wind ramps for
real data. The data is then used to propose a simple model for
characterizing ramp events as well as outline future directions
for ramp event based short term forecasting.
R EFERENCES
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8
Raffi Sevlian received his BS in Mechanical Engineering at the University of California Berkeley in 2007 and is currently persuing a PhD in
Electrical Engineering at Stanford University. He
has worked in wireless communications on both
network software as well physical layer design. His
current interests include statistical signal processing
applications and developing sensing methodologies
in infastructure systems.
Ram Rajagopal is an Assistant Professor of Civil
and Environmental Engineering at Stanford University. He also directs the Stanford Sustainable
Systems Laboratory. Prior to this, he was a DSP
research engineer at National Instruments and a visiting researcher at IBM Research, where he worked
on embedded control systems, machine learning
and image processing. His current research interests
include developing large scale sensing and market
platforms, statistical models and stochastic control
algorithms for integrating renewable energy into the
grid, increasing building energy efficiency and optimizing transportation
energy use. He holds a Ph.D. in Electrical Engineering and Computer Science
and an M.A. in Statistics, both from the University of California Berkeley,
and a Bachelors in Electrical Engineering from the Federal University of Rio
de Janeiro. He is a recipient of the Powell Foundation Fellowship.