 Open Access
 Authors : Kodji Deli, Etienne Tchoffo Houdji, Noel Djongyang, Ahmat Tom
 Paper ID : IJERTV13IS040309
 Volume & Issue : Volume 13, Issue 4 (April 2024)
 Published (First Online): 18052024
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Performance Study of Thirtyfive 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 NgaounderÃ©, 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 0.7987.12, while MBE% and R2, range respectively between 0.00 to 6.52E01 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.

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 [13]. 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 longterm 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, [1014]. 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 PrescottPage 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 [1830]. 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 Sunshinebased models, Cloudbased models, Temperaturebased models, Relative Humiditybased models, Precipitationbased models, Hybrid Parameter based models [5, 29, 3139], 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 angstromPrescott 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, thirtyfive (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).

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 SudanoGuinean 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 Â° C, Maximum daily temperature in Â° C, Minimum daily temperature in Â° 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 
19802013 
13 
21 
Minimum daily temperature 
19802013 
13 
21 
Mean daily temperature 
19802013 
13 
21 
Mean Soil Temperature (ST) 
19802013 
13 
21 
Mean daily visibility 
19802013 
13 
21 
Mean relative humidity 
19802013 
13 
21 
Mean precipitation 
19802013 
13 
21 
Effective day length 
19612015 
21 
33 
Solar radiation 
19842015 
29 
4 

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, thirtyfive (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, cloudbased, temperaturebased, and hybrid parameterbased 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
NÂ°
Models (equation type)
Authors
Mathematical relations
M01
AngstromPrescottPage
(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
ElMetwally Model (linear)
ElMetwally 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
SwartzmanOgunlade 1
(power)
Swartzman and Ogunlade 1967[56]
= ( )
0
M30
SwartzmanOgunlade 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 [ 24, 47, 6163]. 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 longterm 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 overestimation in the calculated value and vice versa. One drawback of this test is that over estimation of an individual observation will cancel underestimation in a separate observation [64, 65]. The RMSE provides information on the shortterm performance of the correlations by allowing a termbyterm 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 Â±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 Â± 5% and the scatter in the results is such that RMSE < 15 %. [33].

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
2.60E02
18
3.31943
23
0.93792
23
0.13460
22
+
3
M02
2.60E02
19
3.31943
24
0.93792
24
0.13460
23
+
4
M03
4.60E02
28
2.86001
16
0.95392
16
0.09517
17
+
5
M04
4.19E02
26
2.98392
18
0.94984
18
0.09929
18
+
3
M05
6.52E01
35
3.38757
28
0.93535
28
0.53040
33
–
6
M06
4.13E02
24
2.98369
17
0.94985
17
0.0994
19
+
4
M07
2.37E02
17
3.42383
29
0.93396
29
0.14724
27
+
3
M08
1.29E02
15
2.98436
19
0.94982
19
0.05306
13
–
3
M09
2.60E02
20
3.31943
25
0.93792
25
0.13460
24
+
5
M10
2.60E02
21
3.31943
26
0.93792
26
0.13460
25
+
4
M11
2.60E02
22
3.31943
27
0.93792
27
0.13460
26
+
4
M12
4.12E01
34
5.28326
34
0.84274
34
0.76649
35
–
3
M13
7.11E02
30
3.97117
33
0.91115
33
0.19643
31
+
3
M14
9.40E02
33
3.92811
32
0.91307
32
0.19876
32
+
3
M15
7.98E02
31
3.92221
31
0.91333
31
0.18737
30
+
3
M16
5.41E15
4
1.24071
7
0.99133
7
0.01300
6
–
4
M17
8.11E15
9
1.13082
4
0.99280
4
0.01164
5
–
6
M18
3.11E14
14
1.03191
3
0.99400
3
0.01164
3
–
6
M19
4.05E15
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
6.01E02
29
2.44475
14
0.96633
14
0.07006
16
+
6
M22
5.41E15
5
1.49512
8
0.98741
8
0.03224
9
+
5
M23
6.76E15
7
2.63538
15
0.96087
15
0.06864
15
–
4
M24
6.76E15
8
1.97014
12
0.97813
12
0.04265
12
–
6
M25
1.35E14
11
2.26247
13
0.97116
13
0.06121
14
–
5
M26
1.22E14
10
1.67897
11
0.98412
11
0.03476
10
–
5
M27
5.41E15
6
1.49898
9
0.98734
9
0.03104
8
–
6
M28
1.49E14
13
1.13881
6
0.99269
6
0.01604
7
+
7
M29
4.17E02
25
3.28640
22
0.93915
22
0.18279
29
+
3
M30
3.57E02
23
3.26011
21
0.94012
21
0.13342
21
+
4
M31
1.66E02
16
3.16308
20
0.94363
30
0.11560
20
+
4
M32
4.31E02
27
7.12188
35
0.71424
35
0.57972
34
+
4
M33
8.75E02
32
3.85647
30
0.91621
30
0.17741
28
+
4
M34
1.35E15
2
1.62270
10
0.98517
10
0.03490
11
–
5
M35
1.35E14
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
NgaounderÃ©
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 NgaounderÃ©
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
6,509E07
X
X
X
X
M34
2,60849
0,01140
0,13761
6,946E03
6,840E06
X
X
X
M35
0,90044
0,08586
2,35685
1,522E03
1,499E01
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(%)=5.27E14, 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
1,22E14
1
0,13889
5
0,99989
5
NM02
= + 0 + + + + + + + +
0
4,05E14
3
0,09373
4
0,99995
4
NM03
= + 0 + + + + + + + + +
0
5,00E14
4
0,03277
2
0,99999
2
NM04
= + ( ) + + + ( ) + + + +
0 0 0
2,26E03
6
0,54489
6
0,99833
6
NM05
= + 0 + + + + + + + + +
0
5,27E14
5
0,01540
1
1,00000
1
NM06
= + 0 + + + + + + + + +
0
1,49E14
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 NgaounderÃ©.
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 overestimation; – is underestimation and is coincident value)
NgaounderÃ© city
Month
Retscreen
pvgis
Solargis
Jan
+
+
+
Feb
+
+
+
Ma
+
+
+
Apr
+
+
May
+
+
Jun
+
Jul
+
Aug
+
+
Sep
+
+
Oct
+
Nov
+
Dec
+

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 KÃ¶ppenGeiger 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 1,22E14% to 2,26E03%. 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 (Â°C)
= mean minimum temperature (Â°C)
= monthly daily mean air temperature ( K.)
= mean soil temperature (Â°C)
= monthly mean temperature (Â°C)
=visibility(Km)
= ( ) = the temperature difference (Â°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 (Â°)
= 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.
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