Time-series modelling of aggregate
wind power output
Alexander Sturt, Goran Strbac
17 March 2011
Introduction
Eastern Wind Integration and Transmission
Study (EWITS) (2010)
• Wind datasets prepared by AWS Truewind over 9 month period
• Created by simulation using mesoscale Numerical Weather Prediction
(NWP) model
• 3 years of synthetic data, 1326 sites (freely available online)
• Hardware used: 80 x dual CPU quad core penguin workstations (640
cores)
• Run time per year of simulation: 21 days (in theory...)
What if this level of detail isn’t needed?
What if we need a model of aggregated wind output?
What if we need to understand the statistical properties?
Modelling strategy
•
•
Univariate model for aggregate wind power, not wind speed
Autoregressive driver: AR(p), hourly (or half-hourly) timesteps
X k  1 X k 1  2 X k 2  ...  k
•
iid N(0,1)
Include diurnal variation with periodic additive term:
X k  X k  k mod n
n = number of data
points per day
•
Fit to long-term distribution with transformation function:
•
Use different models for the different seasons
Pk  W X k 
Model calibration
1. Choose these to satisfy
long-term distribution and
diurnal variation, assuming
X~N(0,1)
1.4
1.2
1
X
0.8
0.6
0.4
0.8
0.2
0.7
1
0
0.6
W
-0.2
-0.4
0.5
Σ
0.4
0.5
0.4
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
-0.5
0.3
μ
0.2
0
-6
-4
-2
0
2
4
6
0.1
0
P
Model calibration
1.4
1.2
1
2. Choose parameters of
AR model to fit shortterm transitional
properties and N(0,1)
asymptotic distribution 0.8
X
0.8
0.6
0.4
0.2
0.7
1
0
0.6
W
-0.2
-0.4
0.5
Σ
0.4
0.5
0.4
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
-0.5
0.3
μ
0.2
0
-6
-4
-2
0
2
4
6
0.1
0
P
Case study: GB2030 model
•
•
•
•
6 years of hourly wind speed data taken from MIDAS
dataset by Olmos (2009)
116 sites (onshore only)
10m anemometer data extrapolated to hub-height and
converted to wind power using turbine curve
Regional weightings chosen to match core 2030 buildout
scenario used by Poyry (2009); offshore capacity mapped
to nearest onshore regions
Olmos
Poyry
GB2030: modelling strategy
•
•
•
•
Weighted regional power output aggregated to produce a
univariate time series
Split into four seasons
For each season, calibrate model to reproduce
asymptotic distribution, diurnal variation and short-term
volatility, using AR(2) model
Tweak to approximate effect of offshore component
GB2030 (untweaked): distribution and volatility
1800
1200
1000
Sim Lower
Hist2001-2
Hist2003-4
Hist2005-6
Hist Av
800
600
400
0.20
RMS change (p.u.)
1400
Sim Upper
Sim Mean
Hist2002-3
Hist2004-5
Hist2006-7
0.15
Sim Upper
Sim Mean
Hist2002-3
Hist2004-5
Hist2006-7
0.10
0.05
200
0
Sim Lower
Hist2001-2
Hist2003-4
Hist2005-6
HistAv
0.00
0-0.05
0.05-0.1
0.1-0.15
0.15-0.2
0.2-0.25
0.25-0.3
0.3-0.35
0.35-0.4
0.4-0.45
0.45-0.5
0.5-0.55
0.55-0.6
0.6-0.65
0.65-0.7
0.7-0.75
0.75-0.8
0.8-0.85
0.85-0.9
0.9-0.95
0.95-1
Occurrences per year
1600
0.25
0
5
10
15
Time horizon (hr)
Power output bucket (p.u.)
Power output distribution
Volatility curve
20
GB2030 (untweaked):
distribution of absolute power output changes
10000
10000
1000
1 hr
Occurrences per year
100
10
1
Sim Upper
Sim Mean
Hist2002-3
Hist2004-5
Hist2006-7
0.1
Sim Lower
Hist2001-2
Hist2003-4
Hist2005-6
HistAv
10
Sim Upper
Sim Mean
Hist2002-3
Hist2004-5
Hist2006-7
1
0.1
0.01
Sim Lower
Hist2001-2
Hist2003-4
Hist2005-6
HistAv
0.6-0.65
0.55-0.6
0.5-0.55
0.45-0.5
0.4-0.45
0.35-0.4
0.3-0.35
0.25-0.3
0.2-0.25
0.15-0.2
0.1-0.15
0.05-0.1
0-0.05
0.25-0.3
0.2-0.25
0.15-0.2
0-0.05
0.05-0.1
0.1-0.15
0.01
Power output change bucket (p.u.)
Power output change bucket (p.u.)
10000
10000
8 hr
100
10
Sim Upper
Sim Mean
Hist2002-3
Hist2004-5
Hist2006-7
1
0.1
24 hr
1000
Occurrences per year
1000
Sim Lower
Hist2001-2
Hist2003-4
Hist2005-6
HistAv
100
10
1
0.1
0.01
Sim Upper
Sim Mean
Hist2002-3
Hist2004-5
Hist2006-7
Sim Lower
Hist2001-2
Hist2003-4
Hist2005-6
HistAv
Power output change bucket (p.u.)
0.75-0.8
0.7-0.75
0.65-0.7
0.6-0.65
0.55-0.6
0.5-0.55
0.45-0.5
0.4-0.45
0.35-0.4
0.3-0.35
0.25-0.3
0.2-0.25
0.15-0.2
0.1-0.15
0.05-0.1
0.01
0-0.05
Occurrences per year
4 hr
100
0-0.05
0.05-0.1
0.1-0.15
0.15-0.2
0.2-0.25
0.25-0.3
0.3-0.35
0.35-0.4
0.4-0.45
0.45-0.5
0.5-0.55
0.55-0.6
0.6-0.65
0.65-0.7
0.7-0.75
0.75-0.8
0.8-0.85
0.85-0.9
0.9-0.95
Occurrences per year
1000
Power output change bucket (p.u.)
GB2030: variation of 4hr volatility with power level
0.18
0.16
0.14
1
0.12
W(x)
0.1
0.08
Power output bucket (p.u.)
x
0
0.9-1.0
0.8-0.9
-6
0.7-0.8
0.2-0.3
0.1-0.2
0
0.6-0.7
0.02
Sim Lower
Hist2001-2
Hist2003-4
Hist2005-6
HistAv
0.4-0.5
0.04
0.3-0.4
Sim Upper
Sim Mean
Hist2002-3
Hist2004-5
Hist2006-7
0.5-0.6
0.06
0-0.1
Mean absolute change (p.u.)
0.2
-4
-2
0
2
4
6
What about turbine cutout?
Denmark, distribution of 4-hour changes
(non-rolling window)
Sim Upper
Sim Mean
Hist2004-5
Hist2006-7
Hist2008-9
1000
Sim Lower
Hist2003-4
Hist2005-6
Hist2007-8
100
8 Jan 2005
10
1
Power output change bucket (p.u.)
0.65-0.7
0.6-0.65
0.55-0.6
0.5-0.55
0.45-0.5
0.4-0.45
0.35-0.4
0.3-0.35
0.25-0.3
0.2-0.25
0.15-0.2
0.1-0.15
0.05-0.1
0.1
0-0.05
Occurrences per year
10000
GB2030: tweaking strategy (1)
Diurnal variation is too great
• Lunchtime wind speed peak at hub height is less pronounced
than at anemometer height (insolation reduces stability)
Mean power ouptut (p.u.)
0.45
0.4
0.35
0.3
0.25
Olmos Summer
0.2
Olmos Winter
0.15
NG Summer
0.1
NG Winter
0.05
0
0
4
8
12
16
20
24
Hour (GMT)
•
Offshore component has no diurnality
=> Reduce μ values by a factor of 4
GB2030: tweaking strategy (2)
Offshore component increases mean capacity factor (28% -> 33%)
=> Stretch W function so as to match duration curves shown in Poyry
(2009). Use same AR parameters as untweaked model
40000
Year 1
35000
Year 2
Year 3
Wind output (MW)
30000
Year 4
25000
Year 5
Year 6
20000
Year 7
Year 8
15000
10000
5000
0
100%
Poyry 2030 data (43GW capacity)
80%
60%
40%
20%
Synthetic data from
tweaked GB2030 model
0%
800
600
1200
1000
Untweaked
Tweaked
400
200
0
Power output bucket (p.u.)
Power output distribution
RMS power output change (p.u.)
0-0.05
0.05-0.1
0.1-0.15
0.15-0.2
0.2-0.25
0.25-0.3
0.3-0.35
0.35-0.4
0.4-0.45
0.45-0.5
0.5-0.55
0.55-0.6
0.6-0.65
0.65-0.7
0.7-0.75
0.75-0.8
0.8-0.85
0.85-0.9
0.9-0.95
0.95-1
Occurrences / year
GB2030: Effect of tweak
1400
0.25
0.2
0.15
Untweaked
0.1
Tweaked
0.05
0
0
4
8
12
Volatility curve
16
Time horizon (hr)
20
24
Wind output (GW)
GB2030: Time history sample (“Turing test”)
Poyry data
45
Wind output (GW)
40
35
30
25
20
15
10
5
0
01-Dec-08
11-Dec-08
21-Dec-08
31-Dec-08
10-Jan-09
Tweaked GB2030 synthetic winter data
20-Jan-09
30-Jan-09
Conclusions
• Non-Gaussian wind power time series can be transformed to a
Gaussian (X) domain and modelled with a Gaussian time
series model
• Synthetic time series reproduce the important long-term and
transitional properties (for power system simulation)
• Simplicity of model makes it possible to write down formulae
for any desired statistic
• Transformation to Gaussian domain simplifies modelling of
correlated RVs:
• Forecast errors (anti-correlated with wind realisation to
prevent forecast biasing)
• Multi-bus models
• Combined demand / wind model
References
• Sturt, A. and Strbac, G. “Time series modelling of power output for
large-scale wind fleets”, Wind Energy, 2011 (to be published)
• Enernex Corporation “Eastern Wind Integration and Transmission
Study”, 2010
http://www.nrel.gov/wind/systemsintegration/ewits.html
• Olmos, P. “Probability distribution of wind power during peak
demand”, MSc dissertation, University of Edinburgh, 2009
• Olmos, P.E., Dent, C., Harrison, G.P. and Bialek, J.W. “Realistic
calculation of wind generation capacity credits”, CIGRE/IEEE
Symposium on integration of wide-scale renewable resources into
the power delivery system, Calgary, 2009
• Poyry Energy Consulting, “Impact of intermittency: how wind
variability could change the shape of the British and Irish electricity
markets: summary report”, 2009 http://www.poyry.com
• Sturt, A. and Strbac, G. “A time series model for the aggregate GB
wind output circa 2030”, 2011
http://www.ee.ic.ac.uk/%20alexander.sturt07/GB2030SOM.pdf