Wind power: Exploratory space-time analysis with M. P. Muñoz*, J. A. Sànchez*, M. Gasulla*, M. D. Márquez** *Department Statistics and Operations Research (UPC) ** Department of Economics and Economic History (UAB) (pilar.munyoz@upc.edu) 1 Outline 1. Introduction & Objectives 2. Exploratory Data Analysis 3. ECF: Empirical-Correlation Function 4. EOF: Empirical Orthogonal Function 5. Conclusions 2 2
Motivation o Knowing the speed and wind direction and its evolution is very important in wind generation planning Statistical models suitable for generation planning take into account the space time components of the wind, These components are highly related to each other and to other meteorological variables 3 3 Evolution of speed and wind direction Statistical models: Space time components of the wind Wind generation planning Pressure Temperature 4 4
Objectives Visualize the spatiotemporal relationships between the variables involved in the problem using the functions we have programmed in R as well as other functions already implemented in R libraries Descriptive techniques: - graphical representations of speed and wind direction - correlation between different space-time variable - autocorrelation functions implemented in several space and time lags - EOF: empirical orthogonal functions... 5 5 6 6
Data description: Data from Hirlam model Period 01/01/2009 to 31/12/2009 Frequency 3 hours Missingdata 01/06/2009 and02/06/2009 Temporal reference system Modeling UTC(~ GMT) Daily Analysis at 00:00 and forecasting at 3, 6, 9, 12, 15, 18, and 21 hours Coordinates LON = W4 o 30 : W6 o 30 LAT = N35 o 3 : N36 o 5 7 Data description: North Component (315 o -45 o ) East Component (45 o -135 o ) South Component (135 o -225 o ) West Component (225 o -315 o ) http://www.amarre.com/html/meteorologia/rosa/index.php 8
R packages:akima, fields and maps Wind speed, Temperature and Pressure 3.00h 2009/01/14 9 R packages:akima, fields and maps Wind speed, Temperature and Pressure 12.00h 2009/02/08 10 10
Marginal plots: Space (1-D)/Time Plots 1 Method for illustrating the wave propagation of climate variables. For example, wind, temperature, Bidimensional plot : x-axis contains time; y-axis represents latitude or longitude They help to find patterns, anomalies, Visual evidence of climate variable behavior 1 Cressie, N., Wikle, C.K. 2011. Statistics for Spatio-Temporal Data. Wiley 11 11 R code: Our implementation Wind: From March 9 to Mach 29 2009, wind direction (lat) ranged around 100º C: East wind (Levante) 12 12
R code: Our implementation Wind: From March 9 to March 29 2009, wind direction (lon) ranged around 100º C: East wind (Levante) 13 13 R code: Our implementation speed-direction speed peed sp Wind (yearly) Highest wind speeds when the wind direction ranges around 100º C: East wind (Levante) 14 14
R package: WindRose Speed and direction The same conclusion as in the previous slide 15 15 Empirical Correlation Function 1 Procedure for detecting spatio-temporal correlations between of climate variables as for example wind It is convenient to present graphically these matrices The space-time observations are Z t ( Z( s1, t),, Z( s, t))' The empirical lag- spatial covariance matrix is Cˆ ( ) Z 1 T T ( Z m ˆ )( Z t Z t t 1 Where the empirical spatial mean is Where s is for space and t for time ˆ Z )', 1 ˆ T Z Z t T t 1 The empirical lag- spatial correlation matrix is Rˆ ( ) ˆ 1/ 2 ( ) ˆ 1/ 2 Z DZ CZ DZ ˆ and (0 D diag( ) 0,1,, T 1 1 Cressie, N., Wikle, C.K. 2011. Statistics for Spatio-Temporal p Data. Wiley ˆ ˆ ) Z C Z 16 16
R our: Own implementation Empirical correlation (lat) at lag 0, for the twelve months 17 17 R code: Our implementation Empirical correlation (lat) at lag 32, for the twelve months 18 18
R code: Our implementation Empirical correlation (lon) at lag 0, for the twelve months 19 19 R code: Our implementation Empirical correlation (lon) at lag 32, for the twelve months 20
Spatio-temporal Kriging 2009-03-01 00:00:00 2009-03-01 01:06:18 2009-03-01 02:12:37 2009-03-01 03:18:56 2009-03-01 03 01 04:25:15 2009-03-01 03 01 05:31:34 2009-03-01 03 01 06:37:53 2009-03-01 03 01 07:44:12 8 7 R packages: sp, mapdata, maps, maptools, rgdal, xts, gstat, spacetime 2009-03-01 08:50:31 2009-03-01 09:56:50 2009-03-01 11:03:09 2009-03-01 12:09:28 2009-03-01 13:15:47 2009-03-01 14:22:06 2009-03-01 15:28:25 2009-03-01 16:34:44 6 Procedure for space-time interpolation from a sample (x in 5 the plot ) Variable: wind speed 4 Date: 2009/03/01. Hours: 20 hours, starting at 0:00 and finishing at 21:00 3 2009-03-01 17:41:03 2009-03-01 18:47:22 2009-03-01 19:53:41 2009-03-01 21:00:00 2 21 21 EOF: Empirical Orthogonal Function EOF is the geophysicist s i terminology for the eigenvectors in the classical eigenvalue/eigenvector decomposition of a covariance matrix 1,2 Reduce the dimensionality (space or time component) in a large spatio-temporal data set. EOF identifies structures in the space dimension 3 Useful to forecast space-time superficies as for example wind or sea surface temperatures 1 Cressie, N., Wikle, C.K. 2011. Statistics for Spatio-Temporal p Data. Wiley 2 Pebesma, E. 2011. Classes and methods for spatio-temporal data in R: the spacetime package (cran.r-project.org) 3 Le,N.D., Zidek, J.V. 2006. Statistical analysis of environmental space-time processes. Springer 22 22
EOF applied to wind speed 66 time series (one in each point), observations every 3 hours for 2 years 36.5 N X21.41 X21.37 X21.33 X21.29 X21.25 X21.21 X21.17 X21.13 X21.9 X21.5 X21.1 X17.41 X17.37 X17.33 X17.29 X17.25 X17.21 X17.17 X17.13 X17.9 X17.5 X17.1 X13.41 X13.37 X13.33 X13.29 X13.25 X13.21 X13.17 X13.13 X13.9 X13.5 X13.1 X9.41 X9.37 X9.33 X9.29 X9.25 X9.21 X9.17 X9.13 X9.9 X9.5 X9.1 X5.41 X5.37 X5.33 X5.29 X5.25 X5.21 X5.17 X5.13 X5.9 X5.5 X5.1 35.5 N X1.41 X1.37 X1.33 X1.29 X1.25 X1.21 X1.17 X1.13 X1.9 X1.5 X1.1 6.5 W 6 W 5.5 W 5 W 4.5 W 23 23 This plot shows the wind speed at 66 points during 6 days (from Aug/07/2009 to Aug/12/2009). Each row has the 8 observations at day. Wind Speed 6.5 W6 W5.5 W5 W4.5 W 6.5 W6 W5.5 W5 W4.5 W 6.5 W6 W5.5 W5 W4.5 W 6.5 W6 W5.5 W5 W4.5 W 2009-08-07 00:00:00 2009-08-07 03:00:002009-08-07 06:00:00 2009-08-07 09:00:002009-08-07 12:00:00 2009-08-07 15:00:00 2009-08-07 18:00:00 2009-08-07 21:00:00 2009-08-08 00:00:002009-08-08 03:00:002009-08-08 06:00:00 2009-08-08 09:00:002009-08-08 12:00:002009-08-08 15:00:00 2009-08-08 18:00:00 2009-08-08 21:00:00 2009-08-09 00:00:002009-08-09 03:00:002009-08-09 06:00:00 2009-08-09 09:00:002009-08-09 12:00:002009-08-09 15:00:00 2009-08-09 18:00:002009-08-09 21:00:00 2009-08-10 00:00:002009-08-10 03:00:002009-08-10 06:00:00 2009-08-10 09:00:002009-08-10 12:00:002009-08-10 15:00:00 2009-08-11 00:00:002009-08-11 03:00:002009-08-11 06:00:00 2009-08-11 09:00:002009-08-11 12:00:002009-08-11 15:00:00 2009-08-12 00:00:002009-08-12 03:00:002009-08-12 06:00:00 2009-08-12 09:00:002009-08-12 12:00:002009-08-12 15:00:00 6.5 W6 W5.5 W5 W4.5 W 6.5 W6 W5.5 W5 W4.5 W 6.5 W6 W5.5 W5 W4.5 W 6.5 W6 W5.5 W5 W4.5 W 2009-08-10 18:00:00 2009-08-10 21:00:00 6 2009-08-11 18:00:002009-08-11 21:00:00 4 2009-08-12 18:00:00 2009-08-12 21:00:00 2 12 10 8 24 24
Spatial EOF EOF's EOF1 6.5 W 6 W 5.5 W 5 W 4.5 W EOF2 The first six EOF EOF3 EOF4 200 100 The first component clearly separates land / sea and the second west / east 0 EOF5 EOF6-100 -200 6.5 W 6 W 5.5 W 5 W 4.5 W 25 25 Temporal EOF First EOF Second EOF EOF 1-40 0 40 80 EOF 2-30 -10 10 30 ene 01 00:00 sep 01 00:00 may 01 00:00 dic 01 00:00 ene 01 00:00 sep 01 00:00 may 01 00:00 dic 01 00:00 Third EOF Fourth EOF EOF 3-20 0 10 EOF 4-15 -5 5 15 30 ene 01 00:00 sep 01 00:00 may 01 00:00 dic 01 00:00 ene 01 00:00 sep 01 00:00 may 01 00:00 dic 01 00:00 Fifth EOF Sixth EOF EOF 5-15 -5 5 15 EOF 6-15 -5 0 5 10 ene 01 00:00 sep 01 00:00 may 01 00:00 dic 01 00:00 ene 01 00:00 sep 01 00:00 may 01 00:00 dic 01 00:00 26 26
Conclusions The relationships detected by the exploratory space time analysis are very useful in the statistical models, obtaining more accurate models The forecast of future wind values will be more accurate. ate The accuracy achieved in predicting wind speed and direction will have a positive impact on the quality of wind generation forecasts. The presented techniques will be useful for diagnosing the quality of fit for the estimated models. 27 27 28 28