National-scale development and calibration of empirical models for predicting daily global solar radiation in China

Energy Conversion and Management 203 (2020) 112236

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National-scale development and calibration of empirical models for predicting daily global solar radiation in China Yu Fenga,b, Daozhi Gongb, Shouzheng Jianga, Lu Zhaoa, Ningbo Cuia, a b

T

⁎

State Key Laboratory of Hydraulics and Mountain River Engineering & College of Water Resource and Hydropower, Sichuan University, Chengdu, China Institute of Environment and Sustainable Development in Agriculture, Chinese Academy of Agriculture Sciences, Beijing, China

A R T I C LE I N FO

A B S T R A C T

Keywords: Empirical models Sunshine duration Air temperature Global solar radiation China

Accurate global solar radiation (Rs) information is pivotal to the design and management of solar energy systems. Nevertheless, the expensive devices for Rs measurements make Rs data always unavailable in many regions around the world. The empirical models that predict Rs using other widely available climatic variables are feasible alternatives when Rs measurements are unavailable. However, the parameters of empirical models are site-specific and always need local calibration. In this context, the present study firstly developed a novel empirical model for accurately predicting Rs at the national scale in China, and then compared the new model with nineteen locally calibrated empirical models that have been largely reported in prior studies, including seven sunshine-based, nine temperature-based, and three complex empirical models. Daily Rs and other meteorological data during 1994–2016 from 96 weather stations in China were used to calibrate/develop and assess the models. The results showed that the newly proposed C4 model generally offered the best prediction accuracy among the models, with average MAE of 1.69 MJ m−2 d−1, RRMSE of 16.2% and NS of 0.895, which can be recommended as the optimal model for predicting Rs in China. The models reviewed and developed in this study improved the prediction accuracy of Rs, which can provide crucial information for the design and implementation of solar photovoltaic and thermal systems.

1. Introduction Solar energy is a promising clean energy source for agricultural and industrial production as conventional fossil fuel is depleting [1]. Global solar radiation (Rs) is a pivotal input to the design and management of solar energy systems [2]. However, the expensive instruments and stringent maintenance make Rs measurements unavailable in many regions around the world [3]. Thus, accurately predicting Rs by using other widely available climatic variables can provide critical knowledge for solar energy-related applications when Rs measurements are unavailable [4–7]. There are currently numerous methods for Rs prediction, including empirical models that describe the correlation between Rs and other climatic variables [7], remote sensing methods that monitor large-scale Rs using satellites [8] and data-driven models that depict the nonlinear and complex relationships between Rs and other climatic variables [9,10]. Previous studies have demonstrated that empirical models are the most commonly applied ones due to their low computational costs and widely available input data [2,11]. During the last decades, a large number of empirical models have been developed for predicting Rs using other commonly available

⁎

meteorological and geographical variables [12,13]. The empirical models can be categorized into three types, namely temperature-based, sunshine-based and complex models. Among these models, temperature-based and sunshine-based models are the most widely used models. Many sunshine-based models are reformed versions of the Angstrom–type equation, which basically relate the sunshine duration (n) with Rs in a specific region [14–16]. Chelbi et al. [17] investigated the performance of different types of Angstrom models for predicting Rs at four stations of Tunisia, and reported that the cubic-type Angstrom model had the best accuracy. Another study in Spain revealed the thirddegree Angstrom model showed better results, but the original Angstrom model was recommended owing to its greater simplicity and wider application [18]. The better performance of the third-degree Angstrom model was also confirmed in China [19]. However, another study in India revealed that the original Angstrom model had a lower error than the second-degree and third-degree Angstrom models [20]. Fan et al. [2] developed four new sunshine-based models in South China, and concluded that the new models provided better prediction accuracy compared with the ten existing models. Paulescu et al. [21] reviewed the physical basis, accuracy and sensitivity of the

Corresponding author at: College of Water Resource and Hydropower, Sichuan University, 610065 Chengdu, China. E-mail addresses: [email protected], [email protected] (N. Cui).

https://doi.org/10.1016/j.enconman.2019.112236 Received 30 July 2019; Received in revised form 6 October 2019; Accepted 28 October 2019 Available online 29 November 2019 0196-8904/ © 2019 Elsevier Ltd. All rights reserved.

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Nomenclature

Tmin z

Minimum air temperature (oC) Altitude (m)

Variables Constants

ΔT φ δ ωs dr Gsc N n P Ra RH Rs Rs0 Ta Tmax

Diurnal temperature range (oC) Latitude (rad) Solar declination (rad) Sunset hour angle (rad) Inverse relative distance Earth-Sun Solar constant (0.0820 MJ m−2 min−1) Maximum possible sunshine duration (h) Sunshine duration (h) Precipitation (mm) Extraterrestrial radiation (MJ m−2 d-1) Relative humidity (%) Global solar radiation (MJ m−2 d-1) Clear-sky radiation (MJ m−2 d-1) Average air temperature (oC) Maximum air temperature (oC)

a, b, c, d, e, f, g, h Empirical coefficients Abbreviations MAE RRMSE NS GPI B-C MPZ TCZ TMZ SMZ

Mean absolute error (MJ m−2 d-1) Relative root mean square error Nash-Sutcliffe coefficient Global performance indicator Bristow-Campbell Mountain plateau zone Temperate continental zone Temperate monsoon zone Subtropical and tropical monsoon zone

by introducing relative humidity into sunshine-based models, and concluded that the new models improved the prediction accuracy of Rs relative to the existing models. Fan et al. [32] established a complex model that incorporated air temperature, vapor pressure deficit, rainfall and relative humidity into sunshine-based models, which outperformed the other studied models. Jahani et al. [33] introduced dew point temperature, ΔT and rainfall as inputs to calibrate the sunshine-based model, and reported that the newly established complex models had the best prediction accuracy. Although empirical models have been extensively reported around the world, these studies generally compared empirical models that have the same formula with different parameters, or proposed new models in a specific region that limit the model application in other regions with different climates. For example, Makade and Jamil [34] compared 300 existing sunshine-based models in Nagpur of India, most of which had a similar model structure with different empirical constants that were locally fitted by other studies in different climates. The 105 models compared by Bayrakçı et al. [35] in Muğla of Turkey can be classified as seven types of models in terms of the model structure. Chen et al. [36] compared 37 groups of empirical models at only three stations in Central China. Fan et al. [28] introduced relative humidity and precipitation into temperature-based models, which can improve model performance in tropical and sub-tropical regions of China, where rainy days and high relative humidity occur frequently. This model, however, may show poor performance in arid regions. China, the biggest photovoltaic manufacture on Earth [37], has great potential for the application of large-scale photovoltaic power systems. However, accurate Rs data, the primary variable affects the development and management of solar photovoltaic systems, are lacking in many regions of China. For instance, there are more than 700 national meteorological stations in China measuring routine meteorological variables (e.g., air temperature, sunshine duration and relative humidity). Nevertheless, only 122 stations recorded Rs data [38] across a land area of ~9.6 million km2. Therefore, developing new empirical models that are universally implementable in different climatic regions of China can provide reliable knowledge for the application of photovoltaic systems [39]. To this end, we firstly developed a complex empirical model that considers the effects of other climatic variables (e.g., relative humidity, rainfall and air temperature) on Rs prediction, then locally calibrated nineteen existing empirical models that have completely different model structures, and finally compared the performance of the newly developed complex model with existing models in China. For this purpose, a large dataset that contains the most complete Rs measurements in China were used for model development/calibration and assessment in different climatic

Angstrom–Prescott model. A case study at 59 stations in Europe was also conducted and showed that the Angstrom–Prescott model with relative sunshine, month index and altitude as inputs could describe ~90% of the Rs variability. Although sunshine-based models are attractive in predicting Rs and their outstanding performance have been extensively reported, their applications are constrained due to the scarcity of sunshine duration data, especially in developing countries. To overcome this shortage, temperature-based models have been developed, because their required input data, air temperature, can be measured easily and widely [22]. Hargreaves and Samani [23] considered that the effects of Rs on diurnal temperature range (ΔT , the difference between daily maximum and minimum air temperature), and developed the well-known Hargreaves model for Rs prediction with only daily air temperature data as input. Also, Bristow and Campbell [24] developed the Bristow-Campbell (B-C) model, in which Rs is an exponential function of ΔT . The model provided reasonable results and explained 70–90% of Rs variability at three places of America. Liu et al. [25] demonstrated the original B-C model had a similar prediction accuracy to the modified Hargreaves model, but it showed significantly better accuracy than the original Hargreaves model. A comparison of five temperature-based models by Almorox et al. [26] also indicated that Rs could be estimated by using easily available meteorological variables, air temperature. Hassan et al. [27] showed the new temperature-based model developed by local parameterization provided more accurate prediction than the existing sunshine-based models. In addition to air temperature and sunshine duration mentioned above, the effects of other climatic variables on Rs prediction were also investigated. Previous studies have developed several complex models to improve the performance of temperature-based and sunshine-based models by incorporating other climatic variables as inputs. Fan et al. [28] reported that the accuracy of single temperature-based models could be enhanced by introducing relative humidity and precipitation data as inputs in the tropical and sub-tropical regions of China. Quej et al. [29] also confirmed the better performance of empirical models based on air temperature, rainfall and relative humidity, compared with models based on air temperature only. Li et al. [30] considered the effects of seasonality and relative humidity on Hargreaves and B-C models and developed six new temperature-based models in Guangzhou of southern China. They found that B-C type models considering the effects of seasonality and/or relative humidity had better results. Regarding the sunshine-based models, extensive studies have been performed to improve their accuracy by incorporating other climatic variables. Yildirim et al. [31] proposed new parametric models 2

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2. Material and methods

coefficients; n is sunshine duration, h; N is maximum possible sunshine duration, h; Ra is extraterrestrial solar radiation, MJ m−2 d−1. Ra is calculated as [43]:

2.1. Study area and data collection

Ra =

Four contrasting climatic zones across China were selected for model comparisons (Fig. 1), which are the temperate continental zone (TCZ), subtropical and tropical monsoon zone (SMZ), mountain plateau zone (MPZ) and temperate monsoon zone (TMZ). Rs was first measured from 1957 in China, and the measurements of Rs were then spread gradually at a total number of 122 stations. However, the measurements at some stations terminated during the 1960s–1990s [40]. In the early 1990s, a large-scale update of the measurement devices was performed, and a new type of radiometer was implemented at the remaining 96 stations across the four climates of China since 1994 [40,41]. Considering the data availability and continuity, daily meteorological variables at the 96 stations during 1994–2016 were collected, including Rs and other widely available variables like maximum (Tmax)/minimum (Tmin) air temperature, relative humidity (RH), sunshine duration (n) and precipitation (P). The data were provided and preliminarily quality-controlled by the China Meteorological Administration. Further quality control was performed following the scheme proposed by Fan et al. [1]. The total data were divided into two parts. The first part (daily data during 1994–2010) was used for establishment/calibration of the models, while the remaining part (daily data during 2011–2016) was utilized for model assessment and comparison.

where Gsc is solar constant, 0.0820 MJ m−2 min−1; dr is inverse relative distance Earth-Sun; ωs is sunset hour angle, rad; φ is latitude, rad; δ is solar declination, rad. Olgeman et al. [44] used a maximum-likelihood quadratic fit to calculate Rs at two stations of Turkey, and derived the quadratic type of Angstrom model (Abbreviated as S2):

zones.

24(60) Gsc dr [ωs sin (φ) sin (δ ) + cos(φ) cos (δ ) sin (ωs )] π

n n Rs = ⎡a + b ( ) + c ( )2⎤ × Ra N N ⎦ ⎣

n n 2 n 3 Rs = ⎡a + b ⎛ ⎞ + c ⎛ ⎞ + d ⎛ ⎞ ⎤ × Ra ⎢ N N N ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎥ ⎣ ⎦

n Rs = ⎡acosφ + b ⎛ ⎞ ⎤ × Ra ⎝ N ⎠⎦ ⎣

(5)

where φ is latitude, rad. Almorox and Hontoria [18] proposed an exponential Angstrom-type model for 16 meteorological stations in Spain (Abbreviated as S5):

n Rs = ⎡a + b exp ⎛ ⎞ ⎤ × Ra ⎝ N ⎠⎦ ⎣

(6)

Bakirci [16] proposed a linear exponential and an exponent Angstrom-type model in Turkey (Abbreviated as S6 and S7, respectively):

()

()

⎧ Rs = ⎡a + b n + c exp n ⎤ × Ra N N ⎦ ⎪ ⎣ ⎨ R = ⎡a n b⎤ × R exponent a ⎪ s ⎣ N ⎦ ⎩

(1)

where Rs is global solar radiation, MJ m

(4)

where d is empirical coefficient. Glover and McCulloch [46] derived an empirical relationship leading to the original Angstrom model at latitude ranging 0°–60°, which is expressed as (Abbreviated as S4):

Seven typical sunshine-based models that were widely utilized in different climatic zones were selected in this study, including linear, quadratic, cubic and exponential types. The most commonly used sunshine-based model was proposed by Angstrom [14], who related the ratio of Rs and clear-sky radiation (Rs0) linearly to the ratio of n and maximum possible sunshine duration (N). However, the Rs0 data was difficult to acquire, so extraterrestrial radiation (Ra) was used to overcome this [42]. The linear Angstrom-Prescott (abbreviated as S1) can be therefore expressed as:

−2

(3)

where c is empirical coefficient. Bahel et al. [45] developed the cubic Angstrom-type correlation at 48 stations around the world (Abbreviated as S3):

2.2. Sunshine-based models

n Rs = ⎡a + b ( )⎤ × Ra N ⎦ ⎣

(2)

linearexponential

()

-1

d ; a and b are empirical

Fig. 1. The geographical distribution of the meteorological stations across different climatic zones of China. 3

(7)

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where z is the altitude, m. Chen et al. [48] calibrated the original Hargreaves model at 48 stations across China, and developed a new model (Abbreviated as T4):

2.3. Temperature-based models The most commonly utilized temperature-based models are Hargreaves (Abbreviated as T1) and B-C (Abbreviated as T2) models. Hargreaves and Samani [23] considered Rs as a function of Ra and ΔT , and derived the following model:

Rs =

a (ΔT )bR

Rs = (a· ΔT + b) × Ra

Hunt et al. [49] calibrated the Hargreaves model and presented the model in an additive equation (Abbreviated as T5):

(8)

a

(11)

Rs = a· ΔT ·Ra + b

o

(12)

where ΔT is diurnal temperature range, ΔT = Tmax − Tmin , C. Bristow and Campbell [24] related Rs and ΔT using an exponential function, and developed the following model (Abbreviated as T2):

Chen et al. [48] developed a logarithmic equation that related Rs/Ra and ΔT (Abbreviated as T6):

Rs = a [1 − exp(−bΔT c )] × Ra

Rs = Ra [a+bln(ΔT)]

(9)

(13)

Annandale et al. [47] analyzed the effects of altitude on Rs and proposed a multiplicative equation (Abbreviated as T3):

Samani [50] proposed a polynomial equation, which related Rs/Ra and ΔT (Abbreviated as T7):

Rs = [a (1 + 2. 7 × 10−5z )· ΔT ] × Ra

Rs = Ra (aΔT 0.5 + bΔT1.5 + c ΔT 2.5)

(10)

(14)

Fig. 2. Boxplots of sunshine-based (S1–S7), temperature-based (T1–T9), and complex (C1–C4) models for estimating daily global solar radiation in the temperate monsoon zone of China. 4

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Fan et al. [28] introduced the average air temperature to enhance the model performance (Abbreviated as T8):

Rs = Ra (a+bΔ T +

c ΔT 0.25

+

d ΔT 0.5)

n Rs = ⎡a + b ⎛ ⎞ + c·Ta⎤ × Ra ⎝N ⎠ ⎣ ⎦

(15)

+ eTa

Chen et al. [48] developed an integrated model, in which Rs had a power correlation and a logarithmic correlation with relative sunshine (n/N) and ΔT , respectively (Abbreviated as C2):

where Ta is average air temperature, Ta = (Tmax + Tmin )/2 , oC. Jahani et al. [33] considered a polynomial relationship between Rs/ Ra and ΔT , and derived the following model (Abbreviated as T9):

Rs = Ra (a+bΔ T + c ΔT 2 + d ΔT 3)

(17)

n c Rs = ⎡a + b ⎛ ⎞ + d ln(ΔT )⎤ × Ra N ⎝ ⎠ ⎣ ⎦

(16)

(18)

Fan et al. [32] considered the effects of P and Ta on Rs, and proposed the following equation (Abbreviated as C3): 2.4. Complex models

n c Rs = ⎡a + b ⎛ ⎞ + d ln(ΔT ) + e ln( P+ 1) + fTa⎤ × Ra N ⎝ ⎠ ⎣ ⎦

El-Sebaii et al. [51] considered an empirical correlation between Rs with n and Ta, and presented the following equation (Abbreviated as C1):

MAE (MJ m-2 d-1)

4

25%~75%

(19)

where e and f are empirical coefficients. In the present study, Rs was considered to have a cubic correlation

Range within 1.5IQR

Mean

Median Line

Outliers

3

2

1 S1 S2 S3 S4 S5 S6 S7 T1

T2 T3 T4 T5

T6 T7 T8

T9 C1 C2 C3 C4

S1 S2 S3 S4 S5 S6 S7 T1

T2 T3 T4 T5

T6 T7 T8

T9 C1 C2 C3 C4

S1 S2 S3 S4 S5 S6 S7 T1

T2 T3 T4 T5

T6 T7 T8

T9 C1 C2 C3 C4

RRMSE (%)

30

20

10

1.0

NS

0.9 0.8 0.7 0.6 0.5 Fig. 3. Boxplots of sunshine-based (S1–S7), temperature-based (T1–T9), and complex (C1–C4) models for estimating daily global solar radiation in the temperate continental zone of China. 5

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where g and h are empirical coefficients; P is precipitation, mm; RH is relative humidity.

with n/N, and ΔT was also considered as Hargreaves and Samani [23] proposed. As for the effects of P, scaled precipitation was utilized using logarithmic transform since the actual amount of daily P varied from 0 to hundreds, which made it challenging to describe. To prevent negative values for the logarithmic function when daily P ranged 0–1, a logarithmic form of ln(P + 1) was applied as proposed by Fan et al. [32]. Further, Ta and RH were also included since they might have linear effects on Rs. Thus, the following complex model was proposed (Abbreviated as C4):

2.5. Performance evaluation Three commonly used statistical indicators, including mean absolute error (MAE), relative root mean square error (RRMSE) and NashSutcliffe coefficient (NS), were used to assess the models [52,53].

MAE =

Rs 2

1 m

+ hRH⎤ × Ra ⎥ ⎦

MAE (MJ m-2 d-1)

(21)

i=1

3

n n n = ⎡a + b ⎛ ⎞ + c ⎛ ⎞ + d ⎛ ⎞ + e ΔT + f ln(P + 1) + gTa ⎢ N N N ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎣

4

m

∑ |Xi − Y|i 1 m

RRMSE=

(20)

25%~75%

Range within 1.5IQR

m

∑ (Yi − Xi)2 i=1

¯ X

Mean

× 100%

(22)

Median Line

Outliers

3

2

1 S1 S2 S3 S4 S5 S6 S7 T1

T2 T3 T4 T5

T6 T7 T8

T9 C1 C2 C3 C4

S1 S2 S3 S4 S5 S6 S7 T1

T2 T3 T4 T5

T6 T7 T8

T9 C1 C2 C3 C4

S1 S2 S3 S4 S5 S6 S7 T1

T2 T3 T4 T5

T6 T7 T8

T9 C1 C2 C3 C4

RRMSE (%)

30

20

10

1.0

NS

0.8

0.6

0.4 Fig. 4. Boxplots of sunshine-based (S1–S7), temperature-based (T1–T9), and complex (C1–C4) models for estimating daily global solar radiation in the mountain plateau zone of China. 6

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∑i = 1 (Yi − Xi)2 m ¯ )2 ∑ (Xi − X

NS = 1 −

3. Results and discussion (23)

i=1

3.1. Comparison of statistical performance of empirical models for predicting global solar radiation

where Xi and Yi are respectively the measured and estimated values. The ranking or comparison of many models with different statistical indicators is of difficulty. Thus, the global performance indicator (GPI) was introduced for overall model ranking [13]. The GPI for model i was estimated as:

Fig. 2 presents the boxplots of the sunshine-based (S1–S7), temperature-based (T1–T9), and complex (C1–C4) models for daily Rs estimation in TMZ. As shown in the figure, complex models generally showed more accurate estimates, followed by sunshine-based and temperature-based models, with average median MAE of 1.59, 1.73 and 2.97 MJ m−2 d−1, average median RRMSE of 15.9%, 17.7% and 28.6%, NS of 0.90, 0.88 and 0.71, respectively. The best sunshine-based, temperature-based and complex models were S3, T3, and C4, respectively, with median MAE of 1.65, 2.13 and 1.55 MJ m−2 d−1, RRMSE of 16.5%, 22.6% and 15.3%, NS of 0.89, 0.81 and 0.91, respectively. The mean values of the empirical coefficients a, b, c and d of S3 were 0.191,

3

GPIi =

∑ αj (gj − yij )

(24)

j=1

where αj is a constant, equals 1 for MAE and RRMSE ( j = 1, 2 ) and −1 for NS ( j = 3); yij is the scaled value of indicator j for model i ; gj is the median of scaled values of indicator j . A higher value of GPI demonstrates the more accurate of the model.

MAE (MJ m-2 d-1)

5

25%~75%

Range within 1.5IQR

Mean

Median Line

Outliers

4 3 2 1 S1 S2 S3 S4 S5 S6 S7 T1

T2 T3 T4 T5

T6 T7 T8

T9 C1 C2 C3 C4

S1 S2 S3 S4 S5 S6 S7 T1

T2 T3 T4 T5

T6 T7 T8

T9 C1 C2 C3 C4

S1 S2 S3 S4 S5 S6 S7 T1

T2 T3 T4 T5

T6 T7 T8

T9 C1 C2 C3 C4

RRMSE (%)

50 40 30 20 10

1.0 0.9

NS

0.8 0.7 0.6 0.5 0.4 Fig. 5. Boxplots of sunshine-based (S1–S7), temperature-based (T1–T9), and complex (C1–C4) models for estimating daily global solar radiation in the subtropical and tropical monsoon zone of China. 7

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0.888, −0.970 and 0.645, mean value of a for T3 was 0.688, and mean values of a, b, c, d, e, f, g and h for C4 were 0.217, 0.757, −0.824, 0.564, 0.015, −0.245, 0.001 and −0.001, respectively. The boxplots of the twenty models for daily Rs estimation in TCZ are shown in Fig. 3. The figure indicates the complex models performed better than sunshine-based and temperature-based models, with average median MAE of 1.62, 1.67 and 2.77 MJ m−2 d−1, average median RRMSE of 13.9%, 14.0% and 23.1%, NS of 0.92, 0.91 and 0.75, respectively. Compared with the models in TMZ (Fig. 2), models in TCZ had better performance. S3, T3 and C4 were the best sunshine-based, temperature-based and complex models respectively, with median MAE of 1.63, 1.92 and 1.60 MJ m−2 d−1, RRMSE of 13.5%, 16.7% and 13.9%, NS of 0.91, 0.88 and 0.92, respectively. The mean values of the empirical coefficients a, b, c and d of S3 were 0.211, 0.647, −0.378 and 0.277, mean value of a for T3 was 0.709, and mean values of a, b, c, d, e, f, g and h for C4 were 0.213, 0.586, −0.339, 0.252, 0.017, −0.020, 0.001 and −0.001, respectively.

MAE (MJ m-2 d-1)

5

25%~75%

Fig. 4 illustrates the boxplots of the twenty models for daily Rs estimation in MPZ. From the figure, the performances of the complex models were better than those of the sunshine-based and temperaturebased models, with average median MAE of 1.84, 1.87 and 2.78 MJ m−2 d-1, average median RRMSE of 14.1%, 14.2% and 21.2%, NS of 0.81, 0.80 and 0.64, respectively. Compared with the boxplots for models in TMZ (Fig. 2) and TCZ (Fig. 3), the distributions of boxplots for models in MPZ showed more variability, indicating that model performance varied more in MPZ, compared with models in TMZ and TCZ. Similar to models in TMZ and TCZ, S3, T3, and C4 were the best sunshine-based, temperature-based and complex models respectively, with median MAE of 1.69, 1.94 and 1.66 MJ m−2 d−1, RRMSE of 12.7%, 15.3% and 12.4%, NS of 0.86, 0.80 and 0.86, respectively. The average values of the empirical coefficients a, b, c and d of S3 were 0.238, 0.745, −0.477 and 0.346, mean value of a for T3 was 0.789, and mean values of a, b, c, d, e, f, g and h for C4 were 0.186, 0.715, −0.472, 0.335, 0.018, −0.007, −0.001 and 0.001, respectively.

Range within 1.5IQR

Mean

Median Line

Outliers

4 3 2 1 S1 S2 S3 S4 S5 S6 S7 T1

T2 T3 T4 T5

T6 T7 T8

T9 C1 C2 C3 C4

S1 S2 S3 S4 S5 S6 S7 T1

T2 T3 T4 T5

T6 T7 T8

T9 C1 C2 C3 C4

S1 S2 S3 S4 S5 S6 S7 T1

T2 T3 T4 T5

T6 T7 T8

T9 C1 C2 C3 C4

50

RRMSE (%)

40 30 20 10 0 1.0 0.9

NS

0.8 0.7 0.6 0.5 0.4 Fig. 6. Boxplots of sunshine-based (S1–S7), temperature-based (T1–T9), and complex (C1–C4) models for estimating daily global solar radiation in China. 8

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The boxplots of the twenty models for daily Rs estimation in SMZ are presented in Fig. 5. As shown in the figure, the complex models also provided higher accuracy than those of sunshine-based and temperature-based models in SMZ, with average median MAE of 1.69, 1.98 and 3.04 MJ m−2 d−1, average median RRMSE of 18.3%, 21.0% and 32.6%, NS of 0.90, 0.86 and 0.68, respectively. Similarly, the distributions of boxplots for models in SMZ had more variability, compared with the boxplots for models in TMZ (Fig. 2) and TCZ (Fig. 3). The best sunshinebased, temperature-based and complex models were S3, T3, and C4 respectively, with median MAE of 1.78, 2.54 and 1.64 MJ m−2 d−1, RRMSE of 19.3%, 27.8% and 17.6%, NS of 0.89, 0.76 and 0.90, respectively. The mean values of the empirical coefficients a, b, c and d of S3 were 0.151, 1.162, −1.536 and 0.976, mean value of a for T3 was 0.679, and mean values of a, b, c, d, e, f, g and h for C4 were 0.113, 0.835, −1.022, 0.663, 0.043, −0.018, 0.002 and −0.001, respectively. Fig. 6 shows the boxplots of the twenty models for daily Rs estimation in China. Overall, the order of the model rank was: complex models > sunshine-based models > temperature-based models, with average median MAE of 1.66, 1.84 and 2.87 MJ m−2 d-1, average median RRMSE of 15.7%, 17.6% and 27.8%, NS of 0.90, 0.88 and 0.71, respectively. Among the complex models, C4 was the best model, while S3 and T3 were the best sunshine-based and temperature-based models, with median MAE of 1.61, 1.72, and 2.21 MJ m−2 d-1, median RRMSE of 15.1%, 16.3% and 22.4%, and median NS of 0.90, 0.89 and 0.80 for the three best models, respectively. The complex models showed better performance than sunshine-based models, indicating that introducing other meteorological variables into the sunshine-based models could further improve the model performance, even though these variables might have no direct effects with Rs.

Table 1 Comparison of overall performance of the empirical models across China. MAE (MJ m−2 d-1)

RRMSE (%)

NS

GPI

Model rank

S1 S2 S3 S4 S5 S6 S7 T1 T2 T3 T4 T5 T6 T7 T8 T9 C1 C2 C3 C4

1.85 1.82 1.79 1.85 1.96 1.82 2.30 3.10 2.95 2.32 3.06 3.21 3.08 2.97 2.91 2.95 1.84 1.74 1.70 1.69

17.7 17.4 17.2 17.7 18.7 17.4 23.0 29.1 28.1 23.2 28.9 29.8 29.0 28.3 27.7 28.2 17.5 16.7 16.3 16.2

0.878 0.882 0.884 0.878 0.865 0.881 0.794 0.670 0.693 0.791 0.676 0.658 0.676 0.689 0.702 0.692 0.880 0.890 0.894 0.895

0.53 0.58 0.62 0.53 0.34 0.57 −0.48 −1.94 −1.69 −0.52 −1.88 −2.10 −1.90 −1.72 −1.60 −1.70 0.55 0.71 0.77 0.79

8 5 4 8 10 6 11 19 14 12 17 20 18 16 13 15 7 3 2 1

d-1, RRMSE of 23.2%, and NS of 0.791. 3.3. Discussion The complex models generally have better estimates than the single sunshine-based models, indicating that introducing other variables into the models can improve the model performance. The newly proposed model has low computation costs, and its required data are available at more than 700 national weather stations across China. Thus, the model can be implemented in most parts of China, which can provide better information on solar energy-related applications. It should be noted that although C1 is a complex model which considers not only sunshine duration but also air temperature, its performance is even worse than that of single sunshine-based S3. This might be attributed to the model structure. C1 was developed based on the liner Angstrom-type model (S1), whose performance can be improved when considering air temperature data, compared with its original form (S1). However, the complex liner Angstrom-type model (C1) showed less accurate results than the sunshine-based cubic Angstrom-type model (S3), indicating that the model performance is more sensitive to model structure. The complex cubic Angstrom-type model further considering the effects of air temperature, rainfall, and relative humidity provided the best results among the twenty selected models, which can be recommended as the optimal model for daily Rs estimation when these climatic data are

3.2. Comparison of overall performance of empirical models for predicting global solar radiation Fig. 7 shows the GPI values of each empirical model at different stations in China. In general, complex models showed the highest GPI at each station, followed by sunshine-based and temperature-based models. Among the twenty empirical models, the newly proposed C4 model showed the highest GPI and thus was ranked as the best model in China, with average MAE of 1.69 MJ m−2 d-1, RRMSE of 16.2%, and NS of 0.895 (Table 1). C3 and C2 were ranked second and third in terms of average GPI. Among the sunshine-based models, S3 was the best model and ranked as the fourth model among the twenty empirical models from the perspective of GPI, with average MAE of 1.79 MJ m−2 d-1, RRMSE of 17.2%, and NS of 0.884. As for temperature-based models, T3 provided the best accuracy and was ranked as the twelfth model among the twenty empirical models, with average MAE of 2.32 MJ m−2

Model

Model

GPI

C4 C3 C2 C1 T9 T8 T7 T6 T5 T4 T3 T2 T1 S7 S6 S5 S4 S3 S2 S1

1.5 1 0.5 0 -0.5 -1 -1.5 -2 -2.5

TMZ

TCZ

MPZ

SMZ

Fig. 7. The GPI values of each empirical model at different stations of China. 9

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Angstrom-type S3 was the best model, with average MAE of 1.79 MJ m−2 d-1, RRMSE of 17.2%, and NS of 0.884. In terms of temperature-based models, T3 provided the best results, with average MAE of 2.32 MJ m−2 d-1, RRMSE of 23.2%, and NS of 0.791. Overall, our proposed C4 that considered the effects of relative humidity, precipitation and air temperature provided the best prediction across different climates of China, and its required inputs are widely available in more than 700 national stations of China. Therefore, the C4 is highly recommended for accurately predicting Rs in various climates of China.

available. The present study found the empirical models based on sunshine data had better performance than models based on air temperature data. Trnka et al. [54] compared seven models for daily Rs estimation in lowlands of the Czech Republic and Austria, and reported that the sunshine-based models provided the best accuracy, followed by cloudbased, rainfall-based and temperature-based models. Mecibah et al. [61] also found sunshine-based models performed better than temperature-based models in Algeria. Their results agree with our findings in China. Sunshine duration is defined as the period during which direct Rs exceeds 120 W m−2, indicating n has a direct association with Rs. Cloud cover also directly affects the amount of Rs reaching the earth surface, thus sunshine-based and cloud-based models have relatively better results. Sunshine-based models are often utilized for Rs estimation, largely owing to the fact that sunshine data are easier to collect, and cloud-based models are sensitive to human biasing [55]. In China, all the national meteorological stations have sunshine duration measurements, but cloud data are not recorded. Among the sunshine-based models, the cubic Angstrom-type model (S3) provided the best performance in China, which can be recommended as an optimal model when only sunshine data are available. Fan et al. [32] found although S3 underestimated daily Rs in humid regions of China, it was recommended for daily Rs estimation with reasonable accuracy when only sunshine duration data were available. Chelbi et al. [17] reported the cubic model (S3 in the present study) had the best regression fit at four stations of Tunisia, which are in accordance with our findings. Although the sunshine-based models provide more accurate results compared with other models [56,57], temperature-based models can be recommended as alternative methods since air temperature data are more widely available compared with sunshine data [28,58]. In this study, the locally calibrated temperature-based models provided reasonable results, especially for T3, which had almost the same estimates as the sunshine-based models (S7). T3 was developed from the original Hargreaves model (T1) by adding geographical data (altitude) as an input parameter. As shown in Fig. 1, China has a wide range of altitude, from < −100 m in Turpan Depression to > 8000 m in Qinghai-Tibetan Plateau, thus adding altitude data can incorporate the effects of altitude on daily Rs estimation. Ajayi et al. [59] reported that previous studies didn’t consider geophysical parameters as the input to Rs models, which limited their applications. Adding these parameters made Rs models capable of describing the differences resulted from the changes in geographical locations. Although the models reported in this study can accurately estimate daily Rs across different climatic zones of China, recent solar energy applications require Rs data on an hourly or shortly basis. Thus, calibration and assessment of these models at hourly or shorter intervals are still needed in future studies. The most complete Rs data were used for model development/calibration and assessment in China, and the newly-developed model showed accurate prediction across different climatic zones of China. However, its potential in other climates outside China remains questionable. Thus, an assessment of the model in different climatic zones across the globe is required to better verify our results.

Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments This study was financially supported by the National Natural Science Foundation of China (No. 51922072, 51779161) and National Key Research and Development Program of China (No. 2016YFC0400206). We thank the Editor and three anonymous reviewers for their valuable suggestions and helpful comments to improve the quality of the manuscript. References [1] Fan J, Wu L, Zhang F, Cai H, Wang X, Lu X, et al. Evaluation and development of empirical models for estimating daily and monthly mean daily diffuse horizontal solar radiation for different climatic regions of China. Renew Sustain Energy Rev 2019;105:168–86. [2] Yao W, Zhang C, Wang X, Zhang Z, Li X, Di H. A new correlation between global solar radiation and the quality of sunshine duration in China. Energy Convers Manage 2018;164:579–87. [3] Zhou Y, Wang D, Liu Y, Liu J. Diffuse solar radiation models for different climate zones in China: Model evaluation and general model development. Energy Convers Manage 2019;185:518–36. [4] Calinoiu D, Stefu N, Boata R, Blaga R, Pop N, Paulescu E, et al. Parametric modeling: A simple and versatile route to solar irradiance. Energy Convers Manage 2018;164:175–87. [5] Hussain S, Alalili A. A hybrid solar radiation modeling approach using wavelet multiresolution analysis and artificial neural networks. Appl Energy 2017;208:540–50. [6] Feng Y, Cui N, Chen Y, Gong D, Hu X. Development of data-driven models for prediction of daily global horizontal irradiance in Northwest China. J Clean Prod 2019;223:136–46. [7] Zhang Y, Cui N, Feng Y, Gong D, Hu X. Comparison of BP, PSO-BP and statistical models for predicting daily global solar radiation in arid Northwest China. Comput Electron Agr 2019;164. 104905. [8] Manju S, Sandeep M. Prediction and performance assessment of global solar radiation in Indian cities: A comparison of satellite and surface measured data. J Clean Prod 2019;230:116–28. [9] Zang H, Cheng L, Ding T, Cheung KW, Wang M, Wei Z, et al. Estimation and validation of daily global solar radiation by day of the year-based models for different climates in China. Renew energy 2019;135:984–1003. [10] Feng Y, Gong D, Zhang Q, Jiang S, Zhao L, Cui N. Evaluation of temperature-based machine learning and empirical models for predicting daily global solar radiation. Energy Convers Manage 2019;198. 111780. [11] Liu Y, Zhou Y, Wang D, Wang Y, Li Y, Zhu Y. Classification of solar radiation zones and general models for estimating the daily global solar radiation on horizontal surfaces in China. Energy Convers Manage 2017;154:168–79. [12] Khorasanizadeh H, Mohammadi K. Prediction of daily global solar radiation by day of the year in four cities located in the sunny regions of Iran. Energy Convers Manage 2013;76:385–92. [13] Despotovic M, Nedic V, Despotovic D, Cvetanovic S. Review and statistical analysis of different global solar radiation sunshine models. Renew Sustain Energy Rev 2015;52:1869–80. [14] Angstrom A. Solar and terrestrial radiation. Quart J Roy Met Soc 1924;50:121–5. [15] Duzen H, Aydin H. Sunshine-based estimation of global solar radiation on horizontal surface at Lake Van region (Turkey). Energy Convers Manage 2012;58(58):35–46. [16] Bakirci K. Models of solar radiation with hours of bright sunshine: a review. Renew Sustain Energy Rev 2009;13:2580–8. [17] Chelbi M, Gagnon Y, Waewsak J. Solar radiation mapping using sunshine durationbased models and interpolation techniques: application to Tunisia. Energy Convers Manage 2015;101:203–15. [18] Almorox J, Hontoria C. Global solar radiation estimation using sunshine duration in

4. Conclusions Accurate prediction of Rs provides important information for the application and management of solar photovoltaic systems. The present study developed a novel empirical model for predicting Rs, and compared the model with nineteen existing empirical models that have been locally calibrated. To calibrate/establish and assess the models on a national scale, daily Rs and other meteorological data during 1994–2016 from 96 stations across four climates in China were used. It was found the newly proposed C4 accurately predicted Rs in various climates, and showed better prediction accuracy than the existing models, with average MAE of 1.69 MJ m−2 d-1, RRMSE of 16.2%, and NS of 0.895. When only sunshine data were available, the cubic 10

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