Performance Study of Thirty-five empirical Models for the Estimation of Global Solar Irradiation in the Tropical Savannah Zone of Cameroon

Performance Study of Thirty-five empirical Models for the Estimation of Global Solar Irradiation in the Tropical Savannah Zone of Cameroon

Kodji Deli, Etienne Tchoffo Houdji, Noel Djongyang

Department of renewable energy, National Advanced School of Engineering,

University of Maroua, P.O.Box 46, Maroua, Cameroon

Ahmat Tom Mechanical engineering department, University Institute of Technology,

University of Ngaounderé, P.O. Box 455, Ngaoundere Cameroon

ABSTRACT

In the present study, thirty five empirical models for estimating monthly horizontal global solar radiation were compared and six new combined models (NM01 to NM06) were developed. Accuracy and applicability of these models were evaluated by using statistical parameters (MBE%, RMSE%, MPE, R2). Monthly meteorological data during more than 20 years were used for model calibration and the data from 1984 to 2015 were used to validate the models. Models have been implemented using MATLAB and Excel tools. This study shows that the model of Ertekin and Yaldiz 1999 (M20), Togrul and Onat 1999 (M28) and Ertekin and Yaldiz 1999 (M20) performed data better than other models for the city of Ngaoundere (MBE%= 0.00E+00; RMSE%=0.797; MPE=-0.00802;

R2=0.996). The six New developed models shown interesting value according to RMSE%, MBE% and R2. Indeed RMSE% range between 0.798-7.12, while MBE% and R2, range respectively between 0.00 to -6.52E-01 and 0.714 to 0.0996. among new developed models the new model 05 (NM05) performed data better for the city of Ngaoundere, these models can be used to evaluate solar radiation in locations with similar climate.

KEYWORDS: empirical models; global solar radiation; correlations; Cameroon; performance.

1. INTRODUCTION

   A precise knowledge of the data of the various components of solar radiation for a particular geographical position is crucial as it allows not only to optimize the design of solar energy conversion systems but also to evaluate their performance [1-3]. Reliable solar radiation data sets are essential for energy planners, engineers and agricultural scientists [4,5]. They are fundamental for the designing of the solar energy systems (solar cookers, solar water heaters,

   solar power). They are essential in agriculture as they allow better analysis of evapotranspiration phenomena and help to better assess the water needs of crops. Techno- economic feasibility of solar projects, thereby allowing the investors, government agencies and the utility operators are well made through a precise knowledge of solar data [6]. Unfortunately we are often confronted with data gaps related to the lack of data records of stations or the continuity of readings. Nowadays there are websites and software (meteoronorm, RETScreen, solargis, PVGIS, homer) that allows us to obtain data of solar resource of a given geographic area [7], but none of them are perfect. In developing countries, lack of financial resources does not allow to have enough measuring station facilities to achieve a precise knowledge of the solar resource [8]. Even in the developed countries there is a dearth of measured long-term solar radiation and daylight data [9] for instance the ratio of weather stations collecting solar radiation data relative to those collecting temperature data worldwide is approximately 1:500. It therefore becomes important to develop calculation procedures to provide radiation estimates for places where measurements are not carried out and for places where there are gaps in the measurement records by using empirical models that, from a number of input meteorological data, will assess how more or less precisely the amount of solar energy received at each point on the surface of a given location. In the open literature, many variables as: extraterrestrial radiation, relative humidity, number of rainy days, altitude, latitude, precipitation, albedo, cloudiness, and evaporation sunshine hours, mean temperature, minimum temperature, maximum temperature, soil temperature, [10-14]. These models appear as hybrid, exponential, logarithmic, power, quartic, quadratic, cubic, and linear forms [15].

   Earlier, the most used parameter to assess solar radiation is the sunshine duration. Using sunshine duration, the simplest model to estimate the average of the global radiation on a horizontal surface is the model of Angstrom 1924 [16], and their modified models known as Angstrom- Prescott-Page model establish by Prescott in 1940 [17]. Many researchers have found the value of the regression coefficients of angstrom model for different locations around the world and demonstrate that the relationship of Angstrom is valid within reasonable degree of accuracy [18-30]. However, for some regions of the globe, assessing accurate value of solar irradiation requires more magnitudes than the sunshine hours [8] thus the models of the solar radiation as function of sunshine data are not entirely valid in all regions. Depending on the available data, several models exist nowadays and are grouped into Sunshine-based models, Cloud-based models, Temperature-based models, Relative Humidity-based models, Precipitation-based models, Hybrid Parameter- based models [5, 29, 31-39], the most frequently used approach has been based on empirical relationships that require the development of a set of equations to estimate solar radiation from commonly measured meteorological variables. The number of such equations that have been published and tested is relatively high, these models have shown a good performance in literature for many sites around the world.

   Despite the amount of work done on the development of empirical correlation for determination of monthly averaged daily global solar radiation in locations around the world, no empirical correlation have been found for this region of Cameroon apart from angstrom-Prescott model, Hargreaves and Samani model, Annandale et al. Model, Bristow and Campbell model and Goodin et al. Model, [28,40,41]. Among different models encountered

   in open literature and depending on the available data, thirty-five (35) models were selected. The objective of this study is to evaluate these models for the Sudanese zone of Cameroon and to develop new models which can perform better solar radiation in this specific location. To achieve this, Matlab, Excel and Sigmaplot tools were used to determine on one hand the regression coefficient of the model and another hand the performance statistics named, mean bias error (MBE), mean percentage error (MPE), root mean square error (RMSE) and determination coefficient (2).

2. STUDY AREA AND WEATHER DATA

The study area is located between the latitude 6N and 8N and between longitude 11E and 16E and covers the administrative region of Adamawa cameroon. It shares its boundaries with Nigeria and the Central African Republic. The Adamawa region is mainly constituted of plateau which is around 1100 m altitude. In its southern part the region is surrounded by volcanic mountains reaching up to 2400 m. the climate is Sudano-Guinean and is under the influence of the African monsoon which brings rains between May and October and by the Harmattan winds coming from Sahara which brings dryness between November and April [42,43].

In this research, weather data are used and contain many parameters recorded daily through several years. These parameters are solar radiation, mean daily sunshine duration, mean daily temperature in ° C, Maximum daily temperature in ° C, Minimum daily temperature in ° C, mean soil Temperature, mean visibility. Mean total precipitable water (m) and Mean relative humidity. These data and their record period time are presented in table 1.

Table 1: Meteorological data recorded and their minimal record time

Parameters	period	Maximum Missing year	Minimum Record time (years)
Maximum daily temperature	1980-2013	13	21
Minimum daily temperature	1980-2013	13	21
Mean daily temperature	1980-2013	13	21
Mean Soil Temperature (ST)	1980-2013	13	21
Mean daily visibility	1980-2013	13	21
Mean relative humidity	1980-2013	13	21
Mean precipitation	1980-2013	13	21
Effective day length	1961-2015	21	33
Solar radiation	1984-2015	29	4

1. METHODOLOGY

   The work began with collection of detailed weather data in the study area. The regression analysis is employed to generate the regression coefficients of different suggested models depending on the meteorological parameters involved in each model selected. Among different models proposed in literature, thirty-five (35) are subject to our

   study. For each model, regression coefficients have to be known as well as percentage of MBE, RMSE, MPE and determination coefficient (R2). This is made possible by using Matlab, Excel and Sigmaplot tools. Using new meteorological parameter as visibility (V) new models have developed and are considered here as modified models.

   4.1- Studied Models

   The number of correlations published and tested to estimate global solar radiations is relatively high, which makes it difficult to select the best method for a particular site and purpose [3].

   Global solar radiation models are classified into four categories (sunshine based, cloud-based, temperature-based, and hybrid parameter-based models). The selection of these models usually takes into account two features: (1) the availability of meteorological and other kind of data used as input by the model and (2) the model accuracy. The models

   selected for the purpose are given into details in Table 2. Here () is the atmospheric precipitable water vapor per unit volume of air (cm) computed according to Leckner 1978 [44]. RH and are respectively the monthly daily mean humidity (in percentage) and air temperature (in Kelvin.).

   = 0.0049 [exp (26.23 5416)] (1)

   Table 2: Equation and types of variables for the 35 empirical models for the estimation of the monthly solar radiation

   N°	Models (equation type)	Authors	Mathematical relations
   M01	Angstrom-Prescott-Page (linear)	Angstrom 1924, Prescott 1940, Page 1961[16,17,45]	= + ( ) 0 0
   M02	Glower and McCulloh model (linear)	Glower and McCulloch 1958[19]	= + ( ) 0 0
   M03	model of Samuel (cubic)	Samuel 1991	= + ( ) + ( )2 + ( )3 0 0 0 0
   M04	Ampratwum and Dorvlo model (logarithmic)	Ampratwum and Dorvlo 1999[25]	= + ( ) 0 0
   M05	Dognimaux and Lemoine model (linear)	Dognimaux and Lemoine 1983[22]	= + [ ( ) + ] + ( ) 0 0 0
   M06	Newland model (logarithmic)	Newland 1989[46]	= + ( ) + ( ) 0 0 0
   M07	Elagib and Mansell model 1 (exponential)	Elagib and Mansell 2000[47]	= + ( ( )) 0 0
   M08	Elagib and Mansell Model 2 (hybrid)	Elagib and Mansell 2000[47]	= + ( ) 0 0
   M09	Elagib and Mansell Model 3 (hybrid)	Elagib and Mansell 2000[47]	= + + + ( ) 0 0
   M10	Elagib and Mansell Model 4 (hybrid)	Elagib and Mansell 2000[47]	= + + ( ) 0 0
   M11	Raja and Twidell model (linear)	Raja and Twidell 1990	= + + ( ) 0 0
   M12	Allen Model (power)	Allen 1997[48]	= ()0.5 0
   M13	Hargreaves model (hybrid)	Hargreaves 1985[49]	= + ()0.5 0
   M14	Bristow and Campbell Model (hybrid)	Bristow and Campbell 1984[50]	= [1 ()] 0
   M15	Chen et al. model 1(logarithmic)	Chen et al. 2004[60]	= + () 0
   M16	Chen et al. model 2 (linear)	Chen et al. 2004[51]	= + ( ) + + 0
   M17	Chen et al. model 3 (linear)	Chen et al. 2004[51]	= + + ( ) + + + 0 0
   M18	Chen et al. model 4 (linear)	Chen et al. 2004[51]	= + + ( ) + + + 0 0
   M19	Chen et al. model 5 (linear)	Chen et al. 2004[51]	= + + ( ) + + + + 0 0
   M20	Ertekin and Yaldiz Model (linear)	Ertekin and Yaldiz 1999[52]	= + + + + + + + 0 0
   M21	Ododo et al. Model (linear)	Ododo et al.1995[53]	= + ( ) + + + ( ) 0 0 0
   M22	El-Metwally Model (linear)	El-Metwally 2004[54]	= + 0 + + +
   M23	Togrul and Onat model 1 (linear)	Togrul and Onat 1999[55]	= + ( ) + + 0
   M24	Togrul and Onat model 2 (linear)	Togrul and Onat 1999[55]	= + + ( ) + + + 0 0
   M25	Togrul and Onat model 3 (linear)	Togrul and Onat 1999[55]	= + ( ) + + + 0
   M26	Togrul and Onat model 4 (linear)	Togrul and Onat 1999[55]	= + + ( ) + + 0 0
   M27	Togrul and Onat model 5 (linear)	Togrul and Onat 1999[55]	= + + ( ) + + + 0 0
   M28	Togrul and Onat model 6 (linear)	Togrul and Onat 1999[55]	= + + ( ) + + + + 0 0
   M29	Swartzman-Ogunlade 1 (power)	Swartzman and Ogunlade 1967[56]	= ( ) 0
   M30	Swartzman-Ogunlade 2 (linear)	Swartzman and Ogunlade 1967[56]	= + ( ) + 0 0
   M31	Garg and garg model 1(hybrid)	Garg and garg 1982[67]	= + ( ) + 0 0
   M32	Garg and garg model 2 (hybrid)	Garg and garg 1982[57]	= + + 0
   M33	De Jong and Stewart model (power)	De Jong and Stewart 1993[58]	= () (1 + + 2) 0
   M34	Hunt et al. Model (hybrid)	Hunt et al. 1998[59]	= + ()0.50 + + + 2
   M35	Coulibaly and Ouedraogo Model (linear)	Coulibaly and Ouedraogo 2016[60]	= + + + + + 0 0

   In table 2 above H is the monthly average daily global radiation, 0 the monthly average daily extraterrestrial radiation, S the day length, 0 the maximum possible sunshine duration. The extraterrestrial radiation 0 is given by:

   = 24 (1 + 0,033 360) ( + ) in Wh/m2 (2)

   0

   365

   180

   = is the solar constant (W/m2)

   =latitude (deg)

   = day of year 1 365

   is the declination (deg)

   = 23.45 [360 (284 + )] (3)

   365

   is the hour angle (deg)

   = 1() (4)

   = 2

   (5)

   0 15

   4.2- Evaluation parameters of the model performance

   All the different models presented above to assess the amount of solar energy reaching a given surface have to be validated. There are many statistical methods available in solar energy literature, which deal with the assessment and comparison of solar radiation estimation models [ 2-4, 47, 61-63]. In the present study statistical indicators, namely mean bias error (MBE), mean percentage error (MPE), root mean square error (RMSE) and determination coefficient (2 ) have been used. MBE helps to have an idea about the long-term performance of the model, a low MBE is desired. Ideally a zero value of MBE should be obtained. A positive value gives the average amount of over-estimation in the calculated value and vice versa. One drawback of this test is that over- estimation of an individual observation will cancel under-estimation in a separate observation [64, 65]. The RMSE provides information on the short-term performance of the correlations by allowing a term-by-term comparison of the deviation between the calculated and measured values, the RMSE is always positive, a zero value is ideal. However, a few large errors in the sum can produce a significant increase in RMSE [5,64]. The coefficient of determination 2 is used to determine how well the regression line approximates the real data points. A model is more efficient when 2 is closer to 1 [65]. These error parameters are defined as follows:

   The Mean Bias Error (MBE) in percentage is:

   (%) = ( (/2)

   1

   ) 100 ; With =

   =1(, ,) ; (6)

   The Root Mean Square Error (RMSE) in percentage is:

   (%) = ( (/2)

   1

   1/2

   2

   ) 100 ; With = (

   =1(, ,) )

   ; (7)

   The Coefficient of determination (2) is:

   2 = 1

   =1

   (,,)2 (,)2

   (8)

   =1

   The mean percentage error is:

   = 1

   ,,

   =1 (

   ,

   ) 100

   (9)

   The models used to compute solar irradiation provides good performance if the MBE and RMSE have as low values as possible. The following quantitative recommendations are sometimes used. For global irradiation, MBE within ±10% and RMSE less than 20% indicate good fitting between model results and measurements [33,66]. Here more stringent criteria for model performance can be adopted. A model to compute solar global irradiation provides good performance if the model is well calibrated with MBE within ± 5% and the scatter in the results is such that RMSE < 15 %. [33].

2. RESULTS AND DISCUSSION

   5.1- Performance statistics of Models

   To appreciate the performance and the accuracy of each model equations (1) to (9) have been used. Statistical analysis has been conducted (MBE,MPE, RMSE, R2, ) using measured data as validation data indeed a model is assumed as the best model when RMSE, MBE and MPE are near zero and R2 is close to one comparison of models is made considering in one side MBE(%) and in other side RMSE(%), MPE(%) and R2 as accuracy criteria. Hence, MBE,MPE, RMSE, R2, and their associated ranking are presented in the table 4 for site, this table contains systematic information on the accuracy of each model involved. This information allows the user to choose the best available estimating model for an application when considering available data and demands for accuracy. Thus, from these tables it is easily seen that the MBE (%), a measure of the overestimation (positive data) or underestimation (negative data) of the computed values with respect to the measured ones, lies between -0.652% and +0.094%. In the same way, the RMSE(%) estimator, which is a measure of the power contained in the estimated values in excess to that possessed by the real ones, lies between 0.796% to 7.121%. The determination coefficient (R2) lies between 0.797 to 0.996 for the site. For the MPE(%) which is the measure of the extent of the error of values in terms of percentage of the observed or measured value, the computed values lies between -0.00802 and 0.76649. Considering MBE(%) as accuracy criteria, the most accurate model is: Ertekin and Yaldiz 1999 (M20) (MBE= 0,00E+00%). Considering RMSE, MPE and R2, the most accurate model is : Ertekin and Yaldiz 1999 (M20) (RMSE=0,79677%, MPE= -0.00802, R2=0,99642).

   Table 3: Percentage root mean square error (RMSE), Mean bias error, Mean percentage error and determination coefficient with their associated ranking for global irradiance for the city of Ngaoundere (+is overestimation and is under estimation)

   Models	MBE(%)	Rank	RMSE(%)	Rank	R^2	Rank	MPE(%)	Rank	Statute	Number of Variables
   M01	2.60E-02	18	3.31943	23	0.93792	23	-0.13460	22	+	3
   M02	2.60E-02	19	3.31943	24	0.93792	24	-0.13460	23	+	4
   M03	4.60E-02	28	2.86001	16	0.95392	16	-0.09517	17	+	5
   M04	4.19E-02	26	2.98392	18	0.94984	18	-0.09929	18	+	3
   M05	-6.52E-01	35	3.38757	28	0.93535	28	0.53040	33	–	6
   M06	4.13E-02	24	2.98369	17	0.94985	17	-0.0994	19	+	4
   M07	2.37E-02	17	3.42383	29	0.93396	29	-0.14724	27	+	3
   M08	-1.29E-02	15	2.98436	19	0.94982	19	-0.05306	13	–	3
   M09	2.60E-02	20	3.31943	25	0.93792	25	-0.13460	24	+	5
   M10	2.60E-02	21	3.31943	26	0.93792	26	-0.13460	25	+	4
   M11	2.60E-02	22	3.31943	27	0.93792	27	-0.13460	26	+	4
   M12	-4.12E-01	34	5.28326	34	0.84274	34	0.76649	35	–	3
   M13	7.11E-02	30	3.97117	33	0.91115	33	-0.19643	31	+	3
   M14	9.40E-02	33	3.92811	32	0.91307	32	-0.19876	32	+	3
   M15	7.98E-02	31	3.92221	31	0.91333	31	-0.18737	30	+	3
   M16	-5.41E-15	4	1.24071	7	0.99133	7	-0.01300	6	–	4
   M17	-8.11E-15	9	1.13082	4	0.99280	4	-0.01164	5	–	6
   M18	-3.11E-14	14	1.03191	3	0.99400	3	-0.01164	3	–	6
   M19	-4.05E-15	3	1.02670	2	0.99406	2	-0.00850	2	–	7
   M20	0.00E+00	1	0.79677	1	0.99642	1	-0.00802	1	–	8
   M21	6.01E-02	29	2.44475	14	0.96633	14	-0.07006	16	+	6
   M22	5.41E-15	5	1.49512	8	0.98741	8	-0.03224	9	+	5
   M23	-6.76E-15	7	2.63538	15	0.96087	15	-0.06864	15	–	4
   M24	-6.76E-15	8	1.97014	12	0.97813	12	-0.04265	12	–	6
   M25	-1.35E-14	11	2.26247	13	0.97116	13	-0.06121	14	–	5
   M26	-1.22E-14	10	1.67897	11	0.98412	11	-0.03476	10	–	5
   M27	-5.41E-15	6	1.49898	9	0.98734	9	-0.03104	8	–	6
   M28	1.49E-14	13	1.13881	6	0.99269	6	-0.01604	7	+	7
   M29	4.17E-02	25	3.28640	22	0.93915	22	-0.18279	29	+	3
   M30	3.57E-02	23	3.26011	21	0.94012	21	-0.13342	21	+	4
   M31	1.66E-02	16	3.16308	20	0.94363	30	-0.11560	20	+	4
   M32	4.31E-02	27	7.12188	35	0.71424	35	-0.57972	34	+	4
   M33	8.75E-02	32	3.85647	30	0.91621	30	-0.17741	28	+	4
   M34	-1.35E-15	2	1.62270	10	0.98517	10	-0.03490	11	–	5
   M35	-1.35E-14	12	1.13082	5	0.99280	5	-0.01218	4	–	6

   Taking into account criteria of performance it is observed that most of the models provide good performance since -5% < MBE <

   +5% and RMSE < 15%. This shows that these models can be used to evaluate global solar irradiation in the sudanese zone of Cameroon. However, goodness of the model and the ranking are essential since they shows how precisely the data are. In table 3, it is observed that models are classified depending on their accuracy for this purpose some models are more accurate than others. It is also observed that models in which more detailed atmospheric information are involved perform data better than those with

   little or no such inputs. The main disturbing fact, however is the ranking disagreements between MBE (%) in one side and RMSE (%), MPE and R2 in order side. Thus two criteria for the evaluation of models accuracy are considered: the best models according to the MBE criterion (RMSE and MPE are fulfilled) and the best model according to RMSE, MPE and R2 criteria. However for the best models selection, criteria according to RMSE, MPE and R2 is more significant indeed, MBE which is the measure of the overestimation and underestimation have a major drawback in its use due to the fact that error effect related to overestimation by the model is cancelled by the model's underestimation that is why, it is characterized by unfair error cancellation. In an ideal scenario where MBE is zero, it implies that the developed model has an excellent long term performance, but also bearing in mind that MBE is not a good statistical tool for evaluating model performance in terms of error computation due to its intrinsic unfair error cancellation. This means that a model with a very small MBE does not really imply that it has a good performance in terms of its prediction. Advantages of the different models depend on the number of variables, on the equation type (Linear, cubic, logarithmic, hybrid, exponential, power), the simplicity and consequent operational efficiency, the facility to compute equations and their accuracy determined by MBE, RMSE, MPE and R2. Models can also be generalised since it can be used for another location elsewhere. Once models are known there is no need for ground solar radiation data. The main limitations of the methods are related to the need of meteorological data and calibration related to these data, the need for ground solar radiation data for validation and the lack of generality. It must be remembered that the same regression equation coefficients, determined for the locations corresponding to the ground solar radiation data, are also used to estimate the solar radiation reaching the ground throughout the region studied. Furthermore, there is no guarantee that they would have the same values in other areas. Limitation can be also related to the space and time since validation data used at different record time and space would not give the same correlations. Complexity of equations are also the main drawback of models. The summary of these result are presented in the table 4.

   Table 4: The two best models according to the MBE and RMSE criteria for each city.

   Cities	Rank	Best model according to MBE	Authors	Best model according to RMSE and R2	Authors
   Ngaounderé	1	M20	Ertekin and Yaldiz 1999 [44]	M20	Ertekin and Yaldiz 1999 [44]
   	2	M34	Hunt et al. in 1998 [51]	M19	Chen et al. 2004

   5.2- Regressions coefficients of Models

   In order to help new comer as well as experienced solar radiation developer, tester, or users, all regression coefficient for different models are presented in table 5.

   Table 5: Regression coefficients of the models for the city of Ngaounderé

   Models	a	b	c	d	e	f	g	h
   M01	0,36325	0,34102	X	X	X	X	X	X
   M02	0,36325	0,00000	0,34102	X	X	X	X	X
   M03	0,36740	0,07490	0,96006	-0,80709	X	X	X	X
   M04	0,67141	0,42293	X	X	X	X	X	X
   M05	-19,47000	-9,41800	2,72300	68,93000	X	X	X	X
   M06	0,65868	0,01418	0,40563	X	X	X	X	X
   M07	-0,62230	0,28770	X	X	X	X	X	X
   M08	-0,92860	1,60300	0,12390	X	X	X	X	X
   M09	0,36325	0,00000	0	0,34102	X	X	X	X
   M10	0,36325	0,00000	0,34102	X	X	X	X	X
   M11	0,36325	0,00000	0,34102	X	X	X	X	X
   M12	0,00000	0,15182	X	X	X	X	X	X
   M13	0,13670	0,11473	X	X	X	X	X	X
   M14	0,79300	0,18220	0,73810	X	X	X	X	X
   M15	0,01983	0,48039	X	X	X	X	X	X
   M16	-0,68757	2,38065	-0,02285	0,16796	X	X	X	X
   M17	-0,90044	0,08586	2,35685	-0,03207	0,14991	-0,00152	X	X
   M18	0,03748	-0,00285	2,19267	-0,00696	-0,10989	0,26142	X	X
   M19	-0,05736	0,00898	2,21654	-0,01148	-0,00639	-0,10122	0,25113	X
   M20	-0,41284	0,25716	0,01041	0,01162	1,11586	-0,11252	0,19089	-0,00409
   M21	0,01323	0,36747	0,01135	0,00063	-0,00223	X	X	X
   M22	-0,24652	-0,07215	0,28391	-0,11333	X	X	X	X
   M23	0,51315	3,73198	-0,05124	0,13402	X	X	X	X
   M24	-0,08373	0,21565	3,10788	-0,05819	0,10047	-0,00685	X	X
   M25	2,25624	2,88746	-0,03405	0,11481	-0,01245	X	X	X
   M26	-1,56887	0,18786	2,52194	0,14034	0,00288	X	X	X
   M27	-3,04847	0,20129	1,83482	0,01405	0,29285	-0,13588	X	X
   M28	-3,95879	0,24528	1,63848	-0,10161	-0,19379	0,34743	0,02045	X
   M29	12,38000	0,28120	-0,15140	X	X	X	X	X
   M30	0,41199	0,30973	-0,00046	X	X	X	X	X
   M31	0,27785	0,39561	0,01850	X	X	X	X	X
   M32	0,77136	-0,00045	-0,07373	X	X	X	X	X
   M33	0,32940	0,22060	-0,00026	-6,509E-07	X	X	X	X
   M34	2,60849	-0,01140	0,13761	-6,946E-03	6,840E-06	X	X	X
   M35	-0,90044	0,08586	2,35685	-1,522E-03	1,499E-01	-0,03207	X	X

   5.3- Prospected Models

   Another goal of this paper is to develop new models and prospect the more accurate model beyond a large number of developed solar radiation models for these reasons six new models were proposed using a call number (NM01 to NM06) to estimate daily global solar radiation. Mathematical equations

   of these models are developed by combining a new meteorological data named Visibility with different forms of other readily available meteorological data. These new models are similar to Ertekin and Yaldiz model [19], Togrul and Onat model [47] and Ododo et al. Model [45] and can be considered as modified models. Among these new prospected models the model (NM05):

   with equation : = + + + + + + + +

   + + ,and statistical parameter

   0 0

   (MBE(%)=5.27E-14, RMSE(%)=0.01540, R2=1.00), appear to be the best model among those prospected. Statistical parameters

   of the model and the associated ranking is presented in table 6.

   Table 6: New models prospects for better MBE, RMSE and R2 in the city of Ngaoundere

   News Models	Equations	MBE(%)	Rank	RMSE(%)	Rank	R2	Rank
   NM01	= + 0 + + + + + + + + 0	-1,22E-14	1	0,13889	5	0,99989	5
   NM02	= + 0 + + + + + + + + 0	4,05E-14	3	0,09373	4	0,99995	4
   NM03	= + 0 + + + + + + + + + 0	5,00E-14	4	0,03277	2	0,99999	2
   NM04	= + ( ) + + + ( ) + + + + 0 0 0	2,26E-03	6	0,54489	6	0,99833	6
   NM05	= + 0 + + + + + + + + + 0	5,27E-14	5	0,01540	1	1,00000	1
   NM06	= + 0 + + + + + + + + + 0	1,49E-14	2	0,05161	3	0,99998	3

   All models prospected perform better RMSE and R2 than 35 models studied above indeed, the smallest value of RMSE (%) is 0.15% (NM05 model) while maximum value is 0.545% for NM04 model, where R2 range from 0.998 (NM04 model) to 1.00 (NM05 model). Table 6, shows a more detailed information about the performance of the six developed models.

   5.4- Comparison of estimated Models with measured and NASA observed data, developed new models with measured data.

   Many other resources are commonly used to design PV solar power in the absence of measured data. These data sources are sometimes used in developing countries due to the lack of measured data. It is therefore important to compare in this paper, data computed from the best models to those obtained from Retscreen, Solargis, and PVgis software with measured data, in order to know how precise are those different models compare to measure one. These data sources are plotted in the figure 1.

   Figure 1: Solar irradiation with measured data, different best models and others resources data for the city of Ngaounderé.

   It is clear that these models predict the trend of the global solar radiation compared to the measured data since there is no visible differences between measured and predicted data. However, when comparing predicted data with others resources data like Retscreen, Solargis, and PVgis which are commonly used for designing solar systems, decision can be easily made on how projects are overdesigned or under designed through the use of any of these data Overestimation and underestimation are presented in table 7.

   Table 7: Comparison of Retscreen, PVGIS and SOLARGIS with measured and best predicted models (+ is over-estimation; – is under-estimation and is coincident value)

   	Ngaounderé city
   Month	Retscreen	pvgis	Solargis
   Jan	+	+	+
   Feb	+	+	+
   Ma	+	+	+
   Apr	+		+
   May	+		+
   Jun	+
   Jul	+
   Aug	+		+
   Sep	+		+
   Oct	+
   Nov	+
   Dec	+

3. Conclusion

The study aimed at comparison of empirical models developed and reported in the literature for the assessment of the monthly global Solar radiation data in tropical savannah (Aw) climate (according to Köppen-Geiger climate classification system) in Cameroon. This comparison is made possible using statistical evaluation of empirical models for predicting monthly mean global solar radiation. A total of thirty five (35) empirical models found in the literature are used in statistical analysis. New models have been developed to perform better solar radiation. In this regard, empirical correlations are developed to estimate the monthly average daily global radiation on a horizontal surface. The accuracy of the models were verified by comparing estimated values with measured values in terms of the following statistical error tests: mean bias error (MBE), root mean square error (RMSE), and the determination coefficient (R2). The values of the determination coefficient for the formulated models are between the ranges of 0.714 to 0.996, when RMSE and MBE range respectively between 0.796% to 7.122% ad 0.0% to -0.652%. For new developed model determination coefficient (R2) range between 0.998 to1.0, when RMSE and MBE range respectively between 0.0145% to 0.138% and -1,22E-14% to 2,26E-03%. It is also observed that for the accurate estimation of the global solar radiation more meteorological data are needed. The results shows that the models of Togrul and Onat 1999 (M28), Ertekin and Yaldiz 1999 (M20) performed data better than the other models. However, for the new models developed the models NM06, NM03 and NM05 are the best models. Results also shows that the formulated models are good enough to be used to predict monthly average daily radiation for tropical savannah zones of Cameroon.

Abbreviations and Nomenclature

= relative humidity in percentage

= precipitation in (mm)

= mean maximum temperature (°C)

= mean minimum temperature (°C)

= monthly daily mean air temperature ( K.)

= mean soil temperature (°C)

= monthly mean temperature (°C)

=visibility(Km)

= ( ) = the temperature difference (°C)

= Altitude (Km)

= the precipitable water vapor from the atmosphere ().

= cloudiness (cloud cover)

= Angstrom sunshine duration(h)

0 = extraterrestrial solar radiation (kWh/m2)

, = calculated solar radiation (kWh/m2)

, = measured solar radiation (kWh/m2)

= mean annual solar radiation (kWh/m2)

= root mean square error (kWh/m2)

= mean percentage error (kWh/m2)

= mean bias error (kWh/m2)

2 = determination coefficient

0 = day length (h)

= sunshine duration (h)

=is the solar constant (W/m2)

=latitude (deg)

= solar declination (°)

= day of year 1 365

is the hour angle (deg)

ACKNOWLEDGEMENTS

The authors of this manuscript are thankful to the Agency for the Safety of Air Navigation in Africa (ASECNA) for providing data which permit to carry out this article.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

REFERENCES

1. K. Bakirci, Models of solar radiation with hours of bright sunshine: A review, Renewable and Sustainable Energy Reviews. 13(2009) 2580 2588.

2. M.S. Adaramola, Estimating global solar radiation using common meteorological data in Akure, Nigeria, Renewable Energy. 47(2012) 38- 44.

3. F. Besharat, A.Dehghan, A.R. Faghih, Empirical models for estimating global solar radiation: A review and case study, Renewable and Sustainable Energy Reviews. 21 (2013) 798821.

4. C. Ertekin, O. Yaldiz, Comparison of some existing models for estimating global solar radiation for Antalya (Turkey), Energy Conversion & Management. 41 (2000) 311-330.

5. M.J. Ahmad, G.N. Tiwari, Solar radiation models-A review, International Journal Of Energy Research. 35 (2011) 271290.

6. T.R. Ayodele, A.S.O. Ogunjuyigbe, Performance assessment of empirical models for prediction of daily and monthly average global solar radiation: the case study of Ibadan, Nigeria. International Journal of Ambient Energy. 2016; DOI: 10.1080/01430750.2016.1222961

7. T.R. Huld, Müller, A. Gambardella, A new solar radiation database for estimating PV performance in Europe and Africa, Solar Energy. 86( 6)(2012) 1803-1815

8. Almorox J, Hontoria C. Global solar radiation estimation using sunshine duration in Spain. Energy Conversion and Management. 45(no. 9)(2004) 15291535.

9. V. Badescu, Modeling Solar Radiation at the Earths Surface, Springer- Verlag Berlin Heidelberg. 2008 ISBN: 978-3-540-77454-9.

10. D.L. Liu, B.J. Scott, Estimation of solar radiation in Australia from rainfall and temperature observations, Agriculture and Forest Meteorology.; 106(2001) 4159.

11. M.Trnka Z.Zalud, J. Eitzinger, M.Dubrovsky, Global solar radiation in Central European

    lowlands estimated by various empirical formulae, Agriculture and Forest Meteorology. 131(2005) 5476

12. H.O. Menges, C.Ertekin, M.H.Sonmete, Evaluation of global solar radiation models for Konya Turkey, Energy Conversion and Management.47 (2006) 31493173.

[13]C.Ertekin, F. Evrendilek, Spatio-temporal modeling of global solar radiation dynamics as a function of sunshine duration for Turkey, Agriculture and Forest Meteorology. 145 (2007), 3647.

[14]F.Evrendilek, C. Ertekin, Assessing solar radiation models using multiple variables over Turkey, Climate Dynamics,31(2008) 131149.

[15] M.H. Sonmete, C. Ertekin, H.O. Menges, H. Hacseferogullari, F. Evrendilek, Assessing monthly average solar radiation models: a comparative case study in Turkey, Environ Monit Assess; 175(2011) 251277.

[16]Angstrom, A Solar and terrestrial radiation, Quarterly Journal of Royal Meteorological Society, 50(1924) 121125.

[17]J.A. Prescott, Evaporation from water surface in relation to solar radiation, Transactions of the Royal Society of Australia. 46 (1940) 114 118.

[18] J.N. Black, C.W. Bonythont, J.A. Prescott, Solar radiation and duration of sunshine, QJ.R. meteor. Soc.; 80(1954) 231-235.

[19]J. Glower J.S.G. McGulloch, The empirical relation between solar radiation and hours of sunshine, Quart J R Met Soc.84(1958) 172-175

[20] J.K. Page, The estimation of monthly mean values of daily total short wave radiation on vertical and inclined surfaces from sunshine records for latitudes 40°N-40°S. In: Proceedings of UN Conference on New Sources of Energy. 598(1961) 378-390.

[21] M. Rietveld, A new method for estimating the regression coefficients in the formula relating solar radiation to sunshine, Agricultural Meteorology. 19(1978) 243252.

[22]R. Dogniaux, M. Lemoine, Classification of radiation sites in terms of different indices of atmospheric transparency, Solar energy research and development in the European Community Series F. 2(1983). Dordrecht: Reidel.

1. P.C. Jain, Global irradiation estimation for Italian locations, Solar and Wind Technology. 3( 4) (1986) 323328,

2. S. Jain, P.C. Jain, A comparison of the Angstrom-type correlations and the estimation of monthly average daily global irradiation, Solar Energy.

   ; 40(2)(1988) 9398.

3. D.B. Ampratwum, A.S.S. Dorvio, Estimation of solar radiation from the number of sunshine hours, Applied Energy. 63(1999) 161167.

4. M. El-Metwally, Simple new methods to estimate global solar radiation based on meteorological data in Egypt. Atmospheric Research. 69(2004) 217239.

5. K. Ulgen, A. Hepbasli, Solar Radiation Models. Part 2: Comparison and Developing New Models, Energy Sources. 26(5) (2004) 521-530.

[28]F. Ayangma, G.E. Nkeng, D.B.Bonoma, J. Nganhou, Evaluation du potentiel solaire au Cameroun: cas du Nord Cameroun, African Journal of Science and Technology. 9 (2) (2008) 32-40.

1. M.S. Okundamiya, J.O. Emagbetere, E.A. Ogujor, Evaluation of various global solar radiation models for Nigeria, International Journal of Green Energy; 13(5)(2016) 505-512.

2. C. Ertekin, O. Yaldiz, Comparison of some existing models for estimating global solar radiation for Antalya (Turkey), Energy Conversion & Management. 41(2000) 311-330.

3. A.A. El-Sebaii, F.S. Al-Hazmi, A.A. Al-Ghamdi, S.J. Yaghmour, Global, direct and diffuse solar radiation on horizontal and tilted surfaces in Jeddah, Saudi Arabia. Applied Energy. 87 (2010)568576

4. A.K. Katiyar, C.K. Pandey, Simple corelation for estimating the global solar radiation on horizontal surfaces in India, Energy. 35(2010) 5043- 5048

5. V. Badescu, C.A. Gueymard, C. Sorin, C. Oprea, Baciu, A. Dumitrescu, F. Iacobescu, I. Milos, C. Rada, Computing global and diffuse solar hourly irradiation on clear sky. Review and testing of 54 models, Renewable and Sustainable Energy Reviews. 16 (2012) 1636

   1656

6. E.Yilmaz, Estimation of Horizontal Solar Radiation in Bolu (Turkey), Energy Sources, Part A: Recovery, Utilization, and Environmental Effects.; 35(11)(2013) 1053-1055.

7. H.B. Tolabi, M.H. Moradi, S. Bin Md Ayob, A review on classification and comparison of different models in solar radiation estimation. Int. J. Energy Res. 38 (2014) 689701.

8. N.I. Sarkar Md, A.I. Sifat, Global solar radiation estimation from commonly available meteorological data for Bangladesh. Renewables. 3(6) (2016) DOI 10.1186/s40807-016-0027-3

9. Y.A. Kaplan, A New Model for Predicting the Global Solar Radiation. Environmental Progress & Sustainable Energy.00(00)( 2017). DOI 10.1002/ep.

10. S.C. Nwokolo, A comprehensive review of empirical models for estimating global solar radiation in Africa. Renewable and Sustainable Energy Reviews.78 (2017) 955995

11. M.H. Soulouknga, O. Coulibaly, D.S. Yamigno, T.C. Kofane, Evaluation of global solar radiation from meteorological data in the Sahelian zone of Chad, Renewables.; 4(4) (2017) DOI 10.1186/s40807- 017-0041-0

12. E. Mboumboue, D. Njomo, M.L. Ndiaye, P.A. N'diaye, M.F. Ndiaye,

    A.K. Tossa, On the applicability of several conventional regression models for the estimation of solar global radiation component in Cameroon and Senegal sub-Saharan tropical regions, Journal of Renewable and Sustainable Energy. 8(2016), 025906 ; doi: 10.1063/1.4947249

13. D.J. Afungchui, N.R. Ebobenow, N.A. Ngwa, Global Solar Radiation of some Regions of Cameroon using the Linear Angstrom Model and Non-linear Polynomial Relations: Part 2, Sun-path Diagrams, Energy Potential Predictions and Statistical Validation. International Journal of Renewable Energy Research. 8(1)( 2018)

14. E.F. Dassou, A. Ombolo, S. Chouto, G.E. Mboudou, J.M.A. Abate Essi,

    E. Bineli, Trends and Geostatistical Interpolation of Spatio-Temporal Variability of Precipitation in Northern Cameroon, American Journal of Climate Change.; 5(2016) 229-244.

15. Z. Aretouyap, P. Njandjock Nouck, D. Bisso, R. Nouayou, B. Lengue,

    A. Lepatio Tchieg. Climate Variability and Its Possible Interactions with Water Resources in Central Africa, Journal of Applied Sciences. 14(2014) 2219-2233.

16. B. Leckner, The spectral distribution of solar radiation at the earths

    surface-elements of a model. Solar Energy. 20 (1978) 143150.

17. J.K. Page, The estimation of monthly mean values of daily total short wave radiation on vertical and inclined surfaces from sunshine records for latitudes 40°N40°S. In Proceedings of UN conference on new sources of energy. (1961) 378390

18. F.J. Newland, A study of solar radiation models for the coastal region of

    South China, Solar Energy.. 31(1988) 227235.

19. N. Elagib, M.G. Mansell, New approaches for estimating global solar radiation across Sudan, Energy Conversion and Management.; 41(2000) 419434.

20. R. Allen, Self calibrating method for estimating solar radiation from air temperature. Journal of Hydrologic Engineering. 2(1997) 5667.

21. G.L. Hargreaves, G.H. Hargreaves, P. Riley, Irrigation water requirement for the Senegal River Basin. Journal of Irrigation and Drainage Engineering ASCE.111(1985) 265275.

22. K.L. Bristow, G.S. Campbell, On the relationship between incoming solar Radiation and daily maximum and minimum temperature.

    Agricultural and Forest Meteorology.31 (1984) 159-166

23. R. Chen, K. Ersi, J. Yang, S. Lu, W. Zhao, Validation of five global radiation models with measured daily data in China, Energy Conversion and Management. 45(2004) 17591769.

24. C. Ertekin, O. Yaldiz, Estimation of monthly average daily global radiation on horizontal surface for Antalya, Turkey, Renewable Energy. 17(1)(1999) 95-102.

25. J.C. Ododo, A.T. Sulaiman, J. Aidan, M.M.Yuguda, F.A. Ogbu, The importance of maximum air temperature in the parameterisation of solar radiation in Nigeria, Renewable Energy. 6 (1995) 751763.

26. M. El-Metwally. Simple new methods to estimate global solar radiation based on meteorological data in Egypt, Atmospheric Research. 69(2004) 217239.

27. I.T. Togrul, E. Onat, A study for estimating solar radiation in Elazig using geographical and meteorological data, Energy Conversion and Management; 40(1999) 15771584.

28. R.K. Swartman, O. Ogunlade, Solar radiation estimates from common parameters. Solar Energy; 11(1967) 170172.

29. H.P. Garg, S.T. Garg, Prediction of global solar radiation from bright sunshine hours and other meteorological Data, Energy conversion management.; 23(2)(1982) 113-118.

30. R. DeJong, D.W. Stewart, Estimating global solar radiation from common meteorological observations in western Canada, Canadian Journal of Plant Science.73(1993) 509-518.

31. L.A. Hunt, L. Kuchar, C.J. Swanton, Estimation of solar radiation for use in crop modelling, Agricultural And Forest Meteorology. 91(1998) 293-300

32. O. Coulibaly, A. Ouedraogo. Correlation of Global Solar Radiation of Eight Synoptic Stations in Burkina Faso Based on Linear and Multiple Linear Regression Methods Hindawi Publishing Corporation Journal of Solar Energy. (2016). ID 7870907, 9 pages http://dx.doi.org/10.1155/2016/7870907

33. H.D. Kambezidis, B.E. Psiloglou. The Meteorological Radiation Model (MRM): Advancements and Applications. In: Badescu, V., editor. Modeling solar radiation at the earth surface. Berlin: Springer (2008) chapter 14: 357392

34. E. Quansah, L.K. Amekudzi, P. Kwasi, J. Aryee, O.R. Boakye, B. Dziewornu , R.S.Mubarick, Empirical Models for Estimating Global Solar Radiation over the Ashanti Region of Ghana. Journal of Solar Energy (2014) ID 897970, 6 pages http://dx.doi.org/10.1155/2014/897970

35. A.A. Osinowo, E.C. Okogbue, S.B. Ogungbenro. F. Olugbenga, Analysis of Global Solar Irradiance over Climatic Zones in Nigeria for Solar Energy Applications, Journal of Solar Energy (2015). ID 819307, 9 pages http://dx.doi.org/10.1155/2015/819307

36. M.J. Ahmad, G.N. Tiwari, Evaluation and comparison of hourly solar radiation models, Int. J. Energy Res.; 33(2009) 538552.

37. B.G. Akinoglu, Recent Advances in the Relations between Bright Sunshine Hours and Solar. In: Badescu V, editor. Modeling solar radiation at the earth surface. Berlin: Springer. (2008) chapter 5:115143

38. C.A. Gueymard, D.R. Myers, Validation and ranking methodologies for solar radiation models. In: Badescu V, editor. Modeling solar radiation at the earth surface. Berlin: Springer (2008) chapter 20: 479509