Poster-Predicting Solar Generation from Weather Forecasts

Report
Predicting Solar Generation from Weather Forecasts
Advisor: Professor Arye Nehorai
Chenlin Wu, Yuhan Lou
Department of Electrical and Systems Engineering
Principal Component Analysis (PCA)
Kernel Trick for SVR
Background

Smart grid: increasing the contribution of renewable in grid
energy

Solar generation: intermittent and non-dispatchable
The kernel trick is a way of mapping observations from a
general set S (Input space) into an inner product space V
(high dimensional feature space)
Φ: ℝ → ℝ
Goals
Creating automatic prediction models
()

Predicting future solar power intensity given weather forecasts
=
≫

 −  ∗ ( , )
Experiments
+
where   ,  = ϕ  , ϕ  .

NREL National Solar Radiation Database 1991-2010

Hourly weather and solar intensity data for 20 years
Gaussian Processes (GP)

Station: ST LOUIS LAMBERT INT’L ARPT, MO
GP regression model:
Input: (combination of 9 weather metrics)
 =   +  , where noise  ~(0,  2 )

Date, Time , Opaque Sky Cover, Dry-bulb Temperature, Dew-point
Temperature, Relative Humidity, Station Pressure,Wind Speed, Liquid
Precipitation Depth
Output :

Amount of solar radiation (Wh/m2) received in a collimated beam on a
surface normal to the sun
Methods


In our research, regression is used to learn a mapping from
some input space of n-dimensional vectors to an output space of
real-valued targets
We apply different regression methods including:

Linear least squares regression

Support vector regression (SVR) using multiple kernel functions

Gaussian processes
Linear Model
 =  X =   + 
where  ∈ ℝ : measurement (solar intensity)
X ∈ ℝ×+1 : each row is a p-dimensional input
 ∈ ℝ+1 : unknown coefficient
 ∈ ℝ : random noise
Loss function(Square error):  −  2 =  −    2
Support Vector Regression (SVR)
Given training data {( , 1 ), ( , 2 )…( ,  )
Linear SVR Model:
  = ,  +  =
minimize
1
2
 2 +
Applying PCA to remove redundant information
The graph shows the MSE with different input
dimensions. The feature set with 8 dimensions performs
the best with the lowest test error.
And as long as we keep more than 5 principle
components, the errors are lower than linear regression

Data Source

Such as: Temperature & Time of the day
 −  ∗ ϕ 
=


Some weather metrics correlate strongly

+
∗
(ξ
+ξ
   )
 − ( ) ≤  + ξ
∗
(
)
−

≤

+
ξ
subject to



ξ , ξ ∗ ≥ 0
Loss function: (epsilon intensive)
0
 ξ ≤ 
ξ≔
ξ −  ℎ.

Predictions are made by proposed methods

20% of data is used to train & 10% of the data is used to test
Linear regression

Assume a zero mean GP prior distribution
MSE is used to evaluate the
result of regression. Followings
are the prediction errors of the 3
different methods:

over inference functions  ∙ . In particular,
Linear Regression
  1 , . . . ,    ~ 0,  , , = (   ,    ) = (  ,   )
215.7884
To make predictions  ∗ at test points  ∗ , where  ∗ =   ∗ + ε
 ∗ : ∗
 , 
~  0,
 ∗, 
∗
It follows that p  , 
,  ∗

 ∗, ∗
∗
,

∗
~  0,
2

130.1537
0
2
0
−1

SVM regression
= (, Σ)
where  =  ,  ∗ [(, ) +  2 ]−1 
Σ =   ∗ ,  ∗ −  ,  ∗  ,  +  2 
SVR
122.9167

 ∗,  .
SPGP
Followings are 24-hour prediction
Sparse Pseudo-input GP (SPGP)
GPs are prohibitive for large data sets due to the inversion
of the covariance matrix.
Consider a model parameterized by a pseudo data set  of
size  ≪ , where n is the number of real data points.
Reduce training cost from  3 to  2  , and prediction
cost from  2 to  2
Pseudo data set :  =  =1… ,  =  =1…
SPGP regression
Prior on Pseudo targets:    = (0,  )
Likelihood:

−1
  , ,  =    ,

 −  
−1
LR
SVR
GP
 +  2
Posterior distribution over  :
  ,  =    −1  ( +  2 )−1 ,   −1 
where  =  +  ( +  2 )−1 
Given new input  ∗ , the predictive distribution:
  ∗ ,  ∗ =    ∗ ∗ , ,    ,  =  ∗ , Σ ∗
−1

∗
where  = ∗   ( +  2 )−1 
Σ ∗ = ∗∗ − ∗   −1 −  −1 ∗ +  2
Predicting Error
191.5258
93.2988
90.2835
Conclusions

Using machine learning to automatically model the function of
predicting solar generation from weather forecast lead to a
acceptable result

Gaussian processes achieved lowest error among all the methods

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