1. DAILY RADIATION FORECASTING BY
STATISTICAL METHODS: PRELIMINARY
RESULTS
Presented by:
LUIS MARTÍN POMARES
ENERGY DEPARTAMENT
Renewable energy division
Plataforma Solar de Almería
3rd Experts Meeting of the IEA SHC Task
“Solar Resource Knowledge Management”
&
MESoR Coordination Meeting
Ispra (VA), Italy
12 – 14 March 2007

2. DAILY RADIATION FORECASTING
1. INTRODUCTION
2. EXPLORATORY DATA
ANALYSIS
3. LINEAR PREDICTION: TAG(p)
4. NON-LINEAR PREDICTION
5. CONCLUSIONS
2

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
3

4. PIRANOMETRIC DATA
•Data Period:
10/7/1996 – 29/12/2003
•#Data:
2304 values
•Daily Goblal Solar
Radiation transformed
to daily Kt Values
4

5. EXPLORATORY DATA
ANALYSIS
8 0
0
0.8
7 0
0
0.7
N m r d m e tr s
ú eo e u s a
6 0
0
0.6
5 0
0
0.5
t ia io
K D r
4 0
0
0.4
3 0
0
0.3
2 0
0
0.2
1 0
0
0.1
0
0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
0 50
0 10
00 1 0
50 20
00 2 0
5 0 K D r
t ia io
Da N
í
Sample Partial Autocorrelation Function
P w rS e a D n
o e p ctr l e sity Estim te v P r d g a
a ia e io o r m
30
Sample Partial Autocorrelations
20
q e cy (d /ra /sa p )
m le
0.8
B d 10
0
0.6
-10
P w r/fre u n
0.4 -20
-30
o e
0.2 -40
-50
0
-60
5
-70
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-0.2 N rm lize F q e cy (× ra /sa p )
o a d re u n
0 2 4 6 8 10 12 14 16 18 20 π d m le

6. EXPLORATORY DATA
ANALYSIS
120
Central Months predominance of
Goods Days. Monthly Histogram
100
External months: Mixture of Kt
(bad and good days) 80
N úm ero de m ues tras
60
40
Kt
20
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
K t D iario
Month 6
Daily Kt for each month

7. SAMPLE PROBABILITY
DISTRIBUTION: BI-EXPONENTIAL
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 Cumulative Daily Distribution Functions
9

8. Partial Autocorrelation
Autocorrelation
•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.
Non Stationary
Non Gaussian Data Data Preprocessing
Non Linear
12

9. LINEAR PREDICTION: TAG(p)
Timedependant Autorregressive Gassuian Model: TAG
Timedependant: Montly Autorregresive Model  12 AR(p)
Autorregresive Model AR(p):
p
∑φ ( X
k =0
k t −k − µ ) = at , t = p + 1, p + 2,...,
Gaussian: Transform Data to Gaussina Distribution using daily
Kt Anomalies
Kti − K T j
Kt _ Anomalyi =
σKT j
13

10. LINEAR PREDICTION: TAG(p)
Enero Enero
0.15 28 85
AR(2)/Persistencia - Enero AR(1) AR(1)
0.1 26 AR(2) AR(2)
Mejora RMSE AR Óptimo frente Persistencia
AR(3) 80
AR(3)
0.05 24 AR(4) AR(4)
AR(5) AR(5)
75
%MBE Predicción Diaria Kt
%RMSE Predicción Diaria Kt
22 AR(6) AR(6)
0
AR(7) AR(7)
20 AR(8) 70 AR(8)
-0.05 AR(9)
AR(9)
18 AR(10)
AR(10)
Persistencia 65
-0.1 Persistencia
16
-0.15 60
14
-0.2 55
12
-0.25 10
1 2 3 4 5 6 1 2 3 4 5 6 50
Horizonte Predicción (Días) 1 2 3 4 5 6
Horizonte Predicción (Días)
Horizonte Predicción (Días)
Julio Julio
0.15 5 28
AR(2)/Persistencia - Julio AR(1) AR(1)
AR(2) AR(2)
0.1 4.5 26
AR(3)
Mejora RMSE AR Óptimo frente Persistencia
AR(3)
AR(4) AR(4)
0.05 4 AR(5) 24 AR(5)
%RMSE Predicción Diaria Kt
%MBE Predicción Diaria Kt
AR(6) AR(6)
AR(7) AR(7)
0 3.5 AR(8) 22 AR(8)
AR(9) AR(9)
-0.05 AR(10) AR(10)
3 20
Persistencia Persistencia
-0.1 2.5 18
-0.15 2 16
-0.2 1.5 14
1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6
Horizonte Predicción (Días)
14
Horizonte Predicción (Días) Horizonte Predicción (Días)

11. LINEAR PREDICTION: TAG(p)
Future Works: Kt Transformation
Predict Kt Differences between days: y (t ) = Kt (t ) − Kt (t − 1)
0.8 800
700
0.6
600
0.4
500
0.2
400
0
300
-0.2
200
-0.4
100
-0.6
0
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
-0.8
0 500 1000 1500 2000 2500
P er S
ow pectral Density Estim te v P
a ia erio ogra
d m
S m le A to r la n F n tio ( C )
a p u co re tio u c n A F 0
1
-10
/ra /sa p )
m le
0.8
u co la n
S m le A to rre tio
-20
0.6 P w r/freq e cy (dB d
-30
0.4
u n
-40
0.2
a p
-50
o e
0 -60
-0.2 -70
-0.4
0 2 4 6 8 10
L g
a
12 14 16 18 20
-80
0 0.1 0.2 0.3
N 17
0.4 0.5 0.6
orm lize F que cy (×π rad
a d re n /sam
0.7
ple)
0.8 0.9 1

12. NON-LINEAR PREDICTION
Model Prediction 1:
Prediction
NEURAL NETWORK (NN)
Model Prediction 2:
Signal Preprocessing:
SPECTRAL SIGNAL ANALYSIS: WAVELET
Prediction
NEURAL NETWORK (NN)
Model Prediction 3:
Future works like Fuzzy Logic, Markov Chain…
Signal Preprocessing:
CLUSTER ANALYSIS: SOM NETWORKS
Prediction:
NEURAL NETWORK (NN) 18

13. Model Prediction 1: RESULTS
Mean Absolute Error (M AE)
Neural Network Model Structure 1
0,9
0,8
NN Model 1 1 Neuron 0,7
0,6
Modelo 1
MAE
Modelo 2
NN Model 2 7-1
0,5
Modelo 3
0,4
Modelo 4
0,3
NN Model 3 5-3-1 0,2
0,1
NN Model 4 7-5-3-1 0
0 2 4 6 8 10 12
NN(X)
Coeficiente Correlación (R) Mean Squared Error (MSE)
0,6 0,5
0,5 0,4
Modelo 1
0,4 Modelo 1
Modelo 2 0,3
0,3 Modelo 2
R
MSE
Modelo 3 0,2
0,2 Modelo 3
Modelo 4 0,1
0,1 Modelo 4
0 0
0 2 4 6 8 10 12 0 2 4 6 8 10 12
-0,1
NN(X)
20
NN(X)

14. 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
Low
Kt
Frequency High
cA1 cD1 Frequency
cA2 cD2
cA3 cD3
21

15. SIGNAL DECOMPOSITION
S ñ O in
e al rig al
1
Kt
0.5
0
0 50 1 0
0 150 200 250 300 350
Se al Apro im ció 3
ñ x a n
1
Kt
0.5
0
0 50 1 0
0 150 200 250 300 350
S ñal D
e eta 1
lle
0.5
Kt
0
-0.5
0 50 1 0
0 150 200 250 300 350
S ñal D
e eta 2
lle
0.5
Kt
0
-0.5
0 50 1 0
0 150 200 250 300 350
S ñal D
e eta 3
lle
0.2
Kt
0
-0.2
0 50 1 0
0 150 200 250 300 350
S ñ R co
e al e nstruida
1
Kt
0.5
0
0 50 1 0
0 150 200 250 300 350
D Ju
ia liano
22

16. Model Prediction 2: WAVENET
aD1(x)
aD1(x-1) DWψ aD1(x+1)
Kt •aD1 .
.
.
DWaD1(x-k)
ψ
aD2(x)…aD2(x-k)
•aD2 aD2(x+1)
aD3(x)…aD2(x-k) IDWψ
•aD3 aD3(x+1)
aD2(x)…aD2(x-k)
•aA3 aA1(x+1) Kt(x+1)
23

17. Model Prediction 2: RESULTS
Mean Absolute Error (MAE)
Neural Network Model Structure
3
2,5
Model 1 1 Neuron 2 Modelo 1
MAE
Modelo 2
Model 2 7-1 1,5
Modelo 3
1 Modelo 4
Model 3 5-3-1(cA) 0,5
7-5-3-1(cD) 0
1 2 3 4 5 6 7 8 9 10
Model 4 7-5-3-1 NN(X)
Coeficiente Correlación (R) Mean Squared Error (MSE)
1,2 0,5
1
Modelo 1 0,4
0,8 Modelo 1
Modelo 2 0,3 Modelo 2
MSE
0,6
R
Modelo 3 0,2 Modelo 3
0,4
Modelo 4 Modelo 4
0,2 0,1
0 0
1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10
NN(X) NN(X)
24

18. Prediction of Wavelet Transform
Coeficientes
Coeficientes Transformada Wavelet Aproximacion 1 Coeficientes Transformada Wavelet Detalle 1
0.9 0.3
0.8 0.2
0.7 0.1
0.6 0
0.5 -0.1
0.4 -0.2
Señal Original
0.3 -0.3
Señal Predecida
0.2 -0.4
0 50 100 150 200 250 300 350 0 50 100 150 200 250 300 350
Día Juliano Día Juliano
Coeficientes Transformada Wavelet Detalle 2 Coeficientes Transformada Wavelet Detalle 3
0.3 0.2
0.2
0.1
0.1
0
0
-0.1
-0.2
-0.1
-0.3
-0.4 -0.2
0 50 100 150 200 250 300 350 0 50 100 150 200 250 300 350
Día Juliano Día Juliano
25

19. Model Prediction 2: Daily Kt Prediction
P redic c ión K t Error Absoluto
1.2 0.25
0.2
1
Error Predicción Kt
0.15
0.8
0.1
0.6 0.05
Kt
0
0.4 0 50 100 150 200 250 300 350
Día Juliano
1
0.9
0.2 0.8
0.7
Predicción Kt
0.6
0 0.5
0.4
0.3
0.2
-0.2
0 50 100 150 200 250 300 350 0.1
Día Juliano 0
0 0.1 0.2
26
0.3 0.4 0.5
Kt Original
0.6 0.7 0.8 0.9 1

20. CONCLUSIONS
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.
28