# Forecasting Solar Irradiance from Time Series with Exogenous Information

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3 the weather data by taking the mean over 24 hours for numerical variables and by normalizing the total counts for dummy variables 31 Statistical analysis We analyzed the stationarity, seasonality, and trend of the solar irradiance time series data First, to study the stationarity, we employed the Variogram [8] which computes G k = Var(y t+k y t )/Var(y t+1 y t ), for k {1, 2,, } We observe from Figure 1 (top) that the value of G k is upper bounded with regular cycles, which suggests stationarity The stationarity property can also be seen from Figure 1 (bottom), where the annual mean is almost constant, and the daily values vary within a bounded range Also, from the same figure, we can see that the raw solar irradiance data exhibits seasonality for a cycle of around one year Its value reaches maximum in the summer, minimum in the winter However, solar irradiance has high fluctuations for a short period of time (ie several days) This suggests, as in [9][10], that solar irradiance is correlated with weather attributes; weather changes with regularity seasonally, but may change drastically within a shorter time frame To investigate the correlation between solar irradiance and weather attributes, for each time step, we plotted the (averaged) weather data and the corresponding solar irradiance, shown in Figure 2 We observe that humidity and sunlight duration have strong correlation with solar irradiance In addition, we notice that in the location chosen for the power plant, it seldomly rains, and that solar irradiance is low as the amount of rain increases Figure 1: Plots of variogram (top), daily observations, weekly average, and yearly average (middle) Figure 2: Correlations between solar irradiance and several weather metrics 32 Smoothing models As an extention of the statistical analysis of the dataset, we investigated several basic autoregressive time series models These models attempt to arrive at a prediction through simply smoothing N most recent observations Our goal is to obtain baseline results for comparison with more complex regression models To this end, we used the simple moving average model and the n th order exponential smoothing model Simple moving average computes a smoothed value at time T by taking the average of N most recent observations before T The nth order exponential smoothing works by recursively applying decreasing weights to observations that are less recent, using a provided discount factor λ: ŷ (n) t = λŷ (n 1) t + (1 λ)ŷ (1) t 1 (1) 3

4 where ŷ (0) t = y t, and y 0 could be specified by the user (eg simply 0) Figure 3 illustrates the extent that these simple models can fit to the data Figure 3: Solar irradiance forecasts by smoothing models 4 Methodology 41 Framework for Learning Solar Irradiance Models The modeling of solar irradiance should consider short-term dependency due to the correlation with weather attributes, as well as dependency on time in the annual cycle due to seasonality To this end, we propose a framework for learning such models In this framework, short-term dependency is built-in to the AR and ARX model, defined as follows AR: y t+h = f(y t u:t ) (2) ARX: y t+h = f(y t u:t, x t u:t ) (3) where u defines the size of the window which contains recent observations, assumed to affect y t+h, the forcast after temporal horizon h The above fomulation implies the assumption that the solar irradiance y t+h is only determined by past observations within a window of the target variable y t u:t (as well as that of the exogenous variables x t u:t if the model is ARX) Dependency on time in the cycle is incorporated into the training data for ARX models as day in the year Training samples are of the form (z t u:t, y t+h ), where z t u:t = z T t u z T t = x T t u x T t y t u y t y t u y t For AR models = wt u T d t u y t u wt T d t y t For ARX models Here, for i {t u,, t}, the exogenous variables x i are specified as x i = [ ] wi T d i, where wi is the weather data in the window, and d i {1,, 365} are days in the year for the time point i The information of day in the year enables the model to learn dependency between the solar irradiance observation and its time point relative to the annual cycle, which is then reflected by d t u:t During training, we can flatten z t u:t such that it becomes a feature vector of u dimensions for AR models, and (2 + w i )u dimensions for ARX models Denote the flattened z t u:t as zt u:t F (4) 4

5 411 LASSO Regression The first ARX model we consider is LASSO regression, that is, linear regression with LASSO regularization Weights β are learned from optimizing l(β) as follows: l(β) = T 1 h (y t+h β T z F 2(T h) t u:t) 2 + λ β 1 (5) where T equals to the total number of time points in the solar irradiance time series 412 Support Vector Regression t=1 Next, we consider the SVR model We can formulate the SVR objective for our context as follows, as described in LibSVM [2] 1 : min α T 1 2 (α h α ) T K(α α ) + ɛ t=1 subject to e T (α α ) = 0 0 α t, α t C, t = 1,, T h (α t + α t ) + T h t=1 y t+h (α t + α t ) where K ij = K(zi u:i F, zf j u:j ) is the kernel function We used RBF kernel in experiments 413 Long Short-Term Memory networks Lastly, we consider Long short-term memory networks (LSTMs)[5], which is a class of neural network Unlike common feedforward neural networks or convolutional neural network, LSTMs allow information to persist inside the model by having loops in their architecture Because of this feature, LSTMs are capable of learning long-term dependencies and have been performing well on timeseries forecasting problems [7] and many more interesting problems including language modeling [12] and image captioning [13] As other neural networks, LSTMs also have many hyperparameters such as number of layers, number of units in a layer, dropout ratio, and so on We show the architecture selected by crossvalidation as described in the next section in Figure 4 5 Experiments In our experiments, we forecasted the solar irradiance of one day ahead (h = 1) given the solar irradiance and weather data of previous u days (window size) We examined models that span multiple levels of complexity, namely, LASSO regression, SVR with RBF kernel, and LSTM The models were trained both autoregressively and with exogenous information, and different window sizes, u {1, 7, 30, 365}, were explored We used the first 70 percent of the timeseries data, which contains approxiamtely 1200 samples, as training dataset and the rest as testing dataset The hyperparameters were chosen through 10-fold crossvalidation We evaluated the models using the testing dataset by mean absolute error (MAE) between the observational data and the forecasted value 51 Results and Discussion The forecasts by autoregressive exogenous models are correlated to the observed solar irradiance as illustrated by Figure 5, and its fit is not shifted unlike the forecasts by the basic smoothing models shown in Figure 3 For window size 1 to 30, adding exogenous information consistently reduced the error as Figure 6 shows This result is reasonable because as we presented in the Dataset section, there are significant short-term correlation between solar irradiance and weather Therefore, by using previous weather data on top of the solar irradiance data itself, estimators should be able to make better forecasts However, the error for LASSO and SVR significantly increased with weather data when window size was set to 365 We reasoned this is because of the lack of model complexity Given a large number 1 This objective corresponds to ɛ SVR (6) 5

6 Figure 4: The architecture of the LSTM network The first dimension is None because it varies depending on the number of samples, and the output dimension is 1 because the network outputs a scalar value prediction of features (365 days 43 features 16, 000), training of the relatively simpler models converged without learning much This claim is supported to some extent by the fact that, a more complex model, namely LSTM, outperformed the other models by far SVR performed the best for small window size like 1 to 7 One possible explanation is that LASSO, as a linear model, lacked the representational power to learn the relationship between solar irradiance and weather On the other hand, LSTM, as a high dimensional non-linear model, lacked the amount of data to fit to the underlying structure that may exist SVR is a medium-sized non-linear model, so it was just good in this particular case for smaller window size 6 Conclusion We have introduced a framework to train models for solar irradiance that consider both short-term dependency as well as dependency on time in the annual cycle Our experiments demonstrate that by adding weather attributes and day in the year as exogenous information, the prediction error is reduced Besides, the best accuracy was obtained when longer time dependency was considered with more complex model, namely the long short-term memory network Indeed, even though our proposed methods are cost-effective and can forcast the general trend of solar irradiance, the 6

7 Figure 5: Solar irradiance forecasts by autoregressive exogenous models with the window size that performed the best on testing dataset The plots are for LASSO, SVR, and LSTM from the top (a) MAE by autoregressive models (b) MAE by autoregressive exogenous models Figure 6: Comparison of autoregressive models vs autoregressive exogenous models prediction error is significantly greater than physics-based methods which has an MAE of around 400 Wh/m 2 Nevertheless, there is plenty of room to improve First, the model can incorporate long-term dependency (one year) by training on samples that contain observations on the same day in year from all other years Second, currently the weather data that we obtained is only an approximation of the weather at the plant location, and we used linear interpolation to fill in missing observations The performance can potentially benefit if weather data is more accurate Finally, we can consider more exogenous information that may have higher correlation with solar irradiance, such as solar activity, satellite images of cloud above the plant location, and so on References [1] A Alzahrani, J Kimball, and C Dagli Predicting solar irradiance using time series neural networks Procedia Computer Science, 36: , 2014 [2] C-C Chang and C-J Lin Libsvm: a library for support vector machines ACM transactions on intelligent systems and technology (TIST), 2(3):27, 2011 [3] J Damon and S Guillas The inclusion of exogenous variables in functional autoregressive ozone forecasting Environmetrics, 13(7): ,

8 [4] E C e Silva, A Borges, M F Teodoro, M A Andrade, and R Covas Time series data mining for energy prices forecasting: An application to real data In International Conference on Intelligent Systems Design and Applications, pages Springer, 2016 [5] S Hochreiter and J Schmidhuber Long short-term memory 9: , [6] R-H Liang and J-H Liao A fuzzy-optimization approach for generation scheduling with wind and solar energy systems IEEE Transactions on Power Systems, 22(4): , 2007 [7] D L Marino, K Amarasinghe, and M Manic Building energy load forecasting using deep neural networks CoRR, abs/ , 2016 [8] D C Montgomery, C L Jennings, and M Kulahci Introduction to time series and analysis forecasting John Wiley & Sons, 2015 [9] V Prema and K U Rao Development of statistical time series models for solar power prediction Renewable Energy, 83: , 2015 [10] N Sharma, J Gummeson, D Irwin, and P Shenoy Cloudy computing: Leveraging weather forecasts in energy harvesting sensor systems In Sensor Mesh and Ad Hoc Communications and Networks (SECON), th Annual IEEE Communications Society Conference on, pages 1 9 IEEE, 2010 [11] N Sharma, P Sharma, D Irwin, and P Shenoy In Smart Grid Communications (SmartGrid- Comm), 2011 IEEE International Conference on, pages IEEE, 2011 [12] M Sundermeyer and y Ralf Schlüter and Hermann Ney, booktitle=interspeech Lstm neural networks for language modeling [13] O Vinyals, A Toshev, S Bengio, and D Erhan Show and tell: A neural image caption generator 2015 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pages ,

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