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A Temperature-Based Model for Estimating Monthly Average Daily Global Solar Radiation in China

Since air temperature records are readily available around the world, the models based on air temperature for estimating solar radiation have been widely accepted. In this paper, a new model based on Hargreaves and Samani (HS) method for estimating monthly average daily global solar radiation is proposed. With statistical error tests, the performance of the new model is validated by comparing with the HS model and its two modifications (Samani model and Chen model) against the measured data at 65 meteorological stations in China. Results show that the new model is more accurate and robust than the HS, Samani, and Chen models in all climatic regions, especially in the humid regions. Hence, the new model can be recommended for estimating solar radiation in areas where only air temperature data are available in China.


Introduction
Solar radiation data are essential for designing solar energy devices. However, the measurement of solar radiation is not easily available due to the cost and techniques involved [1]. The limited coverage of the measurement indicates that there is a need to establish theoretical methods for estimating solar radiation. Among the methods developed, those based on empirical correlations using commonly measured meteorological elements have attracted great attention owing to lower data requirement and computation cost [2].
The widely used correlations for estimating solar radiation are mainly based on sunshine duration and air temperature. In fact, the models estimating solar radiation from sunshine duration are generally more accurate than those involving other meteorological observations [3][4][5][6]. However, sunshine duration is not as readily available as air temperature data at standard meteorological stations [7,8]. So, it is meaningful to elaborate models that estimate solar radiation based on air temperature as an alternative.
Two common approaches estimating solar radiation from air temperature use the methods of Hargreaves and Samani [9] (HS) and Bristow and Campbell [10] (BC). Since the establishment of the two models, many investigations concerning the HS and BC models have been carried out on the improvement in prediction accuracy and general validity, which were reviewed in detail by Liu et al. [2]. The HS model is primarily intended for application in monthly calculation [11]. Although the BC model is superior to the HS model on daily global solar radiation calculation in some studies [2,3,12], however, it is not as good as the HS model in estimation of monthly average solar radiation [13,14]. The report from Bandyopadhyay et al. [13] that estimates solar radiation for 29 stations across India showed that the HS model and its modifications (Annandale et al. [15], Samani [16], and Allen [11,17] models) outperform the BC model in monthly calculation. Similarly, Meza and Varas [14] demonstrated that the revised HS correlation, namely, Allen [11,17] model, has a larger coefficient of determination than the BC model based on the monthly measured data from 21 stations in Chile. In addition, the HS model has been widely used because of its simplicity, and it is recommended in FAO-56 for solar radiation estimation [2].  However, the performance of the HS and its modifications varies significantly in different locations [2,9]. This limits the application of these models in a large country like China with diversities in climate and geography. The present work aims to propose a new simple and practical method that gives good estimates of monthly average daily global solar radiation from air temperature for all climatic regions. The performance of the proposed model is validated by comparing with the original HS model and its two modifications against the measured data at 65 meteorological stations in China using statistical error tests.
The measured data of monthly average daily global solar radiation ( , MJ/m 2 ), monthly average maximum temperature ( max , ∘ C), and minimum temperature ( min , ∘ C) at 65 meteorological stations in China from 1971 to 2000 are used in the present paper. These stations cover the four climate zones and have a diverse range in latitude and altitude with the annual mean temperature difference between 6.20 ∘ C and 16.08 ∘ C. The information of these stations is given in Table 1.
Note that the Δ in the table is according to the definition in (1) as follows.

2.2.
Models. The HS model [9] is the first procedure that calculates global solar radiation from max and min and defined as follows: where Δ = max − min , is monthly average daily extraterrestrial radiation (MJ/m 2 ), and 1 is empirical coefficient.
Following Hargreaves and Samani's pioneer work, Samani [16] and Chen et al. [5] suggested the modifications in the form of (2) and (3), respectively, where 2 , 2 , and 2 are empirical coefficients. Consider where 3 and 3 are empirical coefficients. The characteristic underlying equations (1)-(3) is that they explicitly account for solar radiation and air temperature and implicitly include the influence of relative humidity by means of Δ , which is linearly related to relative humidity [9]. While these models succeed in some areas, the assumption in the HS model as well as its modifications could lead to a reduction in estimation accuracy in some conditions [16]. The HS model assumes that Δ is directly related to the fraction of received at the ground level. However, in fact, many other factors besides solar radiation, such as latitude, altitude, cloudiness, humidity, wind speed, precipitable water, aerosol, and proximity to a large body of water, can influence Δ in a given location [11,16].
Among these factors, precipitable water has a considerable effect on solar radiation and then affects Δ , especially 4 The Scientific World Journal  in humid regions. The ways that precipitable water in the atmosphere affects solar radiation can be found in Garrison [20]. On the other hand, precipitable water is closely related to ambient temperature and relative humidity [21]. In view of this, to improve estimation in simplicity, air temperature is added together with the relative humidity implicitly presented in the HS model to exert precipitable water's effects on calculating solar radiation, and the HS model is revised as follows.
where is monthly average air temperature ( ∘ C) and defined as = ( max + min )/2 and 4 , 4 , and 4 are empirical coefficients.

Calibration and Comparison.
A common method to calculate global solar radiation that is used by many models is to first determine . In this paper, is calculated according to Duffie and Beckman [22]. The empirical coefficients of the four models (1)-(4) are, respectively, calibrated against the measured data of , max , and min (in terms of Δ and ) together with the calculated using a solver that minimizes the square error of estimation with an iterative process.
The models' performance is assessed by four common statistical indicators, namely, mean percentage error (MPE, %), mean bias error (MBE, MJ/m 2 ), root mean square error (RMSE, MJ/m 2 ), and Nash-Sutcliffe equation (NSE), calculated from the estimated and measured values of . These indicators are the ones that are applied most commonly in comparing the models of solar radiation estimation [23,24] and can be calculated as follows: where and are, respectively, the th calculated and measured values (MJ/m 2 ), is the average of the measured values (MJ/m 2 ), and is the number of observations.

Results and Discussion
The empirical coefficients of the four models at each station are reported in Table 2. The table shows that the coefficients of the four models are site-dependent due to the use of local data bases. It should be mentioned that although a sitedependent model requires a data set with the measured for determining the coefficients, this approach is frequently simpler to follow and may be more accurate than complicated mechanistic ones [25]. Figure 1 allows the values of MPE, MBE, RMSE, and NSE from the analysis of the measured and calculated to be compared for the four models at the 65 stations, and the corresponding minimum, maximum, and average values of these statistical indicators are summarized in Table 3. Figure 1 shows that the performance of these temperature-based models improves with Δ in general, except for the Samani model in terms of MPE and MBE. Overall, the new model (4) produces more accurate estimates than the three existing models examined. This can be seen from the fact that (4) has smaller values of MPE, MBE, and RMSE and higher value of NSE compared with the others at all stations. The average values of MPE, MBE, RMSE, and NSE for (4) are 0.2199%, 0.0318 MJ/m 2 , 0.5408 MJ/m 2 , and 0.9724, respectively (in Table 3). Besides, the minimum value of NSE of (4) exceeds 0.80, which shows the superiority of the new model. Moreover, it is also found that compared with (4), the performance of the HS, Samani, and Chen models varies significantly in different climate regions.
For clarity, the estimates of (4) and the three existing models are compared against the measured data at eight representative stations in Figure 2. These stations include Guangzhou (humid), Wuhan (humid), Kunming (humid), Beijing (semihumid), Harbin (semihumid), Lanzhou (semiarid), Lasa (semiarid), and Wulumuqi (arid) stations. As a rule of thumb, apart from the effects of topography, precipitable water in humid regions is generally larger than that in arid regions. Figure 2 shows that from the humid region to the arid region, the performance of the HS model and its modifications generally increases with Δ . This fact also supports the Δ sensitivity of the temperature-based models. The exception at Lasa station that deviates from the Δ sensitivity results from the effects of altitude, which is in accordance with the results reported in the literature that the HS model and its modifications perform poorly at highaltitude sites [11,13]. More importantly, it is interesting to find that the incorporation of in the model can significantly relieve the sensitivity of Δ and altitude associated with the temperature-based models. For example, the minimum NSE

Max.
Ave.   value of (4) at Kunming, Lanzhou, and Lasa stations with higher altitude is 0.9418 and at Guangzhou, Wuhan, and Kunming stations with lower Δ is 0.9380. The two NSE values indicate that the new model can successfully account for the variation of at sites having higher altitude or lower Δ . Consequently, (4) is more robust than the three existing models examined here. Table 3 shows that the MPE, MBE, RMSE, and NSE of (4) range from −0.4718 to 1.4145%, from −0.0867 to 0.1681 MJ/m 2 , from 0.1524 to 1.1429 MJ/m 2 , and from 0.8324 to 0.9988, respectively. They are all the narrowest variation range for the statistical indicators among the four models.

MPE
Also, it can be found that with precipitable water increasing, namely, from the arid region to the humid region, the advantage of the new model over the HS, Samani, and Chen models becomes more prominent. In terms of NSE, (4) outperforms the HS, Samani, and Chen models approximately by 1.44%, 8.95% and 0.29% at Lasa station in the semiarid region whereas by 19.10%, 7.25%, and 11.40% at Wuhan station in the humid region. Note that the difference will be larger if Guangzhou station instead of Wuhan is used in the comparison, as evidently shown in Figure 2. Consequently, the modification of the HS model with the addition of is reasonable.

Conclusions
This work stems from the fact that air temperature is commonly measured at many stations around the world, and the performance of the HS and its modifications varies significantly in different locations. To estimate monthly average daily global solar radiation from air temperature with better accuracy in all climatic regions, a new modification to the HS model is proposed. The new model is validated by comparing with the HS model and its two modifications against the measured data at 65 meteorological stations in China. The study demonstrates that the new model is more accurate and robust than the HS, Samani, and Chen models in all climatic regions, especially in the humid regions. Therefore, it can be recommended for estimating monthly average daily global solar radiation.
Admittedly, a limitation of this study is that the new model developed here is site-dependent, so when it is utilized in locations other than its based region, it is better to calibrate the empirical coefficients against the local data first. Future efforts should be directed to explore the correlation of the model's empirical coefficients with common factors and then develop a model for general application.