International Journal of Marine, Atmospheric & Earth Sciences, 2017, 5(1): 1-19 International Journal of Marine, Atmospheric & Earth Sciences

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1 Article International Journal of Marine, Atmospheric & Earth Sciences, 2017, 5(1): 1-19 International Journal of Marine, Atmospheric & Earth Sciences Journal homepage: ISSN: Florida, USA A Comparative Analysis of Empirical Models for the Estimation of Monthly Mean Daily Global Solar Radiation Using Different Climate Parameters in Sokoto, Nigeria Sulaiman M. Y. * and Umar A. B. Department of Physics, Kebbi State University of Science and Technology Aliero, Nigeria * Author to whom correspondence should be addressed; yushersule@gmail.com Article history: Received 19 November 2016, Received in revised form 27 January 2017, Accepted 27 January 2017, Published 31 January Abstract: Monthly average daily global solar radiation data are essential in the design and study of solar energy conversion devices. In this regard, different empirical models based on Angstrom-Prescott model were selected to estimate the monthly average daily global solar radiation (H), on a horizontal surface for Sokoto State using sunshine duration, relative humidity and temperature. The hourly solar radiation data measured at Sultan Abubakar III International Airport station during the period ( ) was used to calculate the monthly mean values of H using selected models. The selected models were compared on the basis of the statistical error tests such as the mean bias error MBE, the mean percentage error MPE, the root mean square error RMSE, Nash Sutcliffe equation NSE, correlation coefficient r coefficient of determination R 2 and the t-test t. Based on the statistical results a new linear model based on modified Angstrom model is recommended to estimate monthly average daily global solar radiation for Sokoto State areas and in elsewhere with similar climatic conditions where the radiation data is missing or unavailable. The present work will help to advance the state of knowledge of global solar radiation to the point where it has applications in the estimation of monthly average daily global solar radiation. Keywords: Measured global solar radiation, calculated global solar radiation, Clearness index, Sunshine duration, Sokoto, regression analysis.

2 2 1. Introduction Solar radiation is the primary source of the Earth s energy, providing about 99.97% of the heat and light energy required for chemo physical processes in the atmosphere, ocean, land and other water bodies. Solar radiation plays an important role as a renewable energy source as solar radiation measurements could be used to estimate potential power levels that can be generated from photovoltaic cells and also necessary for determining cooling loads for buildings (Gopinathan, et al., 1992). Solar radiation thus has many useful applications in architectural design, evapotranspiration estimates, agriculture, atmospheric, land, ocean, and hydrologic models (Felayi et al., 2011). The acquisition and the development of database on the long term solar radiation will facilitate the evaluation of solar energy potential as an input to the country s energy budget and other modeling applications mentioned earlier. The development of solar technology in Nigeria will minimize its over reliance on wood fuel consumption, estimated at 18 million tons per annum, especially in the rural communities. The immediate short term to the long term solutions to address the problems of inadequate infrastructure in the area of solar energy data acquisitions have been the application of empirical models that rely on the correlation between the parameters obtained from measured meteorological data and the estimated values of solar radiation (Muzathik et al., 2011). In this regard, several empirical formulae have been developed to predict the global solar radiation using various meteorological variables such as sunshine hours, cloud cover, relative humidity, maximum temperature, and water vapour pressure. However, the reliability and usability of these models depends largely on the strength of the correlation between the estimated and measured variables (Ahmad et al., 2011). Sunshine hours has continued to be one of the most widely measured and applied meteorological parameters. This is so because it plays an important role in the determination of global solar radiation data. It is also the parameter with the best correlation with global solar radiation, air temperature, relative humidity and other climatic factors. Accurate solar radiation resource data are necessary at various steps of the design, simulation, and performance evaluation of any project involving solar energy (Gueymard, 2008). Inadequate installations of materials such as solar water heater, solar catering stills, solar dryer and solar cooker which require solar radiation and sunshine hour duration in the research area is a major problem. Also, the use of low efficiency of solar cells and high cost of high intensity solar cells limits the use of solar radiation as an alternative energy source in the study area. Therefore, it is very important if not compulsory to validate several empirical models for the prediction of monthly average global solar radiation on horizontal surface and determine the monthly variation of solar energy to get better view of monitoring the performance of solar energy conversion system in the study area,

3 3 because the world s population continues to grow for several decades, energy demand is likely to increase even faster, the proportion of energy supplied by electricity will also grow at the same rate and the main sources of energy (fossil fuels: coal, petroleum and gas) that we use are believed to be running out. Moreover, these sources of energy can cause harm to our environment. The essence of this study is to compare the predictive efficiency of eight empirical models for the estimation of monthly mean daily global solar radiation using meteorological variables in Sokoto so that the the result obtained could be considered as a reference for any solar energy utilization technology, which is now considered as a possible solution to the energy problems of the remote areas of Nigeria. Also, the results of the study can be used for future energy plan by government, entrepreneurs and NGO s. 2. Materials and Methods In this study, data of the monthly mean of daily global solar radiation, sunshine duration relative humidity, minimum and maximum temperatures from Nigerian Meteorological Agency, Sultan Abubakar III International Airport, Sokoto for Sokoto location were collected and utilized. The data obtained cover a period of eleven years ( ) for Sokoto (Latitude 13 o 3 39 N, Longitude 5 o E, Altitude 285m above sea level). The station is a standardized weather station in which climatic data are measured primarily for the purpose of aviation. The missing and invalid measurements in the database were replaced with the values derived from those of preceding and subsequent days by interpolation. Then the data were averaged to obtain the monthly mean daily values by taking the data for the average day of each month Data Analysis Techniques The first correlation proposed for estimating the monthly average daily global radiation is based on the model of Angstrom (1924). The original Angstrom-Prescott type regression equationrelated monthly average daily radiation to clear day radiation in a given location and average fraction of possible sunshine hours is given by the equation: (1) Where H is the monthly average daily global radiation on a horizontal surface (MJ/m² /day), Ho is the monthly average daily extraterrestrial radiation on a horizontal surface (MJ/m² /day), S the monthly average daily hours of bright sunshine, So is the monthly average day length, and a and b values are known as Angstrom constants and they are empirical. The monthly average daily extraterrestrial radiation on a horizontal surface (Ho) can be computed from the following equation (Iqbal, 1983; Zekai, 2008):

4 4 (2) Where Isc is the solar constant in (MJm -2 day -1 ) expressed as: (3) Eo is the eccentricity correlation factor, expressed as: (4) and δ are the latitude and solar declination angle of the site respectively. ws is the Mean Sunrise Hour Angle for the given month. where N is the characteristic day number for each month of the year starting from N = 1 on 1 st January to 365 on 31 st December. The solar declination (δ) and the mean sunrise hour angle (ws) can be calculated using equation (5) and (6) below (Iqbal, 1983; Zekai, 2008). Usually, the solar declination angle is calculated on the15 th day of each month. (5) (6) For a given month, the maximum possible sunshine duration (monthly average day length So) can be computed by using the following equation (Iqbal, 1983; Zekai, 2008): (7) Then, the monthly mean of daily global radiation H was normalized by dividing with monthly mean of daily extraterrestrial radiation Ho. We can define clearness index (KT) as the ratio of the observed/measured horizontal terrestrial solar radiation (H), to the calculated/predicted horizontal/extraterrestrial solar radiation (Ho). Clearness index (KT) gives the percentage deflection by the sky of the incoming global solar radiation and therefore indicates both level of availability of solar radiation and changes in atmospheric conditions in a given locality. (8) 2.2. Comparison Techniques There are numerous works in literature which deal with the assessment and comparison of monthly mean daily solar radiation estimation models. The most popular statistical parameters are the mean bias error (MBE) and the root mean square error (RMSE). In this study, to evaluate the accuracy of the estimated data, from the models described above, the following statistical tests were used, MBE, RMSE, mean percentage error (MPE) and coefficient of correlation (r), to test the linear relationship between predicted and measured values. For better data modeling, these statistics should be closer to zero, but coefficient of correlation, r, should approach to 1 as closely as possible. The Nash Sutcliffe

5 5 equation (NSE) is also selected as an evaluation criterion. A model is more efficient when NSE is closer to 1. However, these estimated errors provide reasonable criteria to compare models but do not objectively indicate whether a model s estimates are statistically significant. The t-statistic allows models to be compared and at the same time it indicates whether or not a model s estimate is statistically significant at a particular confidence level, so, t-test of the models was carried out to determine statistical significance of the predicted values by the models. i. The Mean Bias Error (9) This test provides information on long-term performance. A low MBE value is desired. A negative value gives the average amount of underestimation in the calculated value. So, one drawback of this mentioned tests is that overestimation of an individual observation will cancel underestimation in a separate observation. ii. The Root Mean Square Error (10) The value of RMSE is always positive, representing zero in the ideal case. The normalized root mean square error gives information on the short term performance of the correlations by allowing a term by term comparison of the actual deviation between the predicted and measured values. The smaller the value, the better is the model s performance. The demerit of this parameter is that a single value of high error leads to a higher value of RMSE. iii. The Mean Percentage Error (11) where, and n are, respectively, the i th measured values and i th calculated values of daily solar radiation and the number of values. A percentage error between 10% and +10% is considered acceptable. iv. The Nash-Sutcliffe Equation (12) where is the arithmetic mean of n measured values of global solar radiation. A model is more efficient when NSE is closer to 1 (Chen et al., 2004). v. The Coefficient of Correlation (Pearson) (13)

6 6 where is the arithmetic mean of n calculated values of global solar radiation. The coefficient of correlation, r can be used to determine the linear relationship between the measured and estimated values (Iqbal, 1983; Akpabio and Etuk, 2003). vi. t-statistic Test (14) The smaller the value of t the better is the performance. To determine whether a model s estimates are statistically significant, one simply has to determine, from standard statistical tables, the critical t value. For the model s estimates to be judged statistically significant at the calculated t value must be less than the critical value. (Iqbal, 1983; Akpabio and Etuk, 2003) Regression Analysis In regression analysis, the first and second order regressions of normal equations were employed to determine the values of a, b and c as: In regression analysis, the first and second order regressions of normal equations were employed to estimate the morning hours of global solar radiation using the following equations: (15) (16) where a, b and c are constants which will be determined, y is the same as H (dependent variable), while x were used to replace any of the meteorological data like relative humidity, temperature ( independent variables) e.t.c. To carry out the regression analysis of the first order, both sides of Equation 15 are multiplied by one and summed on both sides to yield Equation 17. Equation 1 also is multiplied by x and summed on both sides to yield Equation 18: (17) (18) The values of a and b in equations (17) and (18) can be determined using Crammers rule or using any methods of solving simultaneous equations. To obtain the second order regression equation, Equation 16 is multiplied by 1, x and x 2 successively and summed to obtain the following equations: (19) (20) (21)

7 7 The values of a, b and c are determined using Crammer s rule. In this research work, Ho and So were computed for each month by using Equations (2) and (7), respectively. The regression coefficient a and b for the estimated values of global solar radiation was obtained from equation (22) and (23) as given by (Tiwari, 1997). To compute the estimated values of the monthly average daily global radiation Hcal, the computed values of a and b were used in Equation (1) Proposed Models for the Study In this research work, eight models were selected based on the Angstrom-Prescott regression coefficients for the estimation of monthly mean of daily horizontal global solar radiation as summarized in table 1 below; Table 1: Proposed models for the study Model No. Regression Equation Model Type Source (22) (23) 1. Linear (single parameter) Angstrom, (1924) and Prescott (1940) 2. Logarithmic Ampratwum and Dorvlo (1999) 3. Exponential Almorox et al., (2005) where: 4. Linear (single Agbo et al., (2013) parameter) 5. Linear (double Garcia, (1994) parameter) 6. Linear (single parameter) Okonkwo, and Nwokoye, (2014) 7 Quadratic Okonkwo, and Nwokoye, (2014) 8 Quadratic Akinoglu and Ecevit (1990) is fraction of sunshine hours, R is relative humidity, is the ratio of change in temperature to day length ( ) and θ is temperature ratio.

8 8 3. Results and Discussion The accuracy of different models was determined using the data obtained from Sultan Abubakar III International Airport Sokoto. The values of monthly mean daily global solar radiation intensity estimated using 8 models. Models 1 to 8 were compared with the corresponding measured values. The statistical tests of MBE, MPE, RMSE, NSE, r and t-test were determined for the entire period. The regression coefficient for the different models were also determined. The results was summarized in tables below: Table 2: Mean value of input climate parameters of Sokoto for ( ) Months Hm Ho S So H/Ho S/So R Jan Feb Mar Apr May Jun Jul Aug Sept Oct Nov Dec Table 3: The Regression Constants of different models for Sokoto in the period of Model No. Model Type a b C

9 Global solar radiation (MJm -2 day -1 ) 9 Table 4: Comparison between monthly mean daily measured and predicted values (1-8) of global solar radiation for Sokoto for the period of eleven years ( ). Months Hm H1 H2 H3 H4 H5 H6 H7 H8 Jan Feb Mar Apr May Jun Jul Aug Sept Oct Nov Dec Table 5: Validation of the models under different statistical tests No. Model Type r R 2 t MBE RMSE MPE (%) NSE 1. Linear Logarithmic Exponential Linear Linear Linear Quadratic Quadratic Hm H1 5 0 J A N. F E B. M A R. A P R. M A Y J U N. J U L. A U G. S E P T. O C T. NOV. D E C. months Fig. 1: Comparison between the measured and model 1 values of solar radiation

10 10 25 Global Solar Radiation (MJm -2 day -1 ) Hm H2 0 J A N. F E B. M A R. A P R. M A Y J U N. J U L. A U G. S E P T. O C T. NOV. D E C. Months Fig. 2: Comparison between the measured and model 2 values of solar radiation Fig. 3: Comparison between the measured and model 3 values of solar radiation

11 11 Fig. 4: Comparison between the measured and model 4 values of solar radiation Fig. 5: Comparison between the measured and model 5 values of solar radiation

12 12 Fig. 6: Comparison between the measured and model 6 values of solar radiation Fig. 7: Comparison between the measured and model 7 values of solar radiation

13 13 Fig. 8: Comparison between the measured and model 8 values of solar radiation Fig. 9: Comparison between the measured and modeled values (1-8) of solar radiation In table 2, it is observed that the relative duration (S/So) is above 50% except in the months of August with 47.69%. The values of measured solar radiation (Hm = 12.67) corresponding to the lowest value of relative duration (S/So = 47.69%) in the month of August is an indication of poor condition as shown in table 2. Also, from table 2 it was observed that the highest values of relative humidity and

14 14 ratio of minimum to maximum temperature occurred in the month of August (79.27% and 73.68%) respectively which is as a result of a poor condition. Table 3 displayed the regression constants of different models used in the study based on sunshine hours, relative humidity, maximum and minimum temperatures. The values of regression constants for first order and second order regression equations was determined using equations (17-21). The measured and estimated values of the monthly average daily global radiation were also compared in table 4 for Sokoto in the period of eleven years ( ). According to the statistical test results from table 5, it can be seen that the estimated values of monthly mean daily global solar radiation are in favourable agreement with the measured values of monthly mean daily global solar radiation for all the models except model 7, whereas the mean percentage error for the model 7 exceeds ±10% for the study area. However, it was found that all models except model 7 shows good results. This is due to the fact that all the models the lowest values of MBE, MPE, RMSE and t-test and highest values of NES and r compared to the model 7. It was found that, the mean percentage errors, MPE, of the models is in the range of acceptable values between 0.345% and % with lower RMSE values that range from to Also, it is evident from table 5 that, the MBE values of all models except model 7 are equal to zero or very close to zero while the values of t-test range from to (tcritical = at 5% confidence level) with the exception of model 7 which has t value of The comparison between the different models according to the t value shows that the calculated t values were less than the critical t value other than model 7, this results show that all the models except model 7 have statistical significance. Also, as seen in table 5, according to the statistical test of the correlation coefficient (r), and coefficient of determination (R 2 ), all the models achieved good results (above 0.83) for the studied area. This means that the models obtained are reasonably compatible with the measured data. Furthermore, as seen from table 5, model 5 has the highest value of NSE value (0.8434), model 1, 2, 3, 6 and 8 has NSE values closest to 0.7 ranging from to while model 7 has the lowest NSE value of Higher values of NSE are considered excellent indicators, therefore, model 5 gives precise estimation for monthly mean daily global solar radiation in Sokoto. Therefore, it was concluded that model 5 which is linear double parameters (Change in temperature and day length) was recommended for use to estimate monthly mean daily global solar radiation in Sokoto. On the other hand, based on statistical test results from table 5, MPE, t-test and NSE, model 7 is not recommended to be used to estimate monthly mean daily global solar radiation of Sokoto. Figures 1 to 8 show the comparison between the measured and predicted values of global solar radiation. It is seen in Fig. 1, 2, 3 and 8 for model 1, 2, 3 and 8 respectively, underestimated the global

15 15 solar radiation in the months of January to June and overestimated in the months of July to November, while in the month of December the models are in good agreement with the measured values. It is observed in Fig. 5 that model 5 indicates very short and little underestimation in the months of June, September and October. It also indicated very little overestimation in the months of March and August. While the other months of the year are in close agreement with the measured values of global solar radiation. A high level of underestimation was observed throughout the months of the year in Fig. 7 which signifies that model seven is not suitable for the prediction of global solar radiation in Sokoto. It is also evident that from Fig. 4 model 4 indicates overestimation from the month of March to August and indicates underestimation from the months September to December. Fortunately in the month of January, the model s estimation is very close to the measured value of solar radiation. Fig. 6 shows overestimation in the months of January, July and August and shows underestimation in the months of April, May, June, October and November. The model s estimation in the months of February, March and December is very close to the measured values of global solar radiation. The measured and estimated values of the monthly average daily global radiation using models 1 to 8 for Sokoto is illustrated in Figure 9. As can be seen from Fig. 9, agreement between the values obtained from models (1 to 8) and the measured data were good for all the months of the year. It is clear that the deviation between the measured and estimated values using model 7 in Fig. 9 is very larger than that of others. Therefore, it has been concluded that models 7 was not recommended for use to estimate monthly mean daily global solar radiation in Sokoto. 4. Conclusions The objective of this project work is to evaluate various models for the estimation of the monthly average daily global radiation on a horizontal surface from bright sunshine hours, relative humidity and temperatures ratio and also to select the most appropriate model for Sokoto. All available empirical models that can be used to estimate monthly average daily global solar radiation over Sokoto state in Nigeria have been collected from literatures to evaluate the applicability of these models. The collected models were compared on the basis of the statistical error tests such as mean bias error (MBE), the mean percentage error (MPE), root mean square error (RMSE), Nash Sutcliffe equation (NSE), correlation coefficient (r), coefficient of determination (R 2 ) and t-test. According to the results, all models showed good estimation of the monthly average daily global solar radiation on a horizontal surface for Sokoto except model 7 which indicates high underestimation of the measured values.

16 16 Therefore, based on the statistical results a new simple linear model based on modified Angstrom model is recommended to estimate monthly average daily global solar radiation for Sokoto State and in other areas with similar climatic conditions where the radiation data is missing or unavailable. The present work will help to advance the state of knowledge of global solar radiation to the point where it has applications. References Agbo, G.A., Alfa, B., Ibeh G.F., and Adamu I.S. (2013), Application of regression and multiple correlation analysis to morning hours solar radiation in Lapai, International Journals of Physical Sciences. 8(27): Agbo, G.A., Ibeh G.F., Ekpe, J.E. and Isikwue, B.C. (2012), Estimation of global solar radiation at Calabar using two models, Journal of Natural Sciences Research, 2(5): Akinoglu, B.G., and Ecevit, A. (1990), Construction of quadratic model using modified Angstrom coefficients to estimate global solar radiation, Journal of Solar energy Resources, 45: Ahmad F. and Ulfat I. (2004), Empirical Models for the Correlation of Monthly Average Daily Global Solar Radiation with Hours of Sunshine on a Horizontal Surface at Karachi, Pakistan, Turkish Journal of Physics. 28: Akpabio, L.E., and Etuk, S.E. (2003), Relationship between Global Solar Radiation and Sunshine duration for Onne, Nigeria, Turkish Journal of Physics, 27: Akpootu, D.O and Gana, N.N (2013), Estimation of global solar radiation using four sunshine based models in Kebbi, North-Western Nigeria, Nigerian Association of Mathematical Physics, 24 th Annual conference, Pelagia Research Library, 4(5): Almorox, J. and Hontoria, C. (2004), Global solar radiation estimation using sunshine duration in Spain, Journal of Energy conservation and management. 45(9-10): Almorox, J., Benito, M. and Hontoria, C. (2008), Estimation of global solar radiation in Venezuela, Communications reports, Journal of Energy conservation and management, 33: 4. Angström, A. (1924), Solar and terrestrial radiation. Quarterly Journal of Royal Meteorological Society, 50: Ampratwum D.B., Dorvlo A.S.S. (1999), Estimation of solar radiation from the number of sunshine hours, Journal of Applied Energy, 63: Bakirci, K. (2009), Correlations for estimation of daily global solar radiation with hours of bright sunshine in Turkey, Journal of Renewable Energy, 34: Chen R., Ersi K., Yang, J. Lu S. and Zhao W. (2004), Validation of five global radiation models with measured daily data in China, Energy Conversion and Management, 45:

17 17 Falayi E.O., and Rabiu A.B (2005), Modelling global solar radiation using sunshine duration data, Niger. J. Phys. 17: Falayi, E.O., Rabiu, A.B. and Teliat, R.O. (2011), Correlation to estimate monthly mean of daily diffuse solar radiation in some selected cities in Nigeria. Pelagie Research Library, 2(4): Falayi, E.O., and Rabiu, A.B (2011), Estimation of global solar radiation using cloud cover and surface temperature in some selected cities in Nigeria, Arch. Phy. Res., 2(3): Falayi, E.O., Adepitan, J.O and Rabiu, A.B. (2008), Empirical Models for the Correlation of Global Solar Radiation with Meteorological data for Iseyin, Nigeria. International Journal of Physical Sciences, 3(9): Gueymard, C. A. (2000), Prediction and performance assessment of mean hourly global radiation. Journal of Atmospheric and Solar Terrestrial Physics, 67: Iqbal, M. (1983), An Introduction to solar radiation. Academic press. New York Iqbal M. (1978), Estimating of the monthly average of the diffuse component of total insolation on a horizontal surface. Solar Energy, 20(1): Mahmoud, M. and Ibrik, I. (2003), Field experience on solar electric power systems and their potential in Palestine. Renewable and Sustainable Energy Review, 7: Muzathik A.M., Nik W.B.W., Ibrahim M.Z., Samo K.B., Sopian K. and Alghoul M.A. (2011), Daily Global Solar Radiation Estimate Based On Sunshine Hours. International Journal of Mechanical and Materials Engineering (IJMME), 6(1): Okonkwo, G.N., and A.O.C Nwokoye (2014), Estimating global solar radiation from temperature data in Minna location, European Scientific Journal, 10(15): Prescott, J.A. (1940), Evaporation from a water Surface in relation to Solar Radiation, Tran. R. Soc. S. Australia. 64: Sanusi, Y.A (2004), Ranking of the performance of some climatological parameters in the estimation of solar radiation in the Minna Environment, Central Nigeria. Nigeria Journal of Renewable Energy, 12(1&2): Tiwari, G.N. and Suleja, S. (1997), Solar Thermal Engineering System, Narosa Publishing House, New Dehli, India, 24(6): Tsoho, U.H., (2008), Growth and History of the establishment of Makera Assada in Sokoto Metropolis. B.A Project, History Department, Usmanu Danfodiyo University Sokoto. Zekai S. (2008), Solar energy fundamentals and modeling techniques: atmosphere, environment, climate change and renewable energy, first ed. Springer, London.

18 18 APPENDICES Appendix A: Nomenclature and Symbols No. Symbol Nomenclature Unit 1. Monthly average daily global radiation MJ/m² /day 2. Monthly average daily extraterrestrial radiation MJ/m² /day 3. Monthly average daily hours of bright sunshine Hours 4. Monthly average day length Hours 5. Isc Solar constant - 6. Eo Eccentricity correlation factor - 7. Latitude of the study area degree ( 0 ) 8. δ Solar declination angle degree ( 0 ) 9. ws Mean Sunrise Hour Angle degree ( 0 ) 10. N Characteristic day number Days 11. KT Clearness index MBE Mean Bias Error RMSE Root Mean Square Error MPE Mean Percentage Error % 15. r Coefficient of Correlation R 2 Coefficient of determination NSE Nash Sutcliffe equation t t-test a, b & c Empirical Constants n Number of observations Change in Temperature 0 C 22. Temperature ratio R Relative humidity -

19 19 Appendix B: Computed input parameters for estimation of monthly daily global solar radiation Months So (hrs) Ho (MJ/m 2 /d) δ ( o ) Ws ( o ) N (days) January February March April May June July August September October November December