Solar Radiation Data Modeling with a Novel Surface Fitting Approach

Transcription

1 Solar Radiation Data Modeling with a Novel Surface Fitting Approach F. Onur Hocao glu, Ömer Nezih Gerek, Mehmet Kurban Anadolu University, Dept. of Electrical and Electronics Eng., Eskisehir, Turkey anadolu.edu.tr Abstract. In this work one year hourly solar radiation data are analyzed and modeled. Using a 2-D surface fitting approach, a novel model is developed for the general behavior of the solar radiation. The mathematical formulation of the 2-D surface model is obtained. The accuracy of the analytical surface model is tested and compared with another surface model obtained from a feed-forward Neural Network(NN). Analytical surface model and NN surface model are compared in the sense of Root Mean Square Error (RMSE). It is obtained that the NN surface model gives more accurate results with smaller RMSE results. However, unlike the specifity of the NN surface model, the analytical surface model provides an intuitive and more generalized form that can be suitable for several other locations on earth. 1 Introduction Solar radiation is the principal energy source for physical, biological and chemical processes. An accurate knowledge and an insightful model of the solar radiation data at a particular geographical location is of vital importance. Such knowledge is a pre-requisite for the simulation and design of solar energy systems. Architects, agriculturalists, air conditioning engineers and energy conscious designers of buildings also require such information. In many cases, the solar energy applications involve tilted surfaces. To compensate for the effect of radiation on tilted surfaces, knowledge of both diffusing and direct components of global radiation falling on a horizontal surface is required [1]. Menges at al [2] reviewed and compared the available solar-radiation models for a region in detail. The majority of the models developed for the prediction of solar radiation are based on existing climatic-parameters, such as sunshine duration, cloud cover, relative humidity, and minimum and maximum temperatures [3 ]. Unfortunately, for many developing countries, solar-radiation measurements are not easily available because of the expensive measuring equipment and techniques required. In this study, using a 2-D approach as mentioned in Section 2, a novel solar radiation model for one year solar radiation data that is acquired and collected between August 1, 2 and July 3, 26 in Iki Eylul campus of Anadolu University,is developed. The model is based on a surface fitting approach using the data rendered in 2-D. It is observed that hourly alteration of solar radiation data within

2 a day has a Gaussian shaped function, hence the 2-D data along the hour axes are fitted to Gaussian functions. Trust-region algorithm is used as mentioned in Section 3 during calculating the parameters of Gaussian functions. Also a NN model is developed for 2-D data as mentioned in Section 4. Finally the models are compared in the sense of RMSE and the results are presented in Section. The NNs provide a more specific model for the data, hence they yield better prediction models. However, the 2-D surface model is more generic and insightful. Therefore it can also be used as a global model for places with similar yearly solar radiation conditions without utilizing data collection and training. 2 Determination and Estimation of Surface Model Structure and Parameters The first stage in data fitting is to determine a plausible model among known mathematical models that characterizes the data accurately. After setting the mathematical model, coefficients of the model must be estimated. Recently, a novel 2-D interpretation approach that was developed by Hocaoglu at al [6] indicated that rendering or interpretation of the data (i.e. transformation) also proves to be critical even before proceeding to the modeling. In this approach the solar radiation data in time series is rendered and presented in 2-D and it is shown that the representation format has significant advantages over 1-D time series approaches. In this work, starting from the mentioned 2-D rendered representation, a novel surface model is proposed. To determine the structure of the model for fitting to the data, transverse sections are taken from the 2-D along the hour and the day axes as given in Fig Fig. 1. Plots of cross sections along hour and days axes, respectively, for a two year data. Examining Fig.1 it can be deduced that the cross section along the hour axis is similar to a Gaussian function for all days. Conversely, the cross section

3 along the days axis exhibits an oscillatory behavior (seasons) that can be modeled with a sinusoidal function. The hourly variation function was chosen to be Gaussian due to its shape-wise resemblence and simple calculation, and the daily variation was chosen as a sinusoid due to its capability of physically explaining the seasonal variation phenomenon. Once the model of the data is determined, the fitting process must be applied. The result of the fitting process is an estimate of the true but unknown coefficients of the mathematical model. Method of least squares is the basic method that can be used for linear estimation. In this method, the sum of squared residuals is minimized. The residual for the i th data point is obtained as the difference between the actual value and the fitted value as given in equation 1. e i = y i ŷ i (1) The summed square error (SSE), therefore, is given by equation 2 SSE = n e 2 i = i=1 n (y i ŷ i ) 2, (2) where n is the number of data points included in the fit and SSE is the sum of squares error estimate. The supported types of least squares fitting include; Linear least squares, Weighted linear least squares, Robust least squares and Nonlinear least squares. Although linear least squares method can be used to fit a linear (polynomial) model to data, nonlinear functions such as Gaussians and sinusoids may not be suitable. In general, any surface model may be a nonlinear model which is defined in matrix form as in equation 3 i=1 y = f(x,β) + ε, (3) where y is an n-by-1 vector of responses, f is a function of β and X, β is m-by-1 vector of coefficients. X is the n-by-m design matrix for the model. ε is an n-by-1 vector of errors. Obviously, nonlinear models are more difficult to fit than linear models because the coefficients cannot be estimated using simple matrix optimization techniques. Instead, an iterative approach is required that follows the following steps: 1. Start with an initial estimate for each coefficient. For some nonlinear models, a heuristic approach is provided that produces reasonable starting values. For other models, random values on the interval [,1] are provided. 2. Produce the fitted curve for the current set of coefficients. The fitted response value y is given by equation 4 3. Adjust the coefficients and determine whether the fit improves. 4. Iterate the process by returning to step 2 until the fit reaches the specified convergence criteria. ŷ = f(x,b) (4)

4 The above iteration involves the calculation of the Jacobian of f(x, b), which is defined as a matrix of partial derivatives taken with respect to the coefficients. The direction and magnitude of the adjustment in step-3 depend on the fitting algorithm. There are several algorithms to find estimations of nonlinear model parameters. Around those, best knowns are trust-region and Levenberg-Marquardt algorithms. The Levenberg-Marquardt [7] algorithm has been used for many years and has proved to work most of the time for a wide range of linear and nonlinear models with relatively good initial values. On the other hand, trust-region algorithm is specifically more powerful for solving difficult nonlinear problems, and it represents an improvement over the popular Levenberg-Marquardt algorithm. Therefore, trust-region method is used for obtaining the Gaussian parameters of surface functions in this study. The days axis is not optimized by any methods, because its behavior is analytically obtained using geographical facts such as its period being 36 days and its extrema corresponding to June XX and Dec. XX. 3 NN Model for 2-D Data To test test and compare the accuracy of the 2-D model, a NN structure is also built. In this structure, the model does not yield a global, unified and analytical surface function. Instead, the result is a surface function that is more specifically trained to the available data. Although the analytical closed form is ambiguous, the NNs provide a dedicated and better surface model with less RMSE. Since the proposed surface model has two inputs (hour and day numbers) and one output (Solar radiation), the NN structure is constructed to be two input-one output. The input-output pairs are normalized as to fall in the range [-1,1]. It is obtained that using neurons in the hidden layer is appropriate according to simulations. Due to its ability of fast convergence the Levenberg-Marquard learning algorithm is used in learning process of NN. The network is trained using 1 year solar radiation data and surface model of the data is obtained by this way. Both hidden and output layer s output from their net input are calculated using Tan-Sigmoid transfer function. The network is trained in epochs. The results are obtained and compared with the global and analytical surface model in Section 4. 4 Numerical Results The hourly solar radiation data along one day is considered as a Gaussian function as in equation g(x) = ae (x b)2 /c 2 () where a is the height of the Gaussian peak, b is the position of the center of the peak and c is related to the full width at half maximum of the peak. ly radiation data are fitted to the Gaussian function for all days by determining the Gaussian parameters a, b and c using the trust-region algorithm. Totally 36

5 parameter stes a,b and c are obtained for one year of recorded data. Then to form the generic and global surface model of the data, variation of the parameter sets a,b and c are explored along days. Since the daily behavior of the data is expected to have a sinusoidal form as explained in Section 2, the parameters a and c are modeled with sinusoidal functions with periods equal 36 days. For each Gaussian function the position of the center of the peak values should be around the 12. value which corresponds the center of the day time for whole year. As a result, the parameter b is judiciously taken to be 12.. The other coefficients a and c are determined as sinusoidals in equations 6 and 7 a(day) = 364 sin( 2 pi day / 72 ) (6) c(day) = sin( 2 pi day / 712 ) (7) Finally the analytical surface that models the data is obtained as given in equation 8. Surface(day,hour) = a(day) e ((hour 12.)/ c(day) )2 (8) As a visual comparison, the obtained surface model and 2-D plot of actual data is given in Fig.2. The error data calculated by subtracting actual data from the analytical surface model for each hour is given in Fig Fig D plot of actual data and obtained analytical surface model The accuracy of the analytical surface model is tested and compared with surface function generated by NNs. A two input - one output feed forward neural network is built and given in Fig. 4. To numerically compare the NN surface with the analytical surface model, the input-output pairs of network are chosen to be compatible with each other as hour - versus - day - versus - Solar radiation. For instance, if it is desired to find the estimation value of solar radiation at th day of the year, at o clock,

6 Fig. 3. Error surface of the model Number of Number of Predicted solar radiation at desired day, desired hour Input Layer Hidden Layer Output Layer Fig. 4. The adopted NN structure

7 the inputs of network the network is taken as (,) which also corresponds to the coordinates of the surface model. Various number of neurons are used in the hidden layer to determine the optimal number of neurons and it is observed that using neurons is experimentally appropriate to find more accurate prediction values. The network is trained epochs. The plot of epoch number versus total RMS error is obtained as in Fig.. 1 Performance is.37894, Goal is Performance Epoch number Fig.. Plot of performance versus epoch number It is obvious from Fig. that a great deal of learning is already archived in 1 epochs. The surface obtained by NN and plot of actual 2-D data are given in Fig Fig D plot of the solar radiation data, and the surface function obtained by NN. The Autocorrelation coefficient and RMSE values between actual and predicted values of solar radiation data obtained from both analytical surface model and the NN surface model are calculated, tabulated, and presented in Table I.

8 Table 1. RMSE values for proposed structures and Autocorrelation coefficients between actual values and predicted values of solar radiation data Model RMSE R Developed Surface Model NN Surface Model Conclusion In this work, using the 2-D interpretation approach, surface models for solar radiation data are developed. The developed models have two inputs that are the number of days beginning from January 1 of the year and the number hours within the days. For these models, the hourly data variation within a day is fitted to Gaussian functions. The parameters of Gaussian functions are obtained for each day. In the analytical attempt of surface modeling, the behavior of the solar radiation data along the days corresponding to the same hour is observed to have a sinusoidal oscillation. Therefore, the parameters related with the height and width of the Gaussian are fitted to separate sinusoidal functions, and finally the analytical model of the surface is obtained. Alternatively, a NN structure is built with the same input-output data pairs in the 2-D form and a nonlinear and nonanalytical surface model of whole data is obtained. Two models are compared using RMSE distortion relative to the original data. Due to its specifity, the NN model provides a more accurate surface model with less RMSE. On the other hand, the NN surface model is not analytical, and it cannot be generalized to other places. Conversely, the analytical surface model is very intuitive with simple seasonal parameters, and it provides a global view of the solar radiation phenomenon. Therefore, it can be easily adapted to other places in the world without a long data collection period. References 1. Muneer T., Younes S., Munawwar S., Discourses on solar radiation modeling, Renewable and Sustainable Energy Reviews, Vol. 11, (27) Menges H. O., Ertekin C., Sonmete M. H., Evaluation of global solar radiation models for Konya, Turkey, Energy Conversion and Management, Vol.47, (26) Trabea AA, Shaltout MA. Correlation of global solar-radiation with meteorological parameters over Egypt. Renew Energy, Vol.21 (2), Badescu V. Correlations to estimate monthly mean daily solar global-irradiation: application to Romania. Energy, Vol.24 (1999), Hepbasli A, Ulgen K., Prediction of solar-radiation parameters through the clearness index for Izmir, Turkey, Energy Source, Vol.24 (22), Hocaoglu F.O., Gerek Ö.N., Kurban M., A Novel 2-D Model Approach for the Prediction of ly Solar Radiation, LNCS Springer, Vol. 47, (27), Marquardt, D., An Algorithm for Least Squares Estimation of Nonlinear Parameters, SIAM J. Appl. Math, Vol. 11, (1963)