Wolf-Gerrit FrühChristina Skittides

With support from SgurrEnergy

Preliminary assessment of wind climate fluctuations and

use of Dynamical Systems Theory for resource assessment

Questions

• How sensitive is the electricity production of a wind farm to the local wind statistics

• How will climate change affect the electricity production from wind farms?

• How large are inter-annual variations in the electricity production due to weather fluctuations?

• How much can a large spatial distribution of wind farms smooth electricity output

• Can we use dynamic information for an improved wind resource assessment and prediction at potential development sites?

2

Data used• Land surface wind data from the MIDAS data

record provided by the British Atmospheric Data Centre, maintained by NERC– Hourly winds from weather stations all over the UK– In particular from two stations in Edinburgh,

Gogarbank and Blackford Hill– Data format

• Wind speed in knots at 10 m above sea level• Wind direction in degrees

• Wind data converted to m/s at a typical turbine hub

3

Assumptions• Standard wind shear profile

• Best-fit Weibull distribution• Typical turbine performance

curve• Ideal turbine response• All generation exported for

use4

U z U0

log z

log z0

From wind distribution to electricityNumber of hours

during which the wind was u: Φ(u) * T

The output from a turbine during that time: P(u)

The electricity generated during those hours: E(u)= P(u) * Φ(u) * T

Total output:Etotal = Σ E(u)

5

Weibull and Rayleigh distributions• Cumulative Weibull

– k: shape factor

• Weibull– c: scale factor

6

uu ' e

u '

c

k

u k

c

u

c

k 1

e

u

c

k

Capacity factor vs c with k = 2

7

Sensitivity analysis

8

6 5 4 3 2

Example:100 MW farm with expectedCapacity factor 30%: Income £26.2 m

Actual Capacity factor 27%: Income £23.6 mDeficit: £ 2.4 m

Does the wind always blow somewhere?

9Fraction of sites producing at full capacity

Fraction of sites producing at part capacity

Fraction of sites not producing

National capacity factor for first 1000 days in 2010

10

Has the wind changed?

11Ccap = 0.14 ± 0.06

Edinburgh Gogarbank

Has the wind changed?

12Ccap = 0.28 ± 0.08

Edinburgh Blackford Hill

Assessing resource from a short measurement campaign

• Short time to measure potential site– Does not give good statistics

• Are the measurements correlated to a site nearby with existing longer record?

• Use ‘MCP’– Measure a short record– Correlate with longer record– Predict resource at location

with shorter record• Christina

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Statistical Modelling of Wind Energy Resource

Christina Skittides Supervisor: Dr. Wolf G. Früh

17th March 201114

MCP Methods• MCP goal: characterize wind speed

distribution and estimate the annual energy capture of a wind farm

• MCP methods: model relationship between wind speed and direction at two sitesMeasurement period: a year or moreInput: wind speed and wind directionOutput: mapping from one site to otherUse: apply mapping to more data from reference

site15

MCP Methods• MCP invariants:

wind speed, directiondistance, eg. time of flight delayseffects of terrain on the flow, eg. local obstructionslarge/small–scale weather, eg. atmospheric stability

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Reference Derrick Woods & Watson

“Variance Ratio”

Method Characterisation

typical MCP method

refinementof typical method

alternative

Approach wind speed linear

regression fit

same wind speed, binned wind direction

relate variances from reference and

target site

Dynamical Systems Theory

• Dynamical systems involve differential equations that depend on position and momentum

• Phase space: describes the system’s variables• Attractor: defines the solution of the system • Orbit: the path the system follows during its

evolution Method needed to define equivalent variables

to the phase space ones Time-delay

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Time-Delay/PCA Theory

Time-Delay:• practical implementation of dynamical systems• Results sensitive to choice of delays PCA

PCA:• non-parametric method to optimize phase space

reconstruction• identifies number of needed time- delays• gives picture of their shape• reduces dimensions so as to extract useful information

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PCA theory

Useful PCA outputs:• Singular Vectors: represent the dimensions of

the phase space, describe optimum way of reconstructing it

• Singular Values: measure total contribution of each dimension to total variance

• Principal Components (PC): describe the system’s time series, separate important variables from noise

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Pendulum Example

• Dynamical system with two inputs x,y Case A (without noise):

x= 3sin(t/0.7) y= x+0.4sin(t/π)

Case B (with noise):

x= 3sin(t/0.7) y= x+0.4sin(t/π) + 0.6ε

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Pendulum ExampleCase A

21

0 200 400 600 800 1000

-3-2

-10

12

3

time

sig

na

l2

-3 -2 -1 0 1 2 3

-3-2

-10

12

3

tarr[,1]

tarr

[,2

]

0 10 20 30 40 50 60 70

02

00

40

06

00

80

01

00

01

20

01

40

0

Index

lam

bd

a

-0.015 -0.010 -0.005 0.000 0.005 0.010 0.015

-0.0

15

-0.0

10

-0.0

05

0.0

00

0.0

05

0.0

10

0.0

15

pc[, 1]

pc[, 2

]

Noisy Pendulum ExampleCase B

220 10 20 30 40 50 60 70

02

00

40

06

00

80

01

00

01

20

01

40

0

Index

lam

bd

a

0 200 400 600 800 1000

-4-2

02

4

time

sig

na

l2

-0.015 -0.010 -0.005 0.000 0.005 0.010 0.015

-0.0

15

-0.0

10

-0.0

05

0.0

00

0.0

05

0.0

10

0.0

15

pc[, 1]

pc[, 2

]

-4 -2 0 2 4

-4-2

02

4

tarr[, (1 + win)]

tarr

[, (

2 +

win

)]

Conclusions

• PCA is robust and useful for time series of multiple inputs

• Noisy or “clean”data: no significant differences

• Choice of time-delay length and gap of entries in the matrix not important

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Gogarbank Data• 10 year (2000-2010) data taken from

Gogarbank station, Edinburgh• Input variables: wind speed, direction,

pressure, temperature• Apply PCA to different models:

all variables wind speed, direction and pressure only wind speed and direction

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hourly every 3 hours daily

weekly × ×

2 weeks × ×

monthly × ×

seasonal ×

Gogarbank Data• Only wind speed and direction,

hourly measurements per week

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0 50 100 150 200 250 300

06

0

Index

lam

bd

a

-0.03 -0.02 -0.01 0.00 0.01 0.02 0.03

-0.0

30

.02

pc[, 1]

pc

[, 2

]

-0.03 -0.02 -0.01 0.00 0.01 0.02 0.03

-0.0

30

.02

pc[, 2]

pc

[, 3

]

-0.03 -0.02 -0.01 0.00 0.01 0.02 0.03

-0.0

20

.04

pc[, 3]

pc

[, 4

]

0 50 100 150 200 250 300

-0.0

80

.00

Index

sv

ec

[, 1

]

0 50 100 150 200 250 300

-0.1

00

.05

Index

sv

ec

[, 3

]

0 2000 4000 6000 8000

-0.0

30.0

1

td

pc

[, 1

]

0 2000 4000 6000 8000

-0.0

30

.02

td

pc

[, 3

]

Conclusions• Gogarbank station wind: dynamic behaviour

found in structure of PCs and singular vectors• Adding pressure no significant difference • Temperature changes results significantly

since PCs concentrate on seasonal cycle• Cyclic behaviour over the year, more windy

around January and from September- December

• Using 1 week or 2 weeks identifies weather (typical predictability of weather ~ 14 days)

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Following Steps

• Apply PCA to simultaneous data from two weather stations (Gogarbank & Blackford Hill, Edinburgh)

Application to a one-year segmentUsing Gogarbank for other years to predict

Blackford HillComparison with actual measurements from

Blackford Hill

27