ispra 2007 luis martín2
TRANSCRIPT
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
Horizonte Predicción (Días)
%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
Horizonte Predicción (Días)
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
Horizonte Predicción (Días)
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
Horizonte Predicción (Días)
%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
Horizonte Predicción (Días)
%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
0.3Coeficientes Transformada Wavelet Detalle 2
Día Juliano0 50 100 150 200 250 300 350
-0.2
-0.1
0
0.1
0.2Coeficientes Transformada Wavelet Detalle 3
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