The extraterrestrial radiation on a horizontal plane is evaluated according to the following relation [24]:

H 0 = 24 π I s c [ 1 + 0.033 cos ( 360 D n 365 ) ] ∗ [ cos φ cos δ sin ω s + 2 π ω s 360 sin φ sin δ ] (1)

where δ and ω_s are given by [25]:

ω s = cos − 1 ( − tan φ tan δ ) (2)

δ = 23.45 sin [ 360 ( 284 + D n ) 365 ] (3)

where I_sc: solar constant (W/m²); ω_s sunset hour angle (˚C); φ latitude of the location (˚) and δ solar declination (˚).

The most practical correlation for assessing global solar radiation and the most commonly used is given by the relation [26] [27]:

H H 0 = a + b ( S S 0 ) (4)

Thus, by Equations (5), (6) and (7), the correlation parameters can be defined as:

a = − 0.110 + 0.235 cos φ + 0.323 ( S S 0 ) (5)

b = 1.449 − 0.553 cos φ − 0.694 ( S S 0 ) (6)

Thus, the duration of sunshine is given by [28]:

S 0 = 2 15 cos − 1 ( − tan φ tan δ ) (7)

where S₀ monthly average day length (h); S monthly average daily hours of bright sunshine (h) and a, b regression coefficients.

The global solar irradiation is estimated by this model using several parameters namely T_M, T_m, K_r as well as atmospheric pressures [29].

H H 0 = K r ( T M − T m ) 0.5 (8)

where K_r is defined as:

K r = K r a ( P P 0 ) 0.5 (9)

where according to Lunde [30], K_ra = 0.17 and the pressures P P 0 is defined as:

P P 0 = exp ( − 0.0001184 h ) (10)

where h is altitude, varies from site to site; T_M maximum temperature and T_m minimum temperature.

This model estimates global solar radiation by taking into account parameters such as relative humidity, sunshine duration, maximum temperature and geographical location [31].

H = 1.530 K ∗ exp φ ( S S 0 − R H 1 / 3 100 − 1 T max ) (11)

with

K = 100 ( n T max + ψ i j cos ( φ ) ) (12)

n = 1 1 + 0.1 φ (13)

Global solar radiation is estimated by this model using parameters such as minimum temperature, maximum temperature as well as extraterrestrial irradiation [32]. ¶

H H 0 = a ( T max − T min ) 0.5 (14)

This model uses parameters such as maximum and minimum temperature to estimate global solar radiation [33].

H H 0 = a ( 1 + 2.7 × 10 − 5 Z ) ( T max − T min ) 0.5 (15)

The global solar radiation is estimated by the Sayigh model using parameters such as RH, T_max, DI and φ [34]. ¶

G H = 11.6 K N exp ( φ ( D I T m ) − ( R H 15 ) − ( 1 T max ) ) (16)

This indicator is estimated by taking the ratio of the RMSE as well as the average value of the data measured. The accuracy of the model according to [35], is poor if RRMSE < 43%; mean if 20% < RRMSE < 30%; good if 10% < RMSE < 20% and excellent RMSE < 10%.

RRMSE = 1 n ∑ i = 1 n ( H ¯ i , m − H ¯ i , c ) 2 ∑ i = 1 n H ¯ i , m × 100 (17)

For a more comprehensive evaluation of models estimating solar radiation, this indicator proposed by Stone [36] can be used in conjunction with RMSE and MBE [37] [38] [39] [40] [41].

t-stat = ( n − 1 ) MBE 2 RMSE 2 − MBE 2 (18)

To compare the forecast errors of different models, the RMSE is a frequently used measure. The model is better when the value of RMSE is low.

RMSE = 1 n ∑ i = 1 n ( H ¯ i , m − H ¯ i , c ) 2 (19)

In order to show more information about the model deviation, this indicator is used after Behar et al. [40] and Gueymard [39]. Thus, the 95% confidence uncertainty is given by:

U 95 = 1.96 ( SD 2 + RMSE 2 ) 1 / 2 (20)

where SD is the standard deviation of the difference between the measured and calculated data and 1.96 is the coverage factor corresponding to the 95% confidence level.

While values of MBE closest to zero are desirable, this indicator therefore expresses the tendency of the model to overestimate (positive value) and to underestimate (negative value) the global radiation.

When the model shows underestimated and overestimated values at the same time, this test does not show the correct performance, which is the drawback; because the values of underestimation and overestimation cancel each other out.

MBE = 1 n ∑ i = 1 n ( H ¯ i , m − H ¯ i , c ) (21)

The MAE is the ratio of the sum of the absolute values divided by the number of observations.

To measure how close, the estimated values are to the measured values, this quantity is often used in statistics. In inter-comparisons of mean model performance error and dimensioned evaluations, the authors pointed out some advantages of MAE over root mean square error (RMSE) [42].

MAE = 1 n ∑ i = 1 n | H ¯ i , m − H ¯ i , c | (22)

MARE is known as Mean Absolute Percentage Error (MAPE) when expressed as a percentage. Between measured and estimated solar radiation, this indicator is expressed as an average absolute value.

MARE = 1 n ∑ i = 1 n | H ¯ i , m − H ¯ i , c H ¯ i , m | (23)

erMAX = max ( | H ¯ i , m − H ¯ i , c H ¯ i , m | ) (24)

RMSRE = 1 n ∑ i = 1 n ( H ¯ i , m − H ¯ i , c H ¯ i , m ) 2 (25)

To estimate the performance of models, this indicator is often used in statistics. Thus, the models are efficient when the coefficient of determination is close to 1.

R 2 = 1 − ∑ i = 1 n ( H ¯ i , m − H ¯ i , c ) 2 ∑ i = 1 n ( H ¯ i , m − H ¯ m , a v g ) 2 (26)

The relative error in percentage (e) of the global solar radiation values can be estimated by the following equation:

e = | ( H m − H e s ) / H m | × 100

Table 1 presents the different values of the methods used to estimate the global solar irradiation for the Abeche site. The values vary from 5.081 kWh/m² in April to 5.832 kWh/m² in August using the Angstrom-P model. Using Allen’s model, they vary from 4.86 in August to 6.396 kWh/m² in February. As for the Sabbagh model, the variation is between 5.189 in August to 7.606 in April. For the Hargreaves model, they vary from 4.574 in August to 6.020 in February. Using the Annandale model, they vary from 1.353 in August to 1.366 in December. As for the Sayigh method, the variation is from 0.049 in August to 2.672 in February.

Thus, for the six models used, the minimum irradiation values are 0.049 kWh/m² using the Sayigh model and the maximum value is 7.606 kWh/m² using the Sabbagh method.

Table 2 presents the values calculated by all the statistical indicators used in this work.

The most accurate model is identified by comparing its statistical errors with those of other models with the largest R² value. The best performance is obtained by the Sabbagh model and its statistical errors, R², t-Stat, MAE, RMSRE, MARE, RRMSE, erMAX, U₉₅, RMSE, MBE are respectively 0.706, 2.866, −0.214 kWh/m², 0.324, −0.037, 0.735%, 0.115, 7.337 kWh/m², 0.542 kWh/m² and 0.037 kWh/m². Allen’s model ranked second, with R² equal to 0.219, followed by Angstrom-Prescott model equal to 0.164, Hargreaves model −0.129, Annandale model and Sayigh model.

Figure 1 below presents the histogram of the different model values estimating the global solar irradiation for the Abeche site. It shows that the model whose value is close to that measured (NASA) is the Sabbagh model, this model is the most suitable.

Table 3 presents the difference between the satellite values (NASA) with the values estimated by the six models. Thus, the Sabbagh model presents the minimum average value of −3.704% while the maximum average value of 80.725% is recorded by the Sayigh model. This allows us to deduce that from the monthly averages, the Sabbagh model is the most appropriate. Proceeding to the monthly minimum values, it is the Angstrom-Prescott model which presents the lowest monthly minimum value of −8.750 and the highest minimum value of 73.731 by the Annandale model. As for the maximum monthly values, it is the Sabbagh model which has the lowest maximum value of 11.451 and the highest maximum monthly value of 99.164 by the Sayigh model.