This paper proposes Grey System modeling (GM(1,1)), then GM(1,1) with Fourier series of error residual to nowcast the weather data. The GM(1,1) with Fourier series of error residual uses the result of the GM(1,1) to increase the prediction accuracy. For nowcasting, the weather data, the duration of a day is divided into 4 different time slots from dawn to dusk considered on different days. The proposed algorithms have been applied to the realtime Weather sensor data published on Kaggle.com ^{19}. This work considers the weather data gathered from a solar plant having Plant_Id 4135001 every 15 minutes. The data is collected from the solar plant installed near Gandikotta, Andhra, India. A single array of sensors placed optimally at the plant is used for gathering ambient temperature, module temperature, and Irradiance at the plant level.
The following details have been used from the solar plant data set.
Date & Time: This field has the Date and time for each observation. The observations are recorded at a time interval of 15 minutes.
Ambient Temperature: This field represents the ambient temperature at the solar plant location.
Module Temperature: This field represents the temperature reading from a solar panel.
Irradiation: This field represents the amount of irradiation for the time interval of 15 minutes.
The Ambient Temperature, Module Temperature, and Irradiation for the next time interval are predicted based on the past four observations.
Following are the steps involved in prediction.
Observe the Ambient Temperature, Module Temperature, and Irradiation values for fourtime slots and generate the Grey sequence for Ambient Temperature, Module Temperature, and Irradiation.
Formulate the grey models for Ambient Temperature, Module Temperature, and Irradiation
Predict the future values of Ambient Temperature, Module Temperature, and Irradiation employing the model of GM(1,1).
Modify the model of GM(1,1) using the Fourier series of error residuals to enhance the prediction accuracy.
The framework given in
The detailed description of Grey Prediction algorithms is as follows.
In this step, the inherent characteristic of the system is extracted by making use of the existing time series data of weather.
To generate the grey sequence from the given data, the following steps are involved.
Represent the actual weather sequence having n samples as
2. Smoothen the weather data with Accumulated Generating Operator (AGO) to obtain a sequence that is increasing monotonically.
The new sequence of weather data is expressed as
Devise GM(1,1) for weather data by applying a firstorder grey differential equation as:
a and b are found using the least mean square estimation method. Calculate
where
and
From Equation 3, the equation for prediction X^{(1)}(t) at time j is computed by applying the Inverse Accumulating Generation Operator (IAGO) as
To predict the weather data at a time (j+1), determine
To predict the primitive weather data at a time (j+H),
By Considering the actual sequence X^{(0)} and the sequence predicted by GM(1,1), find out the error sequence as:
The error residuals of Equation 8 are represented in the Fourier series as
Equation 9 could be written as
P and C are denoted as follows
By applying the Least Squares method, solve equation 10
The rectification for the Fourier series is computed as
To prove the precision of prediction of GM(1,1) and GM(1,1) with Fourier series of error residual, the results found from the proposed models have been compared with Auto Regression and Double Exponential time series forecasting algorithms.
This work utilizes the weather data gathered from a solar plant having Plant_Id 4135001 every 15 minutes. To prove the correctness of the proposed algorithms for predicting the weather data, irrespective of the time of the day, 4 different periods have been considered. Python is used for implementing the proposed methodology and to make comparisons with the existing models. The details of the data gathered are shown in
5/15/20 17:15 33.300255 35.852207 0.173899 5/15/20 17:30 33.032122 34.338400 0.087162 5/15/20 17:45 32.311871 31.548539 0.038403 5/15/20 18:00 31.132414 30.225331 0.022545 5/15/20 18:15 29.243111 28.682507 0.009150 5/16/20 14:45 32.350198 46.203597 0.514964 5/16/20 15:00 32.436880 45.770209 0.503626 5/16/20 15:15 32.524149 45.840538 0.419736 5/16/20 15:30 32.328025 43.170093 0.349962 5/16/20 15:45 32.103391 39.814439 0.261409 5/19/20 8:15 25.188797 30.107466 0.194092 5/19/20 8:30 25.615072 33.972492 0.297789 5/19/20 8:45 25.791661 34.815735 0.273515 5/19/20 9:00 26.121533 36.858085 0.386892 5/19/20 9:15 26.495522 39.636977 0.424118 5/20/20 7:45 23.921140 28.769306 0.325195 5/20/20 8:00 24.471149 31.939571 0.390585 5/20/20 8:15 25.124652 35.992171 0.437920 5/20/20 8:30 25.562463 38.252356 0.498368 5/20/20 8:45 26.212557 40.396549 0.510470
28.900515 0.085125 0.093385 28.734040 0.579937 0.624064 0.011749 0.003203 0.003441
29.958853 0.229067 0.323966 28.276088 0.274693 0.307097 0.023303 0.012030 0.012091
30.307670 0.080641 0.099466 28.102382 0.224817 0.272384 0.010096 0.002113 0.002482
29.985109 0.000000 0.000000 29.001651 0.000000 0.000000 0.018547 0.000000 0.000000
To show the effectiveness of the proposed algorithms, the algorithms were applied to three more data sets. Data Set2 includes weather data collected on 16052020, between 2.45 PM to 3.30 PM at 15 minutes time intervals as input and predicts the weather at 3.45 PM. Data Set3 uses the weather data measured on 15052020, between 5.15 PM to 6.00 PM at 15 minutes time intervals as input and predicts the weather at 6.15 PM. Data Set4 utilizes the weather data measured on 20052020, between 7.45 AM to 8.30 AM at 15 minutes time intervals as input and predicts the weather at 8.45 PM. The results of the prediction and corresponding error of prediction for Data Set2, Data Set3, and Data Set4 are shown in
















32.498125 
0.062894 
0.066710 
38.738158 
1.025793 
1.167822 
0.253465 
0.024427 
0.028305 

32.426239 
0.069801 
0.076999 
43.002688 
0.561525 
0.668501 
0.303062 
0.014754 
0.017652 

32.321133 
0.047246 
0.057866 
42.411428 
0.462263 
0.567065 
0.291398 
0.000659 
0.000805 

32.132151 
0.000000 
0.000000 
40.562378 
0.000000 
0.000000 
0.292460 
0.000000 
0.000000 
















26.291776 
0.076313 
0.039293 
37.126748 
0.596165 
0.674343 
0.332893 
0.043740 
0.048532 

26.299993 
0.083224 
0.044045 
38.247619 
0.423772 
0.510398 
0.420368 
0.026648 
0.029906 

26.353658 
0.025186 
0.030782 
38.212342 
0.191276 
0.232681 
0.426237 
0.022045 
0.026748 

26.454401 
0.000000 
0.000000 
38.977447 
0.000000 
0.000000 
0.514416 
0.000000 
0.000000 
















26.003278 
0.068617 
0.072869 
40.568467 
0.566864 
0.602743 
0.549479 
0.006693 
0.007129 

26.039219 
0.032520 
0.036284 
40.904388 
0.273143 
0.305622 
0.554076 
0.005257 
0.006313 

26.161331 
0.037671 
0.046459 
42.098806 
0.338640 
0.423381 
0.561193 
0.001257 
0.001455 

26.010646 
0.000000 
0.000000 
40.744246 
0.000000 
0.000000 
0.566219 
0.000000 
0.000000 
It is obvious from the results that for all the Data Sets used, variation between the actual values and the predicted values are very marginal for Ambient temperature, Module temperature, and Irradiation using the GM(1,1) & GM(1,1) with Fourier series of error residual. The MAE and RMSE values computed through the GM(1,1) & GM(1,1) with Fourier series of error residual prove that the variation between the actual and predicted weather data is very low when compared to Autoregression and Double Exponential Smoothing Models.
To further strengthen our claim that GM(1,1) with Fourier series of error residual outperforms in terms of accuracy of prediction compared to the Auto regression model and Double Exponential Smoothing model, the accuracy of prediction is computed for all four data sets using Mean Absolute Percentage Error (MAPE) of prediction and are shown in the following figures.
From
Hence, this study found that in comparison with the Auto regression model and Double Exponential Smoothing model, prediction accuracy is high in GM(1,1). However, the GM(1,1) with Fourier series of error residual outperforms the GM(1,1) in terms of accuracy of prediction.
Nowcasting in realtime demands simple and fast computation of results. Even though many methods are existing for predicting weather parameters that affect solar output, most of these algorithms involve a complex procedure and require a huge volume of data for accurate prediction. GM (1,1), as well as GM (1,1) with Fourier series of error residuals are simple and hence require only a minimal amount of time for Prediction. Another advantage of the proposed model is that the precision of prediction is very high even with fewer measures of input data. The proposed algorithms show more accurate prediction results when compared to conventional methods like Auto regression and Double Exponential Smoothing. The accuracy of prediction of weather data (on which the solar power output depends), confirms that the GM (1,1) with Fourier series of error residuals is the most suitable method for nowcasting. As the increasing penetration of PV systems into the present power systems poses new problems for the stability of electricity grids, there is a demand for energy management methods including precise PV production forecasting which in turn depends on prevailing weather conditions. Hence, the GM (1,1) with Fourier series of error residuals shall be used as a decision aid tool for power system operators, in load dispatch centers. The future work will focus on the applicability of other variations of the Grey model and consider the additional parameters impacting the Solar PV output.