Modelling the Earth’s Solar Irradiance Across Some Selected Stations in Sub-Sahara Region of Africa

Solar radiation is an important parameter in the study of electricity and/or thermal system installation. Direct monthly irradiance data measurements of the earth’s horizontal surface irradiance for the year 1985 to 2019 for nine stations (Sokoto, Birnin Kebbi, Maiduguri, Ilorin, Calabar, Port-Harcourt, Enugu, Iwo, and Ikeja) were collected from the achieve of HelioClim satellite website. The stationarity of the series was determined using the time plot of the irradiance data between the periods under study but the formal test of stationarity was carried out using the Augmented Dickey-Fuller (ADF) test with a p-value less than 0.05 signifying stationarity. Different autoregressive integrated moving average (ARIMA) models were fitted to the irradiance data for each of the selected stations. The results revealed that Sokoto, Maiduguri, and Kebbi have their maximum solar irradiance at about 450 Wm and their minimum at about 340 Wm, while Ilorin has its maximum solar irradiance at about 440 Wm and its minimum solar irradiance at about 370 Wm. Ikeja, Iwo and Port-Harcourt, Calabar and Enugu have their maximum solar irradiance < 440 Wm and their minimum solar irradiance at about 390 Wm. It is therefore concluded that for Kebbi, Iwo and Maiduguri, the best model was ARIMA (3,0,3), for Calabar and Sokoto, it was ARIMA (2,0,2) while for other locations like Enugu, Ikeja, Ilorin and Port Harcourt, ARIMA (2,0,3) was the best model for forecasting irradiance in these study areas. The forecasted values of irradiance between January till December 2020 with its corresponding 95% confidence levels indicate good prediction of solar irradiance for future occurrence. doi: 10.5829/ijee.2020.11.03.05


A B S T R A C T
Solar radiation is an important parameter in the study of electricity and/or thermal system installation. Direct monthly irradiance data measurements of the earth's horizontal surface irradiance for the year 1985 to 2019 for nine stations (Sokoto, Birnin Kebbi, Maiduguri, Ilorin, Calabar, Port-Harcourt, Enugu, Iwo, and Ikeja) were collected from the achieve of HelioClim satellite website. The stationarity of the series was determined using the time plot of the irradiance data between the periods under study but the formal test of stationarity was carried out using the Augmented Dickey-Fuller (ADF) test with a p-value less than 0.05 signifying stationarity. Different autoregressive integrated moving average (ARIMA) models were fitted to the irradiance data for each of the selected stations. The results revealed that Sokoto, Maiduguri, and Kebbi have their maximum solar irradiance at about 450 −2 and their minimum at about 340 −2 , while Ilorin has its maximum solar irradiance at about 440 −2 and its minimum solar irradiance at about 370 −2 . Ikeja, Iwo and Port-Harcourt, Calabar and Enugu have their maximum solar irradiance < 440 −2 and their minimum solar irradiance at about 390 −2 . It is therefore concluded that for Kebbi, Iwo and Maiduguri, the best model was ARIMA (3,0,3), for Calabar and Sokoto, it was ARIMA (2,0,2) while for other locations like Enugu, Ikeja, Ilorin and Port Harcourt, ARIMA (2,0,3) was the best model for forecasting irradiance in these study areas. The forecasted values of irradiance between January till December 2020 with its corresponding 95% confidence levels indicate good prediction of solar irradiance for future occurrence.

INTRODUCTION 1
This is a known fact that solar radiation data are acquired for the designing and installation of solar equipment Babar and Bostrom [1]. Solar radiation information is widely used in agriculture, forestry and biological processes Badescu [2]. As reported by Babar and Bostrom [1], solar radiation data is not easily obtained even though it is of great importance. However, in this case, the data for solar irradiance were considered for statistical modelling. Babar and Bostrom [1] state that, owing to constraints all over the world, solar radiation equipment is not installed at any weather stations, but for the betterment of research, an internet data or satellite data can perform the same function with the installed equipment data. In Nigeria, meteorological stations lack solar radiation equipment for the collection of solar data; however, satellite data can help in the study of solar radiation. Globally, the percentage of weather stations recording the solar radiation is small, roughly 10% as compared to the stations recording other climatological *Corresponding Author E-mail: francisaweda4@gmail.com (F. O. Aweda) quantities like temperature, precipitation, humidity, rainfall etc. Babar and Bostrom [1]. As reported by Badescu [2], weather stations recording short wave solar radiation to station recording temperature is 1:500. Analytically, the modelling of solar irradiance using statistical package sometimes is difficult and required some basic information for the completion of the solar irradiance model.
Mohandes et al. [3], Trajekovic et al. [4], Reddy and Ranjan [5] have used artificial neural network (IA) technique to estimate the solar radiation. The more solar radiation, induces more evaporation of water molecules As reported by Aweda et al. [6]. Aweda et al. [7], demonstrated that seasonal effect on evaporation rate occurs as a result of the dry or wet temperature of the environment. The technical modelling approaches along with performance of the fabricated solar systems for photovoltaic solar power generation and visible solar irradiation were extensively discussed in literature [8][9][10][11][12]. This fact comes as a result of solar radiation reaching the surface of the earth. The research questions that are essential to be answered are (1) comparative analysis of the modelled solar irradiance and the actual values in the prediction of the solar irradiance (2) the significance of the forecasting of solar irradiance. The locations of study area are shown in Figure 1. In addition, geographical characteristics of all stations are summarized in Table 1.

Data collection
The monthly data of solar irradiance of the earth horizontal surface for the year 1985 to 2019 for nine stations (Sokoto, Birnin Kebbi, Maiduguri, Ilorin, Calabar, Port-Harcourt, Enugu, Iwo and Ikeja), located in Nigeria as shown above, were collected from the archive of HelioClim website of soda (http://www.sodapro.com). These stations are spread over Nigeria. The assessment of the data was done on the 1st of February, 2020 at about 8:00 pm local time of Nigeria. Based on literature [9] that is MERRA-2 Meteorological Re-Analysis. The parameter height is at 2m above sea level.

Data usage
The data used in this study were monthly irradiance data between 1985 and 2019. The stationarity of the series was determined using the time plot of the irradiance data between the periods under study but the formal test of stationarity was carried out using the Augmented Dickey-Fuller (ADF) test with a p-value less than 0.05 signifying stationarity. Different ARIMA models were fitted to the irradiance data for each of the selected locations namely: Kebbi, Calabar, Enugu, Ikeja, Ilorin, Iwo, Maiduguri, Port Harcourt and Sokoto. The performance of these time series models was compared using Root Mean Square Error (RMSE) and the model with the least RMSE was considered as the best forecasting model. The model diagnostic checking was carried out using the Ljung Box statistics and p-values greater than 0.05 indicates a good fit. The best forecasting ARIMA model among the proposed models for each of the location was used to forecast irradiance for January to December 2020.
The autoregressive integrated moving average model [ARIMA (p,d,q)] in terms of B operator can be expressed mathematically as follows: where, Also, = −1 and = −1 .

RESULTS AND DISCUSSIONS
The graphs of irradiance for each of the selected location were plotted and the figures obtained are shown in Figures 2-10. These graphs showed that the irradiance was stationary as there is no trend in the data. Result of stationary using ADF test is presented in Table 2 and the     indicates that irradiance series in all the selected locations were stationary. Therefore, different Autoregressive Moving Average (ARMA) models were fitted to the data for each of the location and the parameter estimates of these models were shown in Table 3. Result revealed that all the AR and MA terms in these models are significant. The significance of the Autoregressive (AR) terms in all the estimated models shows that the previous values of irradiance have significant effect on the present value of irradiance while the significance of the Moving Average (MA) terms is an indication that the past errors in irradiance also have a significant effect on the present value of irradiance. Result of model selection and model diagnostic checking is presented in Table 4. Result in Table 4 revealed that the p-values of the Ljung-Box statistic were all greater than 0.05 (p>0.05) meaning that all the models adequately predict the irradiance. This also implies that the models are of good fit to the irradiance data. The selection of the best model among the proposed models was based on the forecasting performance meaning that the model with the least value of Root Mean Square Error, is the best model with results as presented in Table 4 which shows that the best model for forecasting irradiance in each of the locations are as follows: In Kebbi , Iwo and Maiduguri, ARIMA (3,0,3) was the best model, for Calabar and Sokoto locations, it was ARIMA (2,0,2) while other locations like Enugu, Ikeja, Ilorin and Port Harcourt have ARIMA(2,0,3) as the best model for forecasting irradiance in their respective study areas. The forecasted values of irradiance between January 2020 and December 2020 with its corresponding 95% confidence levels are presented in Table 5.

CONCLUSION
The study modelled solar irradiance in nine locations in Nigeria using different forms of ARIMA models. The study made use of satellite data from the satellite application facility on MERRA-2 Meteorological Re-Analysis data of the HelioClim website. Findings revealed that in Kebbi, Iwo and Maiduguri, the best forecasting ARIMA model was ARIMA (3,0,3), for Calabar and Sokoto, it was ARIMA (2,0,2) while for other locations like Enugu, Ikeja, Ilorin and Port Harcourt, ARIMA (2,0,3) was the best model for forecasting irradiance in these study areas. However, it is therefore concluded that the forecasted values of irradiance between January 2020 and December 2020 with its corresponding 95% confidence levels indicate good prediction of solar irradiance for future occurrence.
More so, the implication shows that the northern part has high-temperature difference as compared to the southern part of the country. It is therefore recommended that adequate precautionary measures and policies be made to help mitigate the effects of high-temperature in the northern part of the country.