Method of forecasting for solar-based power systems

Abstract

The method of forecasting for solar-based power systems (10) recognizes that no single solar irradiance forecasting model provides the best forecasting prediction for every current weather trend at every time of the year. Instead, the method trains a classifier to select the best solar irradiance forecasting model for prevailing conditions through a machine learning approach. The resulting solar irradiance forecast predictions are then used to allocate the solar-based power systems (10) resources and modify demand when necessary in order to maintain a substantially constant voltage supply in the system (10).

Claims

1. A computer-implemented method of forecasting for solar-based power systems, comprising the steps of: (a) providing a non-transitory computer-readable medium having embodied thereon a program, which when executed by a computer, causes the computer to execute a method of forecasting for solar-based power systems, the method comprising the steps of: i. measuring a first series of current solar irradiance parameters with sensors for a defined geographical region over predetermined time intervals to form a data set, wherein the measurements comprise measuring direct normal irradiance (DNI), global horizontal irradiance (GHI) and diffuse horizontal irradiance (DHI); ii. selecting a window size defining a number of past measurements and future forecast predictions to be made from the number of past measurements; iii. partitioning the data set into successive and adjacent time series training data sequences of the selected window size; iv. applying a plurality of different forecasting methods to the time series training data sequences to obtain future forecast predictions from each of the forecasting methods; v. comparing the future forecast predictions of each of the forecasting methods to measured data to obtain a corresponding error rate associated with each of the methods, given the time series training data sequences; vi. assigning the forecasting method with the lowest error rate as the forecasting class for the time series training data sequences; vii. repeating steps i through vi to train a classifier to determine an optimal forecasting class for different time series training data sequences; viii. using the sensors to measure a second series of current solar irradiance parameters; ix. using the classifier to determine the optimal forecasting class for the second series of current solar irradiance parameters; x. making future forecast predictions from the second series of current solar irradiance parameters using the optimal forecasting class; xi. predicting solar-based power system demands and generating capacities based upon the future forecast predictions made in step (j); and xii. making adjustments in the solar-based power system demands and stored energy in order to maintain a substantially constant voltage supply for the defined geographic region.

2. The method of forecasting for solar-based power systems as recited in claim 1, wherein the step of measuring the first and second series of current solar irradiance parameters comprises measuring in one minute intervals.

3. The method of forecasting for solar-based power systems as recited in claim 1, further comprising the step of applying data filtering to the data set generated in step i.

4. The method of forecasting for solar-based power systems as recited in claim 1, wherein the plurality of forecasting methods comprise a persistence method and a support vector regression method.

5. A computer-implemented method of forecasting for solar-based power systems, comprising the steps of: providing a non-transitory computer-readable medium having embodied thereon a program, which when executed by a computer, causes the computer to execute a method of forecasting for solar-based power systems, the method comprising the steps of: measuring solar irradiance parameters with sensors for a defined geographical region over predetermined time intervals to form a data set, wherein the measurements comprise measuring direct normal irradiance (DNI), global horizontal irradiance (GHI) and diffuse horizontal irradiance (DHI); selecting a window size defining a number of past measurements and future forecast predictions to be made from the number of past measurements; partitioning the data set into successive and adjacent time series training data sequences of the selected window size; applying a different autoregressive forecasting method to each of the time series training data sequences to obtain future forecast predictions; training a classifier to determine an optimal forecasting class for the different time series training data sequences; predicting solar-based power system demands and generating capacities based upon the future forecast predictions; and making adjustments in the solar-based power system demands and stored energy in order to maintain a substantially constant voltage supply for the defined geographic region.

6. A non-transitory computer-readable medium having embodied thereon a program, which when executed by a computer, causes the computer to execute a method of forecasting for solar-based power systems, the method comprising the steps of: measuring solar irradiance parameters with sensors for a defined geographical region over predetermined time intervals to form a data set; selecting a window size defining a number of past measurements and future forecast predictions to be made from the number of past measurements; partitioning the data set into successive and adjacent time series training data sequences of the selected window size; applying a different autoregressive forecasting method to each of the time series training data sequences to obtain future forecast predictions; training a classifier to determine an optimal forecasting class for the different time series training data sequences; predicting solar-based power system demands and generating capacities based upon the future forecast predictions; and making adjustments in the solar-based power system demands and stored energy in order to maintain a substantially constant voltage supply for the defined geographic region.

Description

BRIEF DESCRIPTION OF DRAWINGS

(1) FIG. 1 is a block diagram illustrating system components for implementing a method of forecasting for solar-based power systems according to the present invention.

(2) FIG. 2A is a graph illustrating autocorrelation of a one-minute time series used in an embodiment of the method of forecasting for solar-based power systems.

(3) FIG. 2B is a graph illustrating partial autocorrelation of the one-minute time series used in the embodiment of the method used in FIG. 2A.

(4) FIG. 3A is a graph illustrating autocorrelation of a five-minute time series used in the embodiment of the method used in FIG. 2A.

(5) FIG. 3B is a graph illustrating partial autocorrelation of the five-minute time series used in the embodiment of the method used in FIG. 2A.

(6) FIG. 4A is a graph illustrating autocorrelation of a ten-minute time series used in the embodiment of the method used in FIG. 2A.

(7) FIG. 4B is a graph illustrating partial autocorrelation of the ten-minute time series used in the embodiment of the method used in FIG. 2A.

(8) FIG. 5 is a graph comparing relative root mean squared deviation (RMSD) of the autoregressive (AR) models of order 1-20 against the persistence (PER) model, and a combination of the autoregressive models in an embodiment of the method of forecasting for solar-based power systems, specifically for each horizon of prediction applied to one-minute average time series data, with the data set being recorded in January of 2014.

(9) FIG. 6 is a graph comparing relative root mean squared deviation (RMSD) of the autoregressive (AR) models of order 1-20 against the persistence (PER) model, and a combination of the autoregressive models in an embodiment of the method of forecasting for solar-based power systems, specifically for each horizon of prediction applied to one-minute average time series data, with a data set recorded in April of 2014.

(10) FIG. 7 is a graph comparing relative root mean squared deviation (RMSD) of the autoregressive (AR) models of order 1-20 against the persistence (PER) model, and a combination of the autoregressive models in an embodiment of the method of forecasting for solar-based power systems, specifically for each horizon of prediction applied to one-minute average time series data, with a data set recorded in June of 2014.

(11) FIG. 8 is a graph comparing relative root mean squared deviation (RMSD, %) of the autoregressive (AR) models of order 1-20 against the persistence (PER) model, and a combination of the autoregressive models in an embodiment of the method of forecasting for solar-based power systems, specifically for the one-minute time series.

(12) FIG. 9 is a graph comparing relative root mean squared deviation (RMSD) of the autoregressive (AR) models of order 1-20 against the persistence (PER) model, and a combination of the autoregressive models in an embodiment of the method of forecasting for solar-based power systems, specifically for each horizon of prediction applied to five-minute average time series data, with the data set being recorded in January of 2014.

(13) FIG. 10 is a graph comparing relative root mean squared deviation (RMSD) of the autoregressive (AR) models of order 1-20 against the persistence (PER) model, and a combination of the autoregressive models in an embodiment of the method of forecasting for solar-based power systems, specifically for each horizon of prediction applied to five-minute average time series data, with a data set recorded in June of 2014.

(14) FIG. 11 is a graph comparing relative root mean squared deviation (RMSD) of the autoregressive (AR) models of order 1-20 against the persistence (PER) model, and a combination of the autoregressive models in an embodiment of the method of forecasting for solar-based power systems, specifically for each horizon of prediction applied to five-minute average time series data, with a data set recorded in November of 2014.

(15) FIG. 12 is a graph comparing relative root mean squared deviation (RMSD, %) of the autoregressive (AR) models of order 1-20 against the persistence (PER) model, and a combination of the autoregressive models in an embodiment of the method of forecasting for solar-based power systems, specifically for the five-minute time series.

(16) FIG. 13 is a graph comparing relative root mean squared deviation (RMSD) of the autoregressive (AR) models of order 1-20 against the persistence (PER) model, and a combination of the autoregressive models in an embodiment of the method of forecasting for solar-based power systems, specifically for each horizon of prediction applied to ten-minute average time series data, with the data set being recorded in February of 2014.

(17) FIG. 14 is a graph comparing relative root mean squared deviation (RMSD) of the autoregressive (AR) models of order 1-20 against the persistence (PER) model, and a combination of the autoregressive models in an embodiment of the method of forecasting for solar-based power systems, specifically for each horizon of prediction applied to ten-minute average time series data, with a data set recorded in August of 2014.

(18) FIG. 15 is a graph comparing relative root mean squared deviation (RMSD) of the autoregressive (AR) models of order 1-20 against the persistence (PER) model, and a combination of the autoregressive models in an embodiment of the method of forecasting for solar-based power systems, specifically for each horizon of prediction applied to ten-minute average time series data, with a data set recorded in December of 2014.

(19) FIG. 16 is a graph comparing relative root mean squared deviation (RMSD, %) of the autoregressive (AR) models of order 1-20 against the persistence (PER) model, and a combination of the autoregressive models in an embodiment of the method of forecasting for solar-based power systems, specifically for the ten-minute time series.

(20) Similar reference characters denote corresponding features consistently throughout the attached drawings.

BEST MODES FOR CARRYING OUT THE INVENTION

(21) The method of forecasting for solar-based power systems recognizes that no single solar irradiance forecasting model provides the best forecasting prediction for every current weather trend at every time of the year. Instead, the present method uses model evaluation data to train a classifier that enables the selection of the best solar irradiance forecasting model for prevailing conditions through a machine learning approach. The resulting solar irradiance forecast predictions are then used to allocate the solar-based power systems resources and modify demand when necessary in order to maintain a substantially constant voltage supply in the system.

(22) Generally, the method of forecasting for solar-based power systems includes the following steps: (a) measuring solar irradiance parameters with sensors for a defined geographical region over predetermined time intervals to form a data set; (b) selecting a window size defining a number of past measurements and future forecast predictions to be made from the number of past measurements; (c) partitioning the data set into successive and adjacent time series training data sequences of the selected window size; (d) applying each of a selected plurality of forecasting methods to the time series training data sequences to obtain future forecast predictions from the forecasting methods applied; (e) comparing the future forecast predictions of each of the forecasting methods to measured data to obtain a corresponding error rate associated with each of the methods, given the time series training data sequences; (f) assigning the forecasting method with the lowest error rate as the forecasting class for the time series training data sequences; (g) repeating steps (a) through (f) for the selected plurality of forecasting methods and use the resulting forecasting evaluation data to train a classifier to determine an optimal forecasting class for different time series training data sequences; (h) using the sensors to measure current solar irradiance parameters; (i) using the classifier to determine the optimal forecasting class for the current solar irradiance parameters; (j) making future forecast predictions from the current solar irradiance parameters using the optimal forecasting class; (k) predicting solar-based power system demands and generating capacities based upon the future forecast predictions made in step (j); and (l) making adjustments in the solar-based power system demands and stored energy in order to maintain a substantially constant voltage supply for the geographic region.

(23) In step (a) above, ground-measured solar radiation data is used to create data files, such as the exemplary data file shown below in Table 1. In Table 1, direct normal irradiance (DNI), global horizontal irradiance (GHI) and diffuse horizontal irradiance (DHI) are each provided as one-minute averages of the measured solar radiation components (measured in W/m.sup.2). Each component in Table 1 was measured by a different sensor mounted on a high-precision solar radiation monitoring station.

(24) TABLE-US-00001 TABLE 1 Measured Solar Radiation Parameters YYYY-MM-DD- HH-MM DNI GHI DHI Kt Kt_p 2014-03-05-06-47 305.34 167.55 102.75 0.65 0.92 2014-03-05-06-48 270.89 156.47 97.45 0.60 0.84 2014-03-05-06-49 221.88 141.53 91.46 0.53 0.74 2014-03-05-06-50 179.62 128.26 86.32 0.47 0.65 2014-03-05-06-51 84.08 105.34 82.82 0.38 0.52 2014-03-05-06-52 108.30 113.37 85.17 0.40 0.55 . . . . . . . . . . . . . . . . . . 2014-03-05-09-46 624.76 688.97 225.32 0.681 0.703

(25) In the above, Kt and Kt_p are clearness indices, calculated from the ground-measured GHI. Kt is the ratio of GHI to the calculated global horizontal radiation at the top of the atmosphere. Kt removes the seasonal dependence of GHI throughout the year. Kt_p is a modified form of Kt, which adds a correction factor for the changing atmospheric air masses traversed by the solar radiation at any moment. Kt and Kt_p are given by:

(26) $\begin{array}{cc}\mathrm{Kt}=\frac{\mathrm{GHI}}{\mathrm{GHItoa}}& \left(1\right)\\ {\mathrm{Kt}}_p=\frac{\mathrm{Kt}}{1.031e^{-1.4/\left(0.9+9.4/\mathrm{am}\right)}+0.1}& \left(2\right)\\ am=\frac{1}{\cos \left(\mathrm{SZA}\right)+0.50573{\left(96.07995-\mathrm{SZA}\right)}^{-1.6364}},& \left(3\right)\end{array}$
where GHI.sub.toa is the extraterrestrial solar radiation on a horizontal surface at the top of the atmosphere, am is the air mass, and SZA is the solar zenith angle in degrees. In order to ensure good accuracy in the modeling results, only measured data that pass some quality control tests should be included. For the data given above, the recommended Baseline Surface Radiation Network quality tests were applied to all entries. Several or just a single measure of solar irradiance can be used. In the present embodiment, all examples will be given with reference to the single measure Kt_p.

(27) Data filtering may be applied to the data set generated in step (a). Any suitable technique, such as normalization, discretization, or wavelet analysis, used alone or in combination, may be used to remove noise from the data. With normalization, all numeric values in the given data set are mapped into a standard numerical interval, typically 0-1. Discretization maps continuous numeric values into discrete counterparts (e.g., high, medium, low). The wavelet transform enables the decomposition of a time series into a time dependent sum of frequency components.

(28) In order to select the window size defining a number of past measurements and future forecast predictions to be made from the number of past measurements (step (b)), the number of steps ahead to be forecasted are first selected. Then, several data points are selected, at the beginning, middle and end of the data collection. For each data point, an initial training data set T is created with a window size, i, given by i=n+m, where m is the number of steps ahead to be forecasted, and n is some function over m, e.g., 3 m. One or several forecasting algorithms are then applied to T and the results are evaluated using an evaluation measure such as the Coefficient of Variation of the Root Mean Square Error (cvRMSE) to compare predicted (pred) against observed (obs) values. Here,

(29) $\mathrm{cvRMSE}=\frac{\sqrt{\frac{\underset{t=1}{\overset{n}{.Math.}}{\left(x_{{\mathrm{pred}}_t}-x_{{\mathrm{obs}}_t}\right)}^2}{n}}}{\mathrm{mean}\left({\mathrm{obs}}_{\mathrm{ti}-n}\right)}.$
The window size, i, is successively increased and decreased until the best evaluation results (e.g., the lowest cvRMSE) are found, to determine the best training window size j.

(30) In step (c), the training data is partitioned into successive and adjacent time series training data sequences of the selected window size. For example, if the time scale is minutes, an ideal training window size may be 180 minutes, and the time series data for Kt_p would be as indicated in Table 2 below.

(31) TABLE-US-00002 TABLE 2 Exemplary Partitioning of Training Data Period for Kt_p Prediction Kt_p at minute 1 . . . Kt_p at minute 180 2014-12-01-06-57.fwdarw. 0.080137 . . . 0.067823 2014-12-01-09-56 2014-12-01-06-57.fwdarw. 0.080137 . . . 0.067823 2014-12-01-09-56 2014-12-01-06-57.fwdarw. 0.080137 . . . 0.067823 2014-12-01-09-56 2014-12-01-06-57.fwdarw. 0.080137 . . . 0.067823 2014-12-01-09-56 2014-12-01-06-57.fwdarw. 0.080137 . . . 0.067823 2014-12-01-09-56 2014-12-01-06-57.fwdarw. 0.080137 . . . 0.067823 2014-12-01-09-56 . . . . . . . . . . . .

(32) In step (d), each of a plurality of forecasting methods is applied to the time series training data sequences to obtain future forecast predictions from the forecasting method applied. The different forecasting models are developed to predict n-steps ahead for the solar irradiance variable of interest, using as training data the time series data developed above. In the present embodiment, two forecasting models are exemplified: Persistence and Support Vector Regression (SVR). The persistence model assumes that no change occurs from the present state throughout the forecasting period. In SVR, within a machine learning approach to forecasting, each training sample is a pair {{right arrow over (x)}, y}, where {right arrow over (x)}.sup.n is a vector for the time-series class to be learned, and y y is the associated value. The aim of the machine learning algorithm is to find a function such that each {right arrow over (x)}.sub.l in the training dataset approximates its value y.sub.i as closely as possible. The resulting function is then used to predict values n-steps ahead of the time series data used for training. When the input data are amenable to linear regression, the SVR prediction function is given by y.sub.i={right arrow over (w)}.Math.{right arrow over (x.sub.l)}+b, where i=1, . . . , n, {right arrow over (w)} is the weight vector (i.e., a linear combination of training patterns that supports the regression function), {right arrow over (x.sub.l)} is the input vector (i.e., the training sample), y.sub.i is the value for the input vector, and b is the bias (i.e., an average over marginal vectors, which are weight vectors that lie within the margins set by the loss function).

(33) The objective of regression is to learn the weight vector {right arrow over (w)} that has the smallest possible length so as to avoid over-fitting. To ease the regression task, a given margin of deviation is allowed with no penalty, and a given margin is specified where deviation is allowed with increasing penalty. The length of the weight vector {right arrow over (w)} is obtained by minimizing the loss function, {right arrow over (w)}.sup.2+C .sub.t=1.sup.n(.sub.i+*.sub.i).sup.1, subject to the constraints y.sub.i({right arrow over (w)}.Math.{right arrow over (x)}.sub.l+b)+.sub.i, or y.sub.i({right arrow over (w)}.Math.{right arrow over (x.sub.l)}+b)*.sub.i, for .sub.i, *.sub.i0. The solution is given by y.sub.i=.sub.i=1.sup.n(a.sub.ia*.sub.i)({right arrow over (w)}.Math.{right arrow over (x.sub.l)})+b, where a.sub.i and a*.sub.i are Lagrange multipliers. The training vectors giving nonzero Lagrange multipliers are called support vectors and are used to construct the regression function. If the input data are not amenable to linear regression, then the vector data are mapped into a higher dimensional features space using a kernel function . One example is the polynomial kernel, according to which ({right arrow over (w)}).Math.({right arrow over (x.sub.l)})=(1+{right arrow over (w)}.Math.{right arrow over (x.sub.l)}).sup.3.

(34) The forecasting models are next evaluated, using an evaluation measure such as cvRMSE. Forecasting evaluation results relative to each prediction step are stored for each training data sequence (e.g., the 180 minutes used above), as shown in Table 3.

(35) TABLE-US-00003 TABLE 3 Forecasting Evaluation Results cvRMSE for cvRMSE for cvRMSE for cvRMSE for SVR model Persistence model SVR model Persistence model Period for Kt_p at 1 at 1 at n at n Prediction step ahead step ahead . . . steps ahead steps ahead 2014-12-01-06-57.fwdarw. 1.941569 3.522824 . . . . . . . . . 2014-12-01-09-56 2014-12-01-06-57.fwdarw. 4.299455 8.545602 . . . . . . . . . 2014-12-01-09-56 2014-12-01-06-57.fwdarw. 7.165063 14.192361 . . . . . . . . . 2014-12-01-09-56 2014-12-01-06-57.fwdarw. 4.637791 14.184008 . . . . . . . . . 2014-12-01-09-56 2014-12-01-06-57.fwdarw. 5.841919 5.199473 . . . . . . . . . 2014-12-01-09-56 2014-12-01-06-57.fwdarw. 25.469365 15.143792 . . . . . . . . . 2014-12-01-09-56 . . . . . . . . . . . . . . . . . . . . . . . . . . .

(36) The forecasting models are then saved as software components that take as input a time series data sequence to output predictions for m steps ahead. Next, classification models are developed to selected the best forecasting method. A time series data set is created that includes the evaluated forecasting models and their time series training data, as shown below in Table 4.

(37) TABLE-US-00004 TABLE 4 Time Series Data Created to Train Classification Models cvRMSE for cvRMSE for SVR model Persistence model Period for Kt_p (average over (average over Kt_p at Kt_p at Prediction all steps ahead) all steps ahead) minute 1 . . . minute 180 2014-12-01-06-57.fwdarw. 1.9416 3.5228 0.0801 . . . 0.2810 2014-12-01-09-56 2014-12-01-06-57.fwdarw. 4.2995 8.5456 0.0801 . . . 0.2810 2014-12-01-09-56 2014-12-01-06-57.fwdarw. 7.1651 14.1924 0.0801 . . . 0.2810 2014-12-01-09-56 2014-12-01-06-57.fwdarw. 4.6378 14.1840 0.0801 . . . 0.2810 2014-12-01-09-56 2014-12-01-06-57.fwdarw. 5.8419 5.1995 0.0801 . . . 0.2810 2014-12-01-09-56 2014-12-01-06-57.fwdarw. 25.4694 15.1438 0.0801 . . . 0.2810 2014-12-01-09-56 . . . . . . . . . . . . . . . . . .

(38) In step (f), the forecasting method with the lowest error rate is assigned as the forecasting class for the time series training data sequences. The best performing class is established as the model with the best evaluation results; e.g., a lower cvRMSE, as shown in Table 5.

(39) TABLE-US-00005 TABLE 5 Model Class Assignment to Training Records cvRMSE cvRMSE for for SVR Persistence model model (average (average Kt_p over all over all at steps steps Kt_p at minute Model Period for Kt_p Prediction ahead) ahead) minute 1 . . . 180 Class 2014-12-01-06-57.fwdarw. 1.9416 3.5228 0.0801 . . . 0.2810 SVR 2014-12-01-09-56 2014-12-01-06-57.fwdarw. 4.2995 8.5456 0.0801 . . . 0.2810 SVR 2014-12-01-09-56 2014-12-01-06-57.fwdarw. 7.1651 14.1924 0.0801 . . . 0.2810 SVR 2014-12-01-09-56 2014-12-01-06-57.fwdarw. 4.6378 14.1840 0.0801 . . . 0.2810 SVR 2014-12-01-09-56 2014-12-01-06-57.fwdarw. 5.8419 5.1995 0.0801 . . . 0.2810 PER 2014-12-01-09-56 2014-12-01-06-57.fwdarw. 25.4694 15.1438 0.0801 . . . 0.2810 PER 2014-12-01-09-56 . . . . . . . . . . . . . . . . . .

(40) Additional data filtering processes may be applied to the above data set. Data can be filtered using any suitable filtering techniques, such as discretization, normalization or wavelet transforms, or a combination of these. The data is then used to train a classification model capable of recognizing the highest ranking model given an input time series training data sequence. A number of machine learning algorithms can be used to train a classifier from data, such as, for example, the decision-tree classification algorithm. In decision-tree classification, the model identifies members of a class as the result of a sequence of decisions. A decision tree typically consists of two types of nodes: test nodes and prediction nodes. The test node describes the condition that must be met in order to make a decision. Several test nodes can occur in a sequence to indicate the number of decisions that must be taken and the order in which these decisions follow one another to reach a prediction outcome.

(41) A decision tree classifier is learned from a training dataset by a model which establishes the sequential order of test nodes according to how informative the nodes' attributes are. The model determines the information content of an attribute by its information gain with respect to the classification tasks. The information gain of an attribute with respect to a class is the reduction in entropy (i.e., the uncertainty) of the value for the class the value of the attribute is known. The test nodes with more informative attributes occur earlier in the decision tree. The model creates test nodes using the available attributes until all data in the training dataset have been accounted for. Typically, not all attributes are used because decision tree learners use pruning strategies to reduce the number of nodes. The number of attributes depends on the specific implementation. The alternating decision tree algorithm uses a machine learning meta-algorithm, called boosting, to minimize the number of nodes without losing accuracy. It should be understood that any suitable classification algorithm may be used, such as, for example, Bayesian nets, Support Vector Machines, boosting, Nave Bayes, bagging, random forest and Model Trees. The classification model is saved as a software component capable of recognizing the highest ranking model given an input time series training data sequence.

(42) It should be understood that the calculations may be performed by any suitable computer system, such as that diagrammatically shown in FIG. 1. Data is entered into system 10 via any suitable type of user interface 16, and may be stored in memory 12, which may be any suitable type of computer readable and programmable memory and is preferably a non-transitory, computer readable storage medium. Calculations are performed by processor 14, which may be any suitable type of computer processor and may be displayed to the user on display 18, which may be any suitable type of computer display. Sensor data is collected from solar irradiance sensors 20, such as, for example, satellite and sky cameras, radiometric sensors, pyrheliometers, pyranometers and the like.

(43) Processor 14 may be associated with, or incorporated into, any suitable type of computing device, for example, a personal computer or a programmable logic controller. The display 18, the processor 14, the memory 12 and any associated computer readable recording media are in communication with one another by any suitable type of data bus, as is well known in the art.

(44) Examples of computer-readable recording media include non-transitory storage media, a magnetic recording apparatus, an optical disk, a magneto-optical disk, and/or a semiconductor memory (for example, RAM, ROM, etc.). Examples of magnetic recording apparatus that may be used in addition to memory 112, or in place of memory 12, include a hard disk device (HDD), a flexible disk (FD), and a magnetic tape (MT). Examples of the optical disk include a DVD (Digital Versatile Disc), a DVD-RAM, a CD-ROM (Compact Disc-Read Only Memory), and a CD-R (Recordable)/RW. It should be understood that non-transitory computer-readable storage media include all computer-readable media, with the sole exception being a transitory, propagating signal.

(45) In the above method, the Persistence technique and the Support Vector Regression (SVR) technique may be used as the forecasting methods. As an alternative, the Persistence model may be replaced by autoregressive (AR) models, particularly for the prediction of solar radiation in the short term (i.e., now-casting) using ground measurements, such as radiometric measurements. This allows for temporal steps of predictions of 1, 5 and 10 minutes. The temporal horizon in all three cases is 15 temporal steps ahead. In the below, due to non-stationary behavior, global solar irradiance has been transformed to a normalized clearness index K*.sub.T. The clearness index is defined as the ratio between ground measured global solar irradiance and extraterrestrial solar irradiance, including a correction for air mass.

(46) Autoregressive models generate lineal predictions from the input and are also known as Infinite Impulse Response Filters (IIRFs). The notation AR(p) refers to the autoregressive model of order p. The AR(p) is expressed by the following equation:
{circumflex over (x)}.sub.t=c+.sub.i=1.sup.pa.sub.ix.sub.ti+.sub.t,
where a.sub.i are the parameters of the model, c is a constant which represents the mean value of the time series, and .sub.t is a white noise signal. As will be discussed in greater detail below, twenty different autoregressive models AR(p) have been tested, with order p=1, . . . , 20 for 1, 5 and 10 minutes average time series. Exemplary data are from the Qatar Environment and Energy Research Institute (QEERI), which has been operating a high precision solar radiation monitoring station since the end of November 2012 in Education City, Doha (25.33 N, 51.43 E). The station is equipped with a solar tracker with a sun sensor kit, for improved tracking accuracy, and a shading ball assembly for diffuse measurements. Mounted on the sun tracker are one first class pyrheliometer for measuring Direct Normal Irradiance (DNI), and two secondary standard pyranometers (one of them shaded) for Global Horizontal Irradiance (GHI) and Diffuse Horizontal Irradiance (DHI) measurements. Both pyranometers are fitted with ventilation units. Data from the monitoring station are sampled every second and recorded as minute averages in W/m.sup.2.

(47) FIGS. 2A and 2B show autocorrelation and partial autocorrelation of a one-minute time series used in the alternative embodiment of the method of forecasting for solar-based power systems, respectively. Similarly, FIGS. 3A and 3B show autocorrelation and partial autocorrelation of a five-minute time series, respectively, and FIGS. 4A and 4B show autocorrelation and partial autocorrelation of a ten-minute time series. The metrics to measure the errors rates of the models are the mean bias deviation (MBD), the relative root mean squared deviation (RMSD) and its relative values rMBD and rRMSD normalized with the mean value of the observed variable to predict for the period under validation.

(48) The present models are compared with a baseline model, with the intention of measuring the improvement achieved. The basic model chosen is the persistence model (PER) because it is the most extended model to contrast new proposed models. As described above, persistence is based on the assumption that the value for the next temporal step is the same as the present value: {circumflex over (x)}.sub.t+k=x.sub.t, where {circumflex over (x)}.sub.t+k is the prediction for the next k steps, and x.sub.t is the observation at the temporal instant t.

(49) FIG. 5 is a graph comparing relative root mean squared deviation (RMSD) of the autocorrelation (AR) model against the persistence (PER) model, and a combination autocorrelation model in the alternative embodiment of the method of forecasting for solar-based power systems, specifically for each horizon of prediction applied to one-minute average time series data (taken from January of 2014). The new AR combined model (AR_COMB_MIN) is detailed below. FIGS. 6 and 7 show similar comparisons for data sets recorded in April of 2014 and June of 2014, respectively.

(50) Analysis of this data shows that there is a dependence on the horizon of prediction and the complexity of the AR models which provide the minimum error. For each horizon n, the best results in terms of minimum rRMSD are obtained with an AR model of order (n+1)[AR (n+1)]. This is the AR combined model (AR_COMB_MIN) noted above. The same behavior is also observed in the 5 and 10 minutes time series. Table 6 shows MBD, RMSD, rMBD and rRMSD results for the AR, PER and AR-combined models with the one-minute time series from 2014.

(51) TABLE-US-00006 TABLE 6 MBD, RMSD, rMBD and rRMSD Results for AR, PER and AR-combined Models with One-minute Time Series Model MBD RMSD rMBD(%) rRMSD(%) AR(1) 0.01 0.06 0.84 9.44 AR(2) 0.01 0.06 0.81 8.76 AR(3) 0.00 0.06 0.59 8.26 AR(4) 0.00 0.05 0.50 7.91 AR(5) 0.00 0.05 0.43 7.63 AR(6) 0.00 0.05 0.39 7.40 AR(7) 0.00 0.05 0.35 7.24 AR(8) 0.00 0.05 0.31 7.12 AR(9) 0.00 0.05 0.28 7.01 AR(10) 0.00 0.05 0.26 6.94 AR(11) 0.00 0.05 0.24 6.90 AR(12) 0.00 0.05 0.23 6.91 AR(13) 0.00 0.05 0.21 6.99 AR(14) 0.00 0.05 0.21 7.07 AR(15) 0.00 0.05 0.20 7.24 AR(16) 0.00 0.05 0.18 7.51 AR(17) 0.00 0.05 0.18 7.78 AR(18) 0.00 0.05 0.17 8.04 AR(19) 0.00 0.06 0.17 8.09 AR(20) 0.00 0.05 0.17 8.04 PER 0.00 0.06 0.13 9.42 AR_COMB_MIN 0.00 0.02 0.26 3.30

(52) FIG. 8 shows model rRMSD results for the one-minute time series for each temporal horizon of prediction. The above data shows that the AR-combined model is better than any other model, including the PER baseline model. The difference is quite considerable during the entire year, except in June. The lower limit of monthly rRMSD is around 6%. FIG. 9 is a graph comparing relative root mean squared deviation (RMSD) of the autocorrelation (AR) model against the persistence (PER) model, and a combination autocorrelation model in the alternative embodiment of the method of forecasting for solar-based power systems, specifically for each horizon of prediction applied to the five-minute average time series data (taken from January of 2014). FIGS. 10 and 11 show similar comparisons for data sets recorded in June of 2014 and November of 2014, respectively.

(53) Table 7 shows MBD, RMSD, rMBD and rRMSD results for the AR, PER and AR-combined models with the five-minute time series from 2014.

(54) TABLE-US-00007 TABLE 7 MBD, RMSD, rMBD and rRMSD Results for AR, PER and AR-combined Models with Five-minute Time Series Model MBD RMSD rMBD(%) rRMSD(%) AR(1) 0.01 0.09 0.96 13.38 AR(2) 0.00 0.08 0.56 12.46 AR(3) 0.00 0.08 0.14 11.89 AR(4) 0.00 0.08 0.05 11.53 AR(5) 0.00 0.08 0.19 11.28 AR(6) 0.00 0.08 0.26 11.14 AR(7) 0.00 0.07 0.32 11.08 AR(8) 0.00 0.08 0.34 11.11 AR(9) 0.00 0.08 0.36 11.22 AR(10) 0.00 0.08 0.37 11.38 AR(11) 0.00 0.08 0.38 11.55 AR(12) 0.00 0.08 0.38 11.77 AR(13) 0.00 0.08 0.38 12.01 AR(14) 0.00 0.08 0.38 12.29 AR(15) 0.00 0.09 0.39 12.62 AR(16) 0.00 0.09 0.41 13.02 AR(17) 0.00 0.09 0.40 13.41 AR(18) 0.00 0.09 0.39 13.82 AR(19) 0.00 0.10 0.38 14.21 AR(20) 0.00 0.10 0.39 14.58 PER 0.01 0.09 1.30 13.40 AR_COMB_MIN 0.00 0.05 0.33 7.72

(55) FIG. 12 shows model rRMSD results for the five-minute time series for each temporal horizon of prediction. From the above data, one can conclude that the AR-combined model outperforms the other models and the PER baseline model for the case of a five-minute average time series. The difference is quite considerable during the entire year, except in June. The lower limit of monthly rRMSD is around 6%.

(56) FIG. 13 is a graph comparing relative root mean squared deviation (RMSD) of the autocorrelation (AR) model against the persistence (PER) model, and the combination autocorrelation model in the alternative embodiment of the method of forecasting for solar-based power systems, specifically for each horizon of prediction applied to the ten-minute average time series data (taken from February of 2014). FIGS. 14 and 15 show similar comparisons for data sets recorded in August of 2014 and December of 2014, respectively.

(57) Table 8 shows MBD, RMSD, rMBD and rRMSD results for the AR, PER and AR-combined models with the ten-minute time series from 2014.

(58) TABLE-US-00008 TABLE 8 MBD, RMSD, rMBD and rRMSD Results for AR, PER and AR-combined Models with Ten-minute Time Series Model MBD RMSD rMBD(%) rRMSD(%) AR(1) 0.00 0.09 0.23 16.26 AR(2) 0.00 0.08 0.26 15.19 AR(3) 0.01 0.08 0.72 14.60 AR(4) 0.01 0.08 0.89 14.23 AR(5) 0.01 0.08 0.95 14.02 AR(6) 0.01 0.08 0.96 13.93 AR(7) 0.01 0.08 0.90 13.90 AR(8) 0.01 0.08 0.88 13.95 AR(9) 0.01 0.08 0.85 14.04 AR(10) 0.01 0.08 0.77 14.18 AR(11) 0.01 0.08 0.73 14.41 AR(12) 0.01 0.08 0.68 14.70 AR(13) 0.01 0.08 0.65 15.07 AR(14) 0.01 0.09 0.61 15.48 AR(15) 0.00 0.09 0.60 15.93 AR(16) 0.00 0.09 0.61 16.40 AR(17) 0.00 0.09 0.62 16.89 AR(18) 0.00 0.10 0.60 17.27 AR(19) 0.00 0.10 0.58 17.69 AR(20) 0.00 0.10 0.55 18.10 PER 0.02 0.10 2.62 16.39 AR_COMB_MIN 0.01 0.06 0.61 10.18

(59) FIG. 16 shows model rRMSD results for the ten-minute time series for each temporal horizon of prediction. From the above data, one can conclude that the AR-combined model is better than any AR model and the PER model, also for the case of 10 minute average time series. The difference is quite considerable during the entire year, except in June. This is due to the fact that June is a month where there are almost no clouds in Qatar and almost no impact in lower variability of solar radiation. The lower limit of monthly rRMSD is around 9%.

(60) It is to be understood that the present invention is not limited to the embodiments described above, but encompasses any and all embodiments within the scope of the following claims.