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