Ispra 2007 luis martín2

Presented by:

LUIS MARTÍN POMARES

ENERGY DEPARTAMENTRenewable energy division

Plataforma Solar de Almería

3rd Experts Meeting of the IEA SHC Task “Solar Resource Knowledge Management”

&MESoR Coordination Meeting

Ispra (VA), Italy12 – 14 March 2007

DAILY RADIATION FORECASTING BY STATISTICAL METHODS: PRELIMINARY

RESULTS

2

DAILY RADIATION FORECASTING

1. INTRODUCTION

2. EXPLORATORY DATA ANALYSIS

3. LINEAR PREDICTION: TAG(p)

4. NON-LINEAR PREDICTION

5. CONCLUSIONS

3

INTRODUCTION

There is a necesity to characterize and predict solar radiation to be used as a energetic resource (RD 436/2004).

Prediction Techniques:• Numerical Prediction Models (NWP)• Statistical Prediction Models

Prediction Horizon:• Nowcasting: less than one hour• Short term: less than a week• Medium term: 1 week – 1 year• Long term: more than a year. Climate

4

PIRANOMETRIC DATA

•Data Period:10/7/1996 – 29/12/2003

•#Data:2304 values

•Daily Goblal Solar Radiation transformed to daily Kt Values

5

EXPLORATORY DATA ANALYSIS

0 500 1000 1500 2000 25000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Kt

Dia

rio

Día N

0 2 4 6 8 10 12 14 16 18 20-0.2

0

0.2

0.4

0.6

0.8

Lag

Sam

ple

Part

ial A

uto

corr

ela

tions

Sample Partial Autocorrelation Function

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

100

200

300

400

500

600

700

800

Núm

ero

de m

uestr

as

Kt Diario

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-70

-60

-50

-40

-30

-20

-10

0

10

20

30

Normalized Frequency ( rad/sample)

Pow

er/

frequency (

dB

/rad/s

am

ple

)

Power Spectral Density Estimate via Periodogram

6

EXPLORATORY DATA ANALYSIS

Central Months predominance of Goods Days.

External months: Mixture of Kt (bad and good days)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

20

40

60

80

100

120

Núm

ero

de m

uest

ras

Kt Diario

Monthly Histogram

Daily Kt for each monthMonth

Kt

9

SAMPLE PROBABILITY DISTRIBUTION: BI-EXPONENTIAL

Cumulative Daily Distribution Functions

Manuel Ibañez, Journal of solar energy engineering, 2002, vol. 124,1,pp. 28-33 Frequency Distribution for Hourly and Daily Clearness Indices.

Daily Probability Density Functions

12

Partial Autocorrelation Autocorrelation

Non StationaryNon Gaussian DataNon Linear

•Low Lag(1) autocorrelation

•Generally authors recomend r1=0.29. [R. Aguiar, 1992, Solar Energy]

•Data analyzed indicates a broad range of values for r1 from 0.17 to 0.65.

Data Preprocessing

13

LINEAR PREDICTION: TAG(p)

Gaussian: Transform Data to Gaussina Distribution using daily Kt Anomalies

Timedependant Autorregressive Gassuian Model: TAG

Timedependant: Montly Autorregresive Model 12 AR(p)

Autorregresive Model AR(p):

,...,2,1,0

pptaX t

p

kktk

j

jii KT

TKKtAnomalyKt

_

14

LINEAR PREDICTION: TAG(p)

1 2 3 4 5 61.5

2

2.5

3

3.5

4

4.5

5

Horizonte Predicción (Días)

%M

BE

Pre

dicc

ión

Dia

ria K

t

Julio

AR(1)AR(2)

AR(3)

AR(4)

AR(5)

AR(6)AR(7)

AR(8)

AR(9)

AR(10)Persistencia

1 2 3 4 5 614

16

18

20

22

24

26

28


%R

MS

E P

redi

cció

n D

iaria

Kt

Julio

AR(1)AR(2)

AR(3)

AR(4)

AR(5)

AR(6)AR(7)

AR(8)

AR(9)

AR(10)Persistencia

1 2 3 4 5 6-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15


Mej

ora

RM

SE

AR

Ópt

imo

fren

te P

ersi

sten

cia

AR(2)/Persistencia - Enero

1 2 3 4 5 6-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15


Mej

ora

RM

SE

AR

Ópt

imo

fren

te P

ersi

sten

cia

AR(2)/Persistencia - Julio

1 2 3 4 5 610

12

14

16

18

20

22

24

26

28


%M

BE

Pre

dicc

ión

Dia

ria K

t

Enero

AR(1)AR(2)

AR(3)

AR(4)

AR(5)

AR(6)AR(7)

AR(8)

AR(9)

AR(10)Persistencia

1 2 3 4 5 650

55

60

65

70

75

80

85


%R

MS

E P

redi

cció

n D

iaria

Kt

Enero

AR(1)AR(2)

AR(3)

AR(4)

AR(5)

AR(6)AR(7)

AR(8)

AR(9)

AR(10)Persistencia

17

LINEAR PREDICTION: TAG(p)Future Works: Kt Transformation

Predict Kt Differences between days: )1()()( tKttKtty

0 500 1000 1500 2000 2500-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.80

100

200

300

400

500

600

700

800

0 2 4 6 8 10 12 14 16 18 20-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Lag

Sam

ple

Auto

corr

ela

tion

Sample Autocorrelation Function (ACF)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-80

-70

-60

-50

-40

-30

-20

-10

0

Normalized Frequency ( rad/sample)

Pow

er/

frequency (

dB

/rad/s

am

ple

)

Power Spectral Density Estimate via Periodogram

18

NON-LINEAR PREDICTION

Model Prediction 2:

Signal Preprocessing:SPECTRAL SIGNAL ANALYSIS: WAVELET

PredictionNEURAL NETWORK (NN)

Model Prediction 3:

Signal Preprocessing:CLUSTER ANALYSIS: SOM NETWORKS

Prediction:NEURAL NETWORK (NN)

Model Prediction 1:

PredictionNEURAL NETWORK (NN)

Future works like Fuzzy Logic, Markov Chain…

20

Model Prediction 1: RESULTS

Mean Absolute Error (MAE)

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

0 2 4 6 8 10 12

NN(X)

MA

E

Modelo 1

Modelo 2

Modelo 3

Modelo 4

Mean Squared Error (MSE)

-0,1

0

0,1

0,2

0,3

0,4

0,5

0 2 4 6 8 10 12

NN(X)

MS

E

Modelo 1

Modelo 2

Modelo 3

Modelo 4

Coeficiente Correlación (R)

0

0,1

0,2

0,3

0,4

0,5

0,6

0 2 4 6 8 10 12

NN(X)

R

Modelo 1

Modelo 2

Modelo 3

Modelo 4

Neural Network Model Structure

NN Model 1 1 Neuron

NN Model 2 7-1

NN Model 3 5-3-1

NN Model 4 7-5-3-1

21

Model Prediction 2: DISCRETE WAVELET TRANSFORM

Piramidal analisys of the signal and decomposition into multiple Layers. It works like a low and high pass filter

LowFrequency High

FrequencycD1cA1

cA2 cD2

cA3 cD3

Kt

22

SIGNAL DECOMPOSITION

0 50 100 150 200 250 300 3500

0.5

1Señal Original

Kt

0 50 100 150 200 250 300 3500

0.5

1Señal Aproximación 3

Kt

0 50 100 150 200 250 300 350-0.5

0

0.5Señal Detalle 1

Kt

0 50 100 150 200 250 300 350-0.5

0

0.5Señal Detalle 2

Kt

0 50 100 150 200 250 300 350-0.2

0

0.2Señal Detalle 3

Kt

0 50 100 150 200 250 300 3500

0.5

1Señal Reconstruida

Kt

Dia Juliano

23

Model Prediction 2: WAVENET

DW

DW

Kt

aD1(x)

aD1(x-1)...aD1(x-k)

aD1(x+1)•aD1

•aD2

•aD3

•aA3

aD2(x)…aD2(x-k)

aD3(x)…aD2(x-k)

aD2(x)…aD2(x-k)

aD2(x+1)

aD3(x+1)

aA1(x+1)

IDW

Kt(x+1)

24

Model Prediction 2: RESULTS

Neural Network Model Structure

Model 1 1 Neuron

Model 2 7-1

Model 3 5-3-1(cA)7-5-3-1(cD)

Model 4 7-5-3-1

Mean Absolute Error (MAE)

0

0,5

1

1,5

2

2,5

3

1 2 3 4 5 6 7 8 9 10

NN(X)

MA

E

Modelo 1

Modelo 2

Modelo 3

Modelo 4

Coeficiente Correlación (R)

0

0,2

0,4

0,6

0,8

1

1,2

1 2 3 4 5 6 7 8 9 10

NN(X)

R

Modelo 1

Modelo 2

Modelo 3

Modelo 4

Mean Squared Error (MSE)

0

0,1

0,2

0,3

0,4

0,5

1 2 3 4 5 6 7 8 9 10

NN(X)

MS

E

Modelo 1

Modelo 2

Modelo 3

Modelo 4

25

Prediction of Wavelet Transform Coeficientes

0 50 100 150 200 250 300 3500.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9Coeficientes Transformada Wavelet Aproximacion 1

Día Juliano0 50 100 150 200 250 300 350

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3Coeficientes Transformada Wavelet Detalle 1

Día Juliano

0 50 100 150 200 250 300 350-0.4

-0.3

-0.2

-0.1

0

0.1

0.2


Día Juliano0 50 100 150 200 250 300 350

-0.2

-0.1

0

0.1


Día Juliano

Señal Original

Señal Predecida

26

Model Prediction 2: Daily Kt Prediction

0 50 100 150 200 250 300 350-0.2

0

0.2

0.4

0.6

0.8

1

1.2Predicción Kt

Día Juliano

Kt

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Kt Original

Pre

dic

ció

n K

t

0 50 100 150 200 250 300 3500

0.05

0.1

0.15

0.2

0.25Error Absoluto

Día Juliano

Err

or P

redi

cció

n K

t

28

Kt data correlates Lag 1 data.

Data to be forecasted is Kt but the signal needs to be preconditioned.

Two general aproaches has been tested: Linear AR (TAG) Non Linear: NN, Wavenets, SOM+NN

Errors range between 20-50% depending on the technique used, forecasting horizon, inputs (Kt-p)

Future Works: AR prediction transforming Kt series Other forecasting techniques: Markov Chain, Fuzzy Logic, Caos Theroy Time Series Forecasting Combination for differents Horizonts. Spatial Forecating with satelite Images NWP with satellite data inputs.

CONCLUSIONS

Ispra 2007 luis martín2

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