Multivariate Regression for Prediction of Solar

Multivariate Regression for Prediction of Solar Irradiance Nalina U., Prema V, Smitha K, and Uma Rao K. Dept. of Electrical & Electronics Engineering RV College of Engineering Bangalore, India Abstract— This paper describes regression models to forecast solar irradiance for a short term (or period). The regression models enable the prediction of solar irradiance in minute values over a period of a few days. A single variate regression model is used and various plots obtained between solar irradiance as dependent variable and air temperature and relative humidity as independent variables have been studied. Optimal range for prediction using regression is decided. To obtain accuracy multivariate regression is carried out. It also presents new multifunctional relationship between solar irradiance, air temperature and relative humidity. This multifunctional regression relationship gives more accurate results compared to other methods having single variable. In this regression model solar irradiance follows an increasing trend upto a particular temperature after which it shows decreasing trend and hence it has been modeled with three equations. Index Terms-- Solar irradiance, regression, relative humidity. prediction

I.

INTRODUCTION

Climate changes combined with a perennial increase in energy demand has forced man to look for renewable energy sources. Encouraged by the benefits of green energy and the governmental financial incentives, PV generation systems established themselves as a reliable source in the past ten years and is a promising energy source for the generations to come. In 2009, China joined the lineage of countries like the US, Germany, Japan and others who have already set ambitious targets in increasing their PV market share. China, with a massive energy demand plans to tap its resourceful northwest to feed its mammoth projects in centralized PV generation in the years to come. [1]. Forecast / prediction involve analysis employing current experiences or knowledge to develop hypotheses on the forthcoming in future. For a precise analysis and prediction of solar energy utilization, it is mandatory to have knowledge of the amount of solar energy that strikes a pre-defined area in the location of study for a specified time period. This enables the researcher to plan the optimal utilization of this immense energy source gifted by the nature. In most systems, forecast or predictions of the future system condition or state are necessary to achieve optimal management

and control. Power predictions for photovoltaic installations are playing bigger role since world get awareness on advantages of solar energy as sustainable energy. Thus, development and research on it has been rising year by year. It is used to optimize usage of the solar energy and provide plausibly accurate knowledge of the solar resource availability at any place it is required [2]. The ability to estimate solar radiation using surrogate measures would provide better analysis of expected performance of technologies. The validation of the average meteorological data for the region will also have applications especially when sizing and deploying solar technology systems [3]. Renewable energy sources, such as hydro, solar, wind systems, geothermal and biomass along with other systems have established their position cater to the ever increasing energy demand and sustainability. This also comes with the disadvantages in the form of their uncertainty, intermittent and non- periodic behavior. This necessitates the integration of fossil- fuel based distributed generation viz. diesel generator systems, battery banks [4]. Solar irradiance prediction is gaining importance due to increasing solar power generation. Solar radiation is an important parameter in solar energy application due to generation from photovoltaic (PV) is directly related to this parameter. Solar radiation varies nonlinearly due to atmospheric events such as cloudy weather, rain, humidity etc. Therefore estimation of solar radiation is an attractive issue in solar energy field. A wide variety of models have been proposed for short term solar forecasting such as regressions in logs, Autoregressive Integrated Moving Average (ARIMA), Unobserved Components model and fuzzy model, Numerical weather prediction model. The parameters used for solar irradiance prediction are air temperature and relative humidity. Regression analysis provides the degree of closeness between variables. It employs several techniques to model and analyze variables. In multivariate regression a single dependent variable’s behavior is analyzed with respect to many independent variables. This model helps to explain or predict the unique significance of each independent variable on the independent variable. Regression analysis helps in causal

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forecasting. Once the relationship is determined, it is possible to build a statistical model and forecast the variables of interest. Forecasting using regression analysis is more reliable and powerful than the time series forecasting. In this paper linear regression analysis is done and relationship between solar irradiance - air temperature and solar irradiance-relative humidity is studied. Various plots have been obtained both including night time data and excluding night time data as well. Multivariate regression model is then implemented by splitting the model into two functions based on air temperature. The mathematical relationship between the variables is hence obtained. The paper is structured as follows. Section 2 describes how the data has been analysed and correlation between the variables has been found. The next section describes about single variate and multivariate regression analysis followed by a section where predicted output is shown. Finally, the paper ends with a conclusion. II.

ANALYSIS OF DATA

The data used for this paper is collected from the Bagalkot weather station for the months of February through June 2013. The data samples are minute wise data of each day in the given period. A portion of this data was used as training set and a portion of it used as test set. By correlation the variables - solar irradiance is 58% positively correlated with air temperature and 51%negatively correlated with relative humidity. Therefore these two variables are taken into consideration for the analysis. III.

ANALYSIS OF PARAMETERS

A. Solar Irradiance (SI) Solar Irradiance is amount of solar energy that strikes a predefined area in the location of study for a specific time period. One of the critical variables in solar irradiance is the angle of incidence. The incident solar energy gets distributed even though the total amount of radiation is dependent on he cross section. This, along with the angle of incidence makes the average incoming solar radiation to be about one fourth of the solar constant (approximately 342 W/m²). The other factors affecting solar irradiance are atmospheric condition and the latitude of the location of study. The earth’s surface receives solar irradiance in the forms of beam (Gb) and diffuse (Gd). The direct irradiance from the sun contributes to the beam component whereas the reflection from the atmosphere or clouds contributes to the diffuse component. The total irradiance that the surface receives can be represented as eqaution(1) G = Gb + Gd

(1)

B. Air Temperature (AT) Air temperature is the temperature of the air in the upper layers of atmosphere, which is one of the important climatic variables that affect solar irradiance. Various human activities have left a detrimental effect on the forecasting of air temperature due to the variance in the contributing factors at different locations.

C. Relative Humidity (RH) Absolute humidity is the ratio mass of water vapor to the mass of dry air in a volume of air at a specified temperature. The hotter air can contain more water. Relative humidity is the ratio of the current absolute humidity to the maximum absolute humidity for a specified air temperature. 100% relative humidity implies that the air is saturated with the maximum amount of water vapor. The relative humidity can be represented in terms of water vapor as well. The amount of water vapor required to saturate the air is usually higher than that present at any given point of time. Relative humidity can also be calculated as a function of the actual and saturated vapor density. Relative Humidity = IV.

A

V

S

D V

D

x 100%

CORRELATION TABLE

The percentage correlation indicates the amount of bearing of each parameter on the variation of the output parameter. Table 1: Values of percentage correlation of two input parameters to the solar irradiance for different ranges of air temperature. % Correlation Range of % Correlation of air of relative temperature temperature humidity less than 330C between 330C and 360C greater than 360C overall correlation

92.37754251

-75.04847559

1.423008093

-14.1232001

-12.61201609

3.2332345

30.33886753

-29.82316846

The correlation between variables can be obtained using the Correlation add-in in Excel. The values in Table 1 were obtained by filtering out the data points based on different temperature ranges, and getting the values of correlation in each group. The overall correlation indicates the % correlation between variables when all the data points, irrespective of the temperature range are considered. It can be observed that the % correlation values between variables greatly vary when the air temperature changes. V.

LINEAR REGRESSION ANALYSIS

Linear Regression models are standard ways to find a better-fit linear model for single or multi-variate scenarios. A multi-variate regression model was done using Microsoft Excel as a tool, to formulate a linear model with inputs as air temperature and relative humidity and output as solar irradiance [5]. A. Singlevariate Regression From the correlation table, it is evident that a functional relationship can be derived out of the data. Single variable analysis essentially involves regression using single input parameter and the desired output parameter is the solar irradiance. The level to which the plots reflect the functional relationship depends on the correlation factor between that input parameter and the solar irradiance.

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Solar Irradiance (W/m^2)

The following single variable analysis is carried out in Microsoft Excel [6] feature called Regression available under Data Analysis add on. In each of this single variable model the input and output parameter columns are fed in and the output is obtained as summarized table having amongst other details the intercept term and the coefficient of the input variable[7,8].

1000 800 600 400 200 0

1) Regression of Solar Irradiance with Air Temperature Here the input variable is Air Temperature while the output is Solar irradiance. ANOVA table obtained from above procedure provides the coefficients and intercepts. The following equation is formed after extracting the data:


1000 800 600 400 200 0

Y

0 Predicted Y

Y

50

Air temperature (deg C)

The input variable is Relative Humidity while the output is solar irradiance which is same as previous model. The equation is obtained as below SI = 761.7112 - 7.78055*RH

(3)

1000 800 600 400 200 0 0

50

100

In the above multivariate analysis it is found that solar irradiance is not following a single functional relationship throughout the range of the input variables because the line plots for the same input variable differs for single variate and multi variate regression models. Further analysis shows that modeling in terms of two or more parts can give a better fitting model. This phenomenon is profoundly noticeable in case of air temperature as input variable. Solar irradiance follows increasing trend below 33°C temperature and decreasing trend above 33°C. To find a better fit, the data for the regression model was split into two functions based on air temperature. Multivariate Regression models were applied to each of these segments to get a linear relationship between the variables. The following plots illustrate the same.

100

Relative humidity (%)

Predicted Y

Fig.2: Line fit plot for Solar Irradiance verses of Relative Humidity.

As it can be seen from the above plots, the single variate analysis does not effectively model the output. The goodness of fit with air temperature and relative humidity are just 12.70% and 15.54% respectively. Hence it is evident that multivariate regression model is to be done to get a better model B. Multivariate Regression Multi variate regression model with inputs variables as air temperature and relative humidity and solar irradiance as the output variable is modeled.[9-11] The linear equation of solar irradiance from the model is given in equation (4). Figures 3 and 4 show the line fit plots for the two variables. SI = 886.7626 – 2.85301*AT – 8.69428*RH

Predicted Y

50 Relative Humidity (%)

Fig.4: Line fit plot for Solar Irradiance verses Air Temperature (input variable 2)

2) Regression of Solar Irradiance with relative humidity

Solar irradiance (W/m^2)

1000 800 600 400 200 0 0

Fig.1: Line fit plot for Solar Irradiance verses of Air Temperature.

Y


(2)


SI = - (846.274) + 35.38621*AT

Y

0 50 Air Temperature (deg C) Predicted Y

1) Below 330C temperature The equation relating solar irradiance, air temperature and relative humidity is given by SI = -1411.03 + 68.386*AT – 1.46*RH (5) 2) Between 330C and 360C temperature For this temperature range, the equation relating solar irradiance, air temperature and relative humidity is given by SI = 1170.094 – 6.632*AT –5.341*RH (6) The following figures 7 and 8 show the line fit plots for this temperature range for the two input variables.

(4)


Fig.5: Line fit plot for Solar Irradiance verses Air Temperrature (input variable 1)


consolidated to form the complette multivariate regression model. The forecast of solar irradiance is carried out using this consolidated model.

Y

34

500

Y

Predicted Y

36

38

40

Air Temperaature (deg C)

42


0 0 -500 Predicted Y

50

100

Relative Humiditty (%)

Fig.6: Line fit plot for Solar Irradiance verses Air Tempeerature (input variable 2)



1000

1000 800 600 400 200 0

1000 800 600 400 200 0 0

Y

Predicted Y

20

Relative Humidity H (%)

40

Fig.10: Line fit plot for Solar Irradiance verses of Air Temperature (input

variable 2)

VI.

PREDICTION OF SOL LAR IRRADIANCE


Fig.7 Line fit plot for Solar Irradiance verses Air Tempeerature (input variable 1)

1000 800 600 400 200 0 0

20

40

60

Relative Humidity ((%)

Y

Predicted Y

Fig.8: Line fit plot for Solar Irradiance verses Air Tempeerature (input variable 2)

3) Above 360C temperature The equation relating solar irradiance, air temperature and relative humidity is given by (7) SI = 1943.887 – 37.757*AT – 2.337*RH From the above set of figures 5 throough 10, it can be seen that the goodness of fit of the regrression model is improved with dividing the entire range of vaalues based on air temperature and fitting a separate model for each section. For temperature ranges till 360C, the goodness of fit is more than 80%. Hence this divide-and-conquer appproach can be

Fig.11: Line plot of predicted Solar Irradian nce values against time for 4 days ahead (288 points)

Fig.12: Scatter plots of actual and prediccted values of Solar Irradiance for 4 days ahead

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The main aim of this paper, to forecast the solar irradiance, is carried out using the multivariate regression model defined by equations (5) through (7). The line plots for the predicted values and their comparison with the actual values of solar irradiance form the data is plotted in figures 11 and 12 respectively. In order to have the forecast, the values of the air temperature and relative humidity are to be obtained from the weather forecast for the required number of days. Then this data is used to compute the solar irradiance values according to the multivariate regression model. For this computation, equation (5) is used to obtain solar irradiance for temperatures below 330C and equation (6) and (7) for temperatures above 330C. It can be seen from the above figures that the prediction is more accurate when air temperature is less than 330C, that is, when the solar irradiance has an increasing trend. For greater temperatures, the predicted values are lesser in agreement with the actual values. Hence other methods such as smoothing methods and time series models can be tried for solar irradiance forecast with better accuracy.

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[5]

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[7]

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VII. CONCLUSION Using multi variable relationship about 85.57% of irradiance is determined in less than 330C air temperature range and 6.96% of irradiance is determined in more than 330C. From the equations (5), (6) and (7), it can be observed that the sign of the coefficients of the equations change for different ranges, indicating the change in functional relationship between the variables for different temperature ranges. With the help of these two equations the predicted values are more accurate and they are considerably equal to the actual values, as illustrated by the plots of solar irradiance forecast for 4 days ahead. t-test for this analysis testifies that predicted values are in good agreement with actual data at 5% level of significance with standard error of 2.242% . From the above findings, we can conclude that multi-functional relationship is suitable for the solar irradiance forecast. ACKNOWLEDGMENT The authors gratefully acknowledge the contributions of Dr. Suresh H Jangamshetti, Dept of EEE, Basaveshwar Engineering

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