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energies Article

Comparison of Solar Radiation Models to Estimate Direct Normal Irradiance for Korea Hyun-Jin Lee 1, *, Shin-Young Kim 2 and Chang-Yeol Yun 2, * 1 2

*

School of Mechanical Engineering, Kookmin University, 77 Jeongneung-ro, Seongbuk-gu, Seoul 02727, Korea New and Renewable Energy Resource Center, Korea Institute of Energy Research, 152 Gajeong-ro, Yuseong-gu, Daejeon 34129, Korea; [email protected] Correspondence: [email protected] (H.-J.L.); [email protected] (C.-Y.Y.); Tel.: +82-2-910-5466 (H.-J.L.); +82-42-860-3746 (C.-Y.Y.)

Academic Editor: Francesco Calise Received: 10 February 2017; Accepted: 20 April 2017; Published: 30 April 2017

Abstract: Reliable solar radiation data are important for energy simulations in buildings and solar energy systems. Although direct normal irradiance (DNI) is required for simulations, in addition to global horizontal irradiance (GHI), a lack of DNI measurement data is quite often due to high cost and maintenance. Solar radiation models are widely used in order to overcome the limitation, but only a few studies have been devoted to solar radiation data and modeling in Korea. This study investigates the most suitable solar radiation model that converts GHI into DNI for Korea, using measurement data of the city of Daejeon from 2007 to 2009. After ten existing models were evaluated, the Reindl-2 model was selected as the best. A new model was developed for further improvement, and it substantially decreased estimation errors compared to the ten investigated models. The new model was also evaluated for nine major cities other than Daejeon from the standpoint of typical meteorological year (TMY) data, and consistent evaluation results confirmed that the new model is reliably applicable across Korea. Keywords: solar radiation model; global horizontal irradiance (GHI); direct normal irradiance (DNI); typical meteorological year (TMY); decomposition model

1. Introduction Continuous global energy issues, such as climate change and energy shortages, have increased the interest in energy-efficient buildings and solar energy systems. The energy simulation of such systems is critical for accurate performance evaluation and, ultimately, for optimal design. As the most important input to the energy simulation, reliable solar radiation data must be given in advance. The most useful solar radiation data are global horizontal irradiance (GHI), but direct normal irradiance (DNI) or diffuse horizontal irradiance (DHI) are also important. For example, irradiance on the surface of a solar collector or solar cell is determined when either DNI or DHI is given in addition to the GHI [1]. Note that the GHI, DNI, and DHI are interdependent and, thus, knowing two irradiances out of three is sufficient. Solar radiation measurements are often limited to a few locations or short-term periods in some countries. Furthermore, in general, availability of DNI (or DHI) data is much lower than that of GHI data because DNI measurement using a sun tracker costs more and needs more careful maintenance. Along with research efforts for energy-efficient buildings and solar energy systems, the demand for DNI data has increased significantly in Korea [2,3]. Even though the Korea Meteorological Administration (KMA) provides GHI data, as well as other meteorological data, such as dry bulb temperature and wind speed, DNI is not included [4]. When DNI data are not available, it is necessary to rely on a solar radiation model that accounts for regional climate characteristics. Many solar radiation models to Energies 2017, 10, 594; doi:10.3390/en10050594

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estimate DNI with GHI have been developed [5,6]. However, only a few studies have been devoted to solar radiation data and modeling for Korea, and even fewer studies to hourly DNI data. A lack of DNI measurement data has been a major obstacle to meaningful studies. Recently, Lee et al. [7] modeled GHI with cloud cover data for major cities in Korea and, successively, Lee et al. [8] reported a solar radiation model developed for estimating DNI with GHI. However, the model of Lee et al. [8] tends to underestimate DNI data such that DNI values exceeding 750 W/m2 seldom occur. Long-term, 20 or 30 years, solar radiation data collected on an hourly basis are desirable to reflect climatic characteristics at a specific location and obtain reliable simulation results [9]. Since a direct handling of massive data is burdensome, representative datasets generated from raw, long-term data are often used. The representative data, usually referred to as the typical meteorological year (TMY) data, contain 8760 hourly values of meteorological elements for the one-year duration [10,11]. The Korean Solar Energy Society (KSES) has shared the TMY data of seven cities in Korea but, unfortunately, its TMY datasets reveal unreasonably low DNI values [12]. As a result, the users relying on the TMY data from KSES have a risk of underestimating DNI effects in their energy simulations. This study aims at investigating solar radiation models, including a newly developed model, for the estimation of DNI from GHI in Korea and providing a guideline for the selection of solar radiation models in energy simulations. In the beginning, ten well-known solar radiation models are evaluated with three years of data from the city of Daejeon in Korea. Then, a new model based upon the quasi-physical approach proposed by Maxwell [13] is presented. Finally, from the standpoint of the TMY data, variations of solar irradiance due to the solar radiation model are analyzed, and the nationwide extension of the new model is investigated. 2. Evaluation of Existing Solar Radiation Models KMA as a national representative provides meteorological data over 100 locations [4]. GHI is also available at some locations, but DNI is not available at all. Meanwhile, Korea Institute of Energy Research (KIER) measured both GHI and DNI in the city of Daejeon for research purpose. In this study, KIER measurement data from 2007 to 2009 were used for evaluation of solar radiation models and for development of a new model. The city of Daejeon is located approximately at the center of Republic of Korea, and its latitude, longitude, and altitude are 36.18◦ , 127.24◦ , and 77.1 m, respectively. The pyranometer for GHI measurement was a CMP 11 model from Kipp & Zonen Company in Delft, the Netherlands whereas the pyrheliometer for DNI measurement was a CHP 1 model with a SOLYS 2 sun tracker from the same company. Both GHI and DNI were measured every minute and averaged over 60 min to get hourly data. The uncertainties originated from both of the sensors are less than 1%. Based on references [14,15] and experiences, the estimated measurement uncertainties of GHI and DNI are generally 3% on the average and 5% at most. For the three years, the average percentages of missing GHI and DNI data were 2.3% and 1.9%, respectively. The data pair that misses either GHI or DNI and was measured when the zenith angle of the sun was larger than 85◦ were excluded. The number of the remained pairs of GHI and DNI measurements totals 11,928. Solar radiation models to estimate DNI can be classified into two categories, parametric and decomposition models [5]. In parametric models, solar radiation is obtained from other meteorological parameters, such as cloud cover, atmospheric turbidity, pressure, and water content. On the other hand, decomposition models rely on correlations between global, direct, and diffuse components of solar radiation. Whereas parametric models require detailed information of the atmospheric conditions, decomposition models are relatively easy to use once the GHI is known. Out of many decomposition models, ten were selected in this study: Orgill and Hollands [16], Vignola and McDaniels [17], Louche et al. [18], Lee et al. [8], Lam and Li [19], CIBSE [20], Erbs et al. [21], Maxwell [13], and two from Reindl et al. [22]. These models are widely used for modeling solar radiation, e.g., in relevant textbooks [1] and in model comparison studies [5,8,23]. The selected models were evaluated by comparing the modeled and the measured DNI data.

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Most of the decomposition models use correlations between global, direct, and diffuse solar radiation. Once the global, direct, diffuse, and extraterrestrial irradiances on a horizontal surface are given by It , In , Id , and I0 , respectively, three non-dimensional parameters—the clearness index, k t = It /I0 , the direct beam transmittance, k n = In /I0 , and the diffuse fraction, k d = Id /It —can be defined. Note that It represents GHI. DNI is denoted by Gn and is related to In by the equation of In = Gn cos θz , in which θz is the zenith angle of the sun. Usually, the correlations of a solar radiation model render k n or k d as a function of k t in separate intervals divided by k t values. If a model yields k d rather than k n , DNI is calculated via Gn = It (1 − k d )/ cos θz after k d is obtained. Only three models among the ten models evaluated in this study are presented in the following for brevity. The Lee model [8] is selected because it was recently developed with measurement data from the city of Daejeon in 2009. The Reindl-2 [22] and Maxwell [13] models are selected because they show good performances compared to the others, which will be demonstrated later. The Reindl-2 model was developed with measurement data at five European and North American locations, and the term ofcos θz is added as the second input parameter besides k t . The Maxwell model will be explained in the next section. The rest of the other models can be found in the corresponding articles. (1) Lee model [8]: k d = 0.92 for k t ≤ 0.2

(1)

k d = 0.691 + 2.4306k t − 7.3371k2t + 4.7002k3t for 0.2 < k t

(2)

(2) Reindl-2 model [22]: k d = 1.02 − 0.254k t + 0.0123 cos θz for k t ≤ 0.3

(3)

k d = 1.4 − 1.749k t + 0.177 cos θz for 0.3 < k t < 0.78

(4)

k d = 0.486k t − 0.182 cos θz for k t ≥ 0.78

(5)

k n = k nc − ∆k n

(6)

k nc = 0.866 − 0.122m a + 0.0121m2a − 0.000653m3a + 0.000014m4a

(7)

∆k n = A + B exp(Cm a )

(8)

A = 0.512 − 1.56k t + 2.286k2t − 2.222k3t for k t ≤ 0.6

(9)

A = −5.743 + 21.77k t − 27.49k2t + 11.56k3t for k t > 0.6

(10)

B = 0.37 + 0.962k t for k t ≤ 0.6

(11)

B = 41.4 − 118.5k t + 66.05k2t + 31.9k3t for k t > 0.6

(12)

C = −0.28 + 0.923k t − 2.048k2t for k t ≤ 0.6

(13)

C = −47.01 + 184.2k t − 222.0k2t + 73.81k3t for k t > 0.6

(14)

(3) Maxwell model [13]:

Except for the Maxwell model, the correlations for k n or k d are a polynomial function of k t . The only difference lies in the coefficients that account for climate characteristics where the solar irradiance data were measured and used for developing each model. Meanwhile, the Maxwell model possesses a different functional form because the quasi-physical approach is applied; that is, it, in part, reflects the physics involved in the atmospheric transmission of solar radiation. In order to identify proper models for Korea, DNI values were calculated with the ten candidate models using the selected GHI data as the input and compared with actual DNI measurement data. Linear regression analysis between measured and modeled DNI data was conducted for each model. Based on the regression analyses, the ten models were divided into three groups from the standpoint

possesses a different functional form because the quasi-physical approach is applied; that is, it, in part, reflects the physics involved in the atmospheric transmission of solar radiation. In order to identify proper models for Korea, DNI values were calculated with the ten candidate models using the selected GHI data as the input and compared with actual DNI measurement 4data. Energies 2017, 10, 594 of 12 Linear regression analysis between measured and modeled DNI data was conducted for each model. Based on the regression analyses, the ten models were divided into three groups from the standpoint that the estimation estimation is is larger larger than than the the measurement measurementand andhigh highDNI DNIvalues valuesof ofmore morethan than750 750W/m W/m22 are properly estimated. estimated. The Thefirst firstgroup, group,which whichincludes includes Orgill, Vignola, Louche, Erbs, thethe Orgill, thethe Vignola, thethe Louche, the the Erbs, the the Reindl-1, Lam models, estimates DNIvalues valueslarger largerthan thanthe the measurement measurement and yields Reindl-1, andand thethe Lam models, estimates DNI yields unacceptably time to time. Figure 1a shows the scatterplot obtained with thewith Vignola unacceptably high highvalues valuesfrom from time to time. Figure 1a shows the scatterplot obtained the 2 , but such model as a representative of the first group. Some estimated DNI values exceeded 1000 W/m Vignola model as a representative of the first group. Some estimated DNI values exceeded 1000 2, butvalues high are very Korea. W/mDNI such high DNI rarely valuesobserved are very in rarely observed in Korea.

Figure 1. Scatterplots of direct normal irradiance (DNI) measurements and estimations, in which the Figure 1. Scatterplots of direct normal irradiance (DNI) measurements and estimations, in which the red curves curves represent represent the the linear linear regression regressionfits: fits: (a) (a) Vignola; Vignola;(b) (b)Lee; Lee;(c) (c)Reindl-2; Reindl-2;and and(d) (d)Maxwell. Maxwell. red

On the contrary, the Lee and the CIBSE models, which belong to the second group, seldom yield thevalues, contrary, the Leeinand the CIBSE models, which belong the to the second group, seldom yield high On DNI resulting underestimation. Figure 1b shows scatter plot obtained with the high DNI values, resulting in underestimation. Figure 1b shows the scatter plot obtained with the Lee model as a representative of the second group. The linear regression trend beyond DNI values Lee model 750 as aW/m representative of the second group.that Theadditional linear regression trend values 2 slopes downward, exceeding suggesting correlation atbeyond large ktDNI should be 2 exceeding 750 W/m slopes downward, suggesting that additional correlation at large k t should be introduced. Even though the Lee model was developed with the DNI measurement data in Korea [8], introduced. Even though the Lee model was developed with the DNI measurement data in Korea [8], its underestimation implies that one-year data used for the model development are not enough for its underestimation implies that one-year data used for the model development are not enough for proper estimation. proper estimation. The third group includes the Reindl-2 and the Maxwell models. These models estimate DNI The third group includes the Reindl-2 and the Maxwell models. These models estimate DNI values values larger than those measured, but they do not pose extreme behaviors, in contrast to those in larger than those measured, but they do not pose extreme behaviors, in contrast to those in the first and the first and the second groups. Lave et al. [23] demonstrated that the Reindl-2 and the Maxwell the second groups. Lave et al. [23] demonstrated that the Reindl-2 and the Maxwell models use the term models use the term of cos θz (via air mass in the Maxwell model) in addition to kt and, thus, they of cos θz (via air mass in the Maxwell model) in addition to k t and, thus, they outperform other models kt this outperform other models that use only, which consistent frequency, with this study. The observation that use k t only, which is consistent with study. Theisobservation normalized by dividing frequency, normalized bydata dividing with thecalculated total number of datatopoints, wasofcalculated to with the total number of points, was according the level the DNI. according Histograms the level of the DNI. Histograms of the normalized data shown in Figure 2 indicate that the Reindl-2 of the normalized data shown in Figure 2 indicate that the Reindl-2 model suitably estimates the model suitably estimates the observation frequency DNI in each bin. On the tends other observation frequency of DNI values in each DNI bin.ofOn thevalues other hand, theDNI Maxwell model 2 hand, the Maxwell overestimate values . in the bins when from the 750scatterplots to 950 W/min2. to overestimate DNImodel valuestends in theto bins from 750 toDNI 950 W/m However, However, the scatterplots in Figure 1c,dthe areReindl-2 compared, it is clear thatiswith the Reindl-2 model Figure 2c,dwhen are compared, it is clear that with model the DNI occasionally estimated the DNI is occasionally estimated too high. too high.

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Figure 2. Histograms of DNI measurement and estimation, in which frequency is normalized by

Figure 2. Histograms of DNI measurement and estimation, in which frequency is normalized by dividing by the total number of data points. dividing by the total number of data points.

The coefficient of determination (R2), the mean bias error (MBE), the root mean square error

The coefficient of determination (R2 ), the mean biascalculated error (MBE), the rootthe mean square error (RMSE), and the median absolute deviation (MAD) were to distinguish goodness-of2 values remarkably, roughly fit of each model and summarized in Table 1. The models do not alter R (RMSE), and the median absolute deviation (MAD) were calculated to distinguish the goodness-of-fit in the variation range of 2%, and the R21. ofThe the Reindl-2 the best. The MBE is defined roughly as the of each model and summarized in Table models model do notisalter R2 values remarkably, in sum of the measurements minus the estimation. Therefore, a negative value of MBE means 2 the variation range of 2%, and the R of the Reindl-2 model is the best. The MBE is defined as the sum of overestimation of a model, and a positive value means the opposite. Whereas the MBE is a good the measurements minus the estimation. Therefore, a negative value of MBE means overestimation of a measure for yearly-based estimation, the RMSE is for hourly-based estimation. If the best model was model, and a positive value means the opposite. Whereas the MBE is a good measure for yearly-based to be selected from the ten investigated models, the first selection criterion is to exclude extreme estimation, the Accordingly, RMSE is for out hourly-based If the statistics best model was and to be selected from the ten behaviors. of the third estimation. group, with better of MBE RMSE the Reindl-2 investigated models, the first selection criterion is to exclude extreme behaviors. Accordingly, model becomes the most suitable for estimating the DNI in Korea. The unacceptable underestimationout of the third with betterthe statistics of MBE and RMSE thethough Reindl-2 model the Note mostthat suitable of thegroup, DNI disqualifies CIBSE and the Lee models even their MBEsbecomes are smaller. the RMSEthe values roughly range fromunderestimation 25% to 35% of of thethe mean value and for estimating DNIininTable Korea.1 The unacceptable DNIDNI disqualifies theare CIBSE significantly larger than the measurement approximately 3%. values in Table 1 roughly and the Lee models even though their MBEsuncertainty are smaller.byNote that the RMSE range from 25% to 35% of the mean DNI value and are significantly larger than the measurement Table 1. Regression analysis of each model with a linear polynomial of y = C1 x + C0 : the coefficient uncertainty by approximately 3%. of determination (R2), the mean bias error (MBE) as the measurement minus estimation, the root mean square error (RMSE), and the median absolute deviation (MAD).

Table 1. Regression analysis of each model with a linear polynomial of y = C1 x + C0 : the coefficient of Group Modelbias error (MBE) Cas 1 the measurement C0 R2minus MBE RMSE determination (R2 ), the mean estimation, the rootMAD mean and Hollands −11.91 75.16 28.14 square errorOrgill (RMSE), and the [16] median absolute0.99 deviation15.70 (MAD). 92.8% 92.3% 93.0% 2 R92.9% 92.8% 92.9% 92.3% 91.8% 93.0% 92.7% 92.9% 92.9% 92.9% 93.7% 91.8% 93.2% 92.7% 94.5%

−19.95 −45.45 MBE −20.92 −11.91 −23.17 −19.95 −39.16 −45.45 16.23 −20.92 6.93 −23.17 −17.82 −39.16 −37.05 16.23 −2.04

Reindl of et al. [22]—2 3. III Development a New Solar Radiation0.97 Model 26.24 1.04 26.78 Maxwell [13]

93.7% 93.2%

3.1. Methodology New

94.5%

I Group

I II III

II

Vignola and McDaniels [17] Louche et al. [18] Model Erbs et al. [21] Reindl et al. [22]—1 [16] Orgill and Hollands Vignola and McDaniels [17] Lam and Li [19] Louche et [8] al. [18] Lee et al. Erbs et al. CIBSE [20][21] Reindl et al. [22]—1 Reindl et al. [22]—2 Lam and Li [19] Maxwell [13] Lee et al. New [8]

CIBSE [20]

0.96 1.06 C1 1.03 0.990.98 0.960.92 1.060.78 1.030.89 0.98 0.97 0.92 1.04 0.780.96

0.89

0.96

30.16 29.33 C012.08 15.70 27.22 30.16 60.77 29.33 42.12 12.08 20.77 27.22 26.24 60.77 26.78 42.12 12.16

20.77

12.16

92.9%

78.11 92.02 RMSE 80.07 75.16 77.10 78.11 86.37 92.02 85.43 80.07 72.63 77.10 70.29 86.37 84.95 85.43 63.37

32.58 31.70 MAD 29.62 28.14 30.20 32.58 37.98 31.70 37.80 29.62 30.07 30.20 26.84 37.98 30.38 37.80 26.57

−17.82 −37.05

70.29 84.95

26.84 30.38

−2.04

63.37

26.57

6.93

72.63

30.07

In the previous section, the Reindl-2 model turned out to be the most suitable model. However,

3. Development ofsatisfactory. a New Solar Radiation Model it is not entirely Above all, the MBE is still large, and some outliers can occur, as indicated

in Figure 1c. An effort to improve the solar radiation model was made. The Reindl-2 and the Maxwell

3.1. Methodology models naturally became good candidates due to the aforementioned comparison results. The ReindlIn the previous section, the Reindl-2 model turned out to be the most suitable model. However, it is not entirely satisfactory. Above all, the MBE is still large, and some outliers can occur, as indicated in Figure 1c. An effort to improve the solar radiation model was made. The Reindl-2 and the Maxwell models naturally became good candidates due to the aforementioned comparison results. The Reindl-2

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model uses a simple curve-fitting approach. Hence, the modification based on the quasi-physical 2 model uses a simple curve-fitting approach. Hence, the modification based on the quasi-physical approach of the Maxwell model [13] is likely to be a better estimate for Korea. approach of the Maxwell model [13] is likely to be a better estimate for Korea. Maxwell’s quasi-physical approach was established upon the three following assumptions: First, Maxwell’s quasi-physical approach was established upon the three following assumptions: First, air mass, m a , ismthe, is dominant parameter affecting the relationship between k and k . Second, using a air mass, the dominant parameter affecting the relationship betweenn kn andt kt . Second, a physical model to calculate clear-sky k n will provide a physically-based reference, from which changes using a physical model to calculate clear-sky kn will provide a physically-based reference, from in k n can be calculated. Third, the seasonal, annual, and climate variations in the relationship between which changes in kn can be calculated. Third, the seasonal, annual, and climate variations in the k n and k t are entirely accounted for by parametric functions in k t that relate ∆k n to m a , cloud cover, relationship between kn and kt are entirely accounted for by parametric functions in kt that relate and precipitable water vapor. The second hypothesis explains Equation (6) above. If clear-sky k n is Δ k to m a , cloud cover, and precipitable water vapor. The second hypothesis explains Equation (6) defined nas k nc for limiting values, ∆k n represents the deviation from it. Maxwell [13] adopted the above. If clear-sky kn is defined as knc for limiting values, Δ k n represents the deviation from it. Bird clear-sky model for k nc , which corresponds to Equation (7). According to the first and third Maxwell [13] adopted the Bird clear-sky model for knc , which corresponds to Equation (7). According hypotheses that were obtained from statistical analyses, ∆k n has a functional form, as in Equation (8). to the first and third hypotheses that were obtained from statistical analyses, Δ k has a functional The coefficients A, B, and C in Equation (8) were determined by fitting solarn radiation data from form, as in Equation (8). The coefficients A, B, and C in Equation (8) were determined by fitting solar Atlanta, Georgia, USA in 1981. radiation data from Atlanta, Georgia, USA in 1981. Development of a new solar radiation model based on the quasi-physical approach starts from Development of a new solar radiation model based on the quasi-physical approach starts from accepting Equations (6)–(8). theremaining remaining to determine the coefficients A,using B, and C accepting Equations (6)–(8).Then, Then, the tasktask is tois determine the coefficients A, B, and C usingsolar solarradiation radiation data from Korea, which willrise givetorise to correlations similar to Equations (9)–(14). data from Korea, which will give correlations similar to Equations (9)–(14). The The first todivide dividektk tinto intothe the intervals whose median values increase 0.05, at starting at 0.25, first step step isisto intervals whose median values increase by 0.05,by starting 0.25, and and in each interval of k the regression analysis between ∆k and m is carried out. For example, n m a isa carried out. For example, in each interval of kt t the regression analysis between Δ k n and Figure 3 shows the scatterplot fromfrom the regression analysis at k t at= 0.25, in which the results represent kt = 0.25 , in which the results Figure 3 shows the scatterplot the regression analysis very represent cloudy conditions, as indicated by the range of 0.225 ≤ k < 0.275. Therefore, they correspond to t 0.225 ≤ kt < 0.275 . Therefore, they very cloudy conditions, as indicated by the range of the limiting caseto of the ∆k nlimiting = k nc , implying the extraterrestrial solar radiation is completely absorbed correspond case of Δkthat n = knc , implying that the extraterrestrial solar radiation is or scattered by the atmosphere and, thus, the direct normal component solar component radiation isofessentially completely absorbed or scattered by the atmosphere and, thus, the directof normal solar zero. radiation The black thin curve in Figure 3 represents k in Equation (7), and the fact that all thethe data in is essentially zero. The black thin curve nc in Figure 3 represents knc in Equation (7),ofand Figure 3 are curve supports this statement. fact that located all of thebelow data inthis Figure 3 are located below this curve supports this statement.

k and the air mass at k = 0.25 , in which the solid red curve represents Figure 3. Scatterplot of Δand Figure 3. Scatterplot of ∆k n n the air mass at k t =t 0.25, in which the solid red curve represents the the regression fit and thethin blackcurve thin curve represents in Equation regression fit and the black represents k nc ink ncEquation (7). (7).

The similar regression analyses (not presented) to determine the coefficients A, B, and C were

The similar regression analyses (not presented) to determine the coefficients A, B, and C were repeated at each interval of kt until kt = 0.70 . Note that Maxwell [13] presented the regression repeated at each interval of k t until kkt = 0.70. Note that Maxwell [13] presented the regression analyses analyses up to kt = 0.8 because t > 0.81 was not available in the solar radiation data from Atlanta. up to k t = 0.8 because k t > 0.81 was not available in the solar radiation data from Atlanta. For Daejeon, For Daejeon, Korea, there are some solar radiation data even when kt > 0.75 , but they are not enough Korea, there are some solar radiation data even when k t > 0.75, but they are not enough to derive to derive statistically meaningful fits. The reason is that the solar radiation in the southeastern region statistically meaningful fits. The reason is that the solar radiation in the southeastern region of the US of the US is more abundant than in Northeastern Asia. After the coefficients A, B, and C were is more abundant thaninterval in Northeastern Asia. After the coefficients A, B, C were at determined at each of kt , another regression analysis was carried outand in order to fitdetermined A, B, and each C interval of k , another regression analysis was carried out in order to fit A, B, and C as a function t as a function of kt . The development procedure can be summarized as follows: of k t . The development procedure can be summarized as follows: • Calculate cos θz , m a , and I0 on an hourly basis ( I0 is calculated based on [13]). kn with • Calculate cos θkzt, m I0 onthe anmeasured hourly basis calculated on [13]). • Calculate and GHI,(II0t isand DNI, G nbased . a , and • Calculate k t and k n with the measured GHI, It and DNI, Gn . • Divide the intervals of k t and the group data by the interval.

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Conduct a regression analysis between ∆k n and m a and determine the coefficients A, B, and C in each group. Energies 2017, 10, 594intervals of k and the group data by the interval. 7 of 12 • Divide the t Conduct another regression analysis to fit the coefficients A, B, and C as a function of k • Conduct a regression analysis between Δ k n and m a and determine the coefficients A, B,t .and •

Divide the intervals of kt and the group data by the interval. C in each group. m a and determine • Conduct a regression analysis between thefunction coefficients B, and Figure 4 shows that the coefficients A and BΔ kcan be expressed as a linear andA, a third-order n and • Conduct another regression analysis to fit the coefficients A, B, and C as a function of kt . polynomial function over the entire range, respectively. Meanwhile, the fitting of the coefficient C with C in each group. Figure 4 shows the coefficients Atoand B separating can be expressed asthe aC linear function 0.25 andkta≤ • Conduct anotherthat regression analysis fitfor the coefficients B, and as a function of . thirda third-order polynomial function is required kA, two ranges, k t ≤ 0.50 t into order polynomial function over the entire range, respectively. Meanwhile, the fitting of the coefficient and 0.50 ≤ k ≤ 0.70. Similarly to the derivation of the original model by Maxwell [13], extrapolation t Figure 4 shows that the coefficients A and can be expressed as a linear function and aranges, thirdk into C with a third-order polynomial function is Brequired for separating the two is also applied in thisfunction study over when > 0.75. Finally, the correlations int the solar radiation model order polynomial thek tentire range, respectively. Meanwhile, the fitting of the coefficient 0.25 ≤ kt ≤ 0.50 and 0.50 ≤ kt ≤ 0.70 . Similarly to the derivation of the original model by Maxwell developed the data from Daejeon, Korea can be written follows: kt into the two ranges, C withwith a third-order polynomial function is required for as separating [13], extrapolation is also applied in this study when kt > 0.75 . Finally, the correlations in the solar 0.25 ≤ kt ≤ 0.50 and 0.50 ≤ kt ≤ 0.70 . Similarly to the derivation of the original model by Maxwell radiation model developed with the data Daejeon, Koreat can be written as follows: A =from 0.3452 − 0.3782k (15) [13], extrapolation is also applied in this study when kt > 0.75 . Finally, the correlations in the solar A = 0.3452 − 0.3782 kt can be written as follows: (15) radiation model developed with the data from Daejeon, Korea 2 3

B = 0.5329 + 0.2676k t − 0.0216k t + 0.1584k t

(16)

2 3 B = 0.5329 +A0.2676 kt −−0.0216 = 0.3452 0.3782kkt t + 0.1584 kt

(16) (15)

C = −0.2117 − 0.0513 kt + 1.2976 kt2 − 3.3222 kt3 forkt3kt ≤ 0.5 B = 0.5329 + 0.2676 kt − 0.0216 kt2 + 0.1584

(17) (16)

CC==0.7221 − 10.2801 ktk+t +30.3285 27.9766 forkt 0.5 < kt −0.2117 − 0.0513 1.2976 ktt2 − 3.3222 kt3ktfor ≤ 0.5

(18) (17)

C = 0.7221 − 10.2801kt + 30.3285 kt2 − 27.9766 kt3 for 0.5 < kt

(18)

C = −0.2117 − 0.0513k t + 1.2976k2t − 3.3222k3t for k t ≤ 0.5 C = 0.7221 − 10.2801k t + 30.3285k2t2 − 27.9766k3 3t for 0.5 < k t

(17) (18)

Figure 4. Regression analysis to fit the coefficients A, B, and C according to Equation (8).

Figure 4. Regression analysis to fit the coefficients A, B, and C according to Equation (8).

3.2. ResultsFigure 4. Regression analysis to fit the coefficients A, B, and C according to Equation (8).

3.2. Results

The scatterplot with the new model in Figure 5 is qualitatively similar to the counterpart in 3.2. Results

The scatterplot with thereduces new model in Figure 5 is estimated qualitatively tomodel the counterpart Figure 1d. It significantly the occasional outliers by thesimilar Reindl-2 shown in in The scatterplot with the new model in Figure 5 is qualitatively similar to the counterpart in in Figure 1c. The goodness-of-fit is greatly improved with the new model. Table 1 shows the MBEshown and Figure 1d. It significantly reduces the occasional outliers estimated by the Reindl-2 model Figure 1d. It significantly reduces the occasional outliers estimated by the Reindl-2 model shown in the1c. RMSE significantly improved respect to thethe Reindl-2 model, by a factor of 8.7 and by Figure The are goodness-of-fit is greatlywith improved with new model. Table 1 shows the MBE and Figure 1c. The goodness-of-fit is greatly improved with in theFigure new model. Table 1 shows the MBE and 9.8%, respectively. However, the normalized frequency 2 demonstrates that the new model the RMSE are significantly improved with respect to the Reindl-2 model, by a factor of 8.7 and by the RMSE poorer are significantly improved withinrespect to the Reindl-2 model, by750 a factor is slightly than the Reindl-2 model estimating high DNI exceeding W/m2.of 8.7 and by 9.8%,9.8%, respectively. However, thethe normalized frequency demonstrates thatthe the new model is respectively. However, normalized frequencyin inFigure Figure 22 demonstrates that new model 2. slightly poorer than the Reindl-2 model in estimating high DNI exceeding 750 W/m 2 is slightly poorer than the Reindl-2 model in estimating high DNI exceeding 750 W/m .

Figure 5. Scatterplot of the DNI measurement and estimation from the new model. Figure 5. Scatterplot of the DNI measurement and estimation from the new model.

Figure 5. Scatterplot of the DNI measurement and estimation from the new model.

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As summarized in Table 2, the error of the yearly sum of DNI supports that the new model remarkably error in Energies improves 2017, 10, 594 estimation performance. For all three years—2007, 2008, and 2009—the 8 of 12 Energies 2017, 10, 594 8 of 12 estimation does not exceed 1.5%. Figure 6 shows variations of errors for estimating the monthly sum. Tableto 2, smaller the errorerrors of thecompared yearly sumto ofthe DNIMaxwell supportsand thatthe the Reindl-2 new model In general,As thesummarized new modelinleads models. As summarized in Table 2, the error of the yearly sumyears—2007, of DNI supports that 2009—the the new model remarkably improves estimation performance. For all three 2008, and errormodel The largest error for the Maxwell model is 35.5% in February 2009 and 24.3% for the Reindl-2 remarkably performance. For all three years—2007, 2008, and 2009—the error in estimationimproves does notestimation exceed 1.5%. Figure 6 shows variations of errors for estimating the monthly in Julyin2007. However, theexceed largest error for 6the newvariations model isofonly 15.7%estimating in Februarymonthly 2009. Errors estimation does 1.5%. Figure shows errors sum. In general, thenot new model leads to smaller errors compared to thefor Maxwell and the the Reindl-2 are remarkably decreased around the winter and the spring seasons, they are increased in sum. In general, theerror new model leads to smaller to 2009 thewhereas Maxwell and models. The largest for the Maxwell model iserrors 35.5%compared in February and 24.3% forthe theReindl-2 Reindllargest error forand the the Maxwell model is in February and 24.3% for the Reindlseveralmodels. the2007. summer the autumn seasons. new model toinestimate DNI 2 months model The inof July However, largest error for35.5% the The new model is2009 onlytends 15.7% February 2009.values 2Errors model in July 2007. However, thearound largestmodels error forregardless theand newthe model is only 15.7% in February smaller thanare the Maxwell and the Reindl-2 ofspring the month, thereby resulting remarkably decreased the winter seasons, whereas they2009. are in the areinremarkably aroundin the winter the spring seasons, increased several of the summer and the autumn new modelwhereas tends to they estimate overallErrors downward shiftsmonths of decreased monthly errors Figure 6. and In seasons. order toThe investigate the effects of are seasonal increased in several months of Maxwell the summer and the autumnmodels seasons.regardless The new model tends to estimate DNI values smaller than the and the Reindl-2 of the month, thereby positions of the sun, the hourly errors in June 2009 and December 2009 against the solar zenith angle, DNI values smaller than the Maxwell and Reindl-2 models regardless of thetomonth, thereby in the overall downward shifts of the monthly errors in Figure 6. In investigate the θz , areresulting presented in Figure 7. The smaller estimation by the new model is order essentially independent of resulting in the overall downward shifts of monthly errors in Figure 6. In order to investigate the effects of seasonal positions of the sun, the hourly errors in June 2009 and December 2009 against the θz , which is generally true for the non-presented months as well. Consequently, it can be concluded effects of seasonal the sun, the June 2009 and December θz , are ofpresented solar zenith angle,positions in hourly Figure errors 7. Theinsmaller estimation by the2009 newagainst modelthe is that the underestimation thepresented new model consistently occurs estimation throughout year and contributes solar zenith angle, θz of , are in Figure 7. The smaller by athe new model is θ essentially independent of z , which is generally true for the non-presented months as well. to decreasing largest monthly error. isSince the months errors are months decreased dominate generally true for where the non-presented as well. essentiallythe independent of θz , which Consequently, it can be concluded that the underestimation of the new model consistently occurs those where errors are increased, performance in the yearly estimation is improved. In spite of better Consequently, it can concluded that the underestimation of the new model throughout a year andbecontributes to decreasing the largest monthly error. Sinceconsistently the months occurs where estimation in yearly irradiances, caution must be paid when the new model is applied forwhere estimating throughout a year and contributes to decreasing the largest monthly error. Since the months errors are decreased dominate those where errors are increased, performance in the yearly estimation errors are decreased dominate those where errors are increased, performance in the yearly estimation monthly irradiances around the summer and the autumn seasons. is improved. In spite of better estimation in yearly irradiances, caution must be paid when the new is improved. In spite of better estimation in yearly irradiances, be paid when the new model is applied for estimating monthly irradiances around thecaution summermust and the autumn seasons. model is appliedTable for estimating monthly around the summer autumn seasons. 2. Relative errors irradiances for estimating the yearly sum ofand the the DNI. Table 2. Relative errors for estimating the yearly sum of the DNI. Table 2. Relative errors for2007 estimating2008 the yearly2009 sum of the DNI. Model Model 2007 2008 2009 Model 2007 2008 12.3% 12.3% 2009 Maxwell 15.3% 13.2% Maxwell 15.3% 13.2% Reindl-2 7.7% 5.7% Maxwell 15.3% 13.2% Reindl-2 7.7% 5.7% 5.5% 12.3% 5.5% New 1.5% − 0.1% Reindl-2 7.7% 5.7% 5.5% New 1.5% −0.1% −0.5% −0.5% New 1.5% −0.1% −0.5%

Figure 6. Relative errors for estimating the monthly sum of the DNI.

Figure 6. Relative errors themonthly monthly sum of the DNI. Figure 6. Relative errorsfor forestimating estimating the sum of the DNI.

Figure 7. Effects of the solar zenith angle on errors for estimating the hourly DNI, in which the dotted Figure 7. Effects of the solar zenith angle on errors for estimating the hourly DNI, in which the dotted

the solidoflines the averages of the Reindl-2 and the newthe models in each interval Figureand 7. Effects the indicate solar zenith angle on errors for estimating hourly DNI,separate in which the dotted and the solid lines angle, indicate the averages the Reindl-2 new models of the solar zenith respectively: (a)ofJune 2009; andand (b) the December 2009. in each separate interval and the lines indicate averages (a) of June the Reindl-2 the new models in each separate interval of solid the solar zenith angle,the respectively: 2009; andand (b) December 2009. of the solar zenith angle, respectively: (a) June 2009; and (b) December 2009.

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4. Variations of Solar Irradiance in TMY Data Since the DNI measurement data of other cities in Korea were not accumulated systematically and sufficiently, the previous results obtained with data from the city of Daejeon cannot be directly validated for the nationwide extension. The ten major cities in Korea, including Daejeon, are considered: Busan, Cheongju, Daegu, Gangneung, Gwangju, Incheon, Jeju, Jeonju, and Seoul. The solar radiation models were compared in terms of the TMY data instead of the data of a specifically-selected year. Since the TMY data represent regional climatic characteristics for the one-year duration, they are generally used for energy simulations and facilitate evaluation [10]. In the following, TMY data of each city were generated after DNI was estimated with solar radiation models. Then, variations of solar irradiance were investigated to identify similar trends across the ten major cities. A TMY dataset consists of the months selected from individual years and concatenated to form a complete year. In this study, the method of National Renewable Energy Lab (NREL) in the US was adopted for generation of TMY data [10]. The first step is to select five candidate months close to the long-term weather characteristics for each month. For the selection, monthly cumulative distribution functions (CDFs) for the daily data of a weather element are compared with the long-term CDF. According to the Finkelstein-Schafer (FS) statistics in Equation (19) [24], the deviation of the CDF of a specific month from the long-term CDF is calculated for the j-th weather element, in which the subscript n indicates the number of days in a month and xi denotes daily data on i-th day: FS j =

1 n CDFlong−term ( xi ) − CDFmonthly ( xi ) ∑ n i =1

(19)

The ten weather elements in Table 3 are judged to be more important than the others, and a weighted sum of the FS statistics is used to select the five candidate months. 10

WFS j =

∑ w j FSj

(20)

j =1

Table 3. Weighting factors for the Finkelstein-Schafer (FS) statistics. Weather Element

Weighting Factor

Max Dry Bulb Temperature

1/20

Min Dry Bulb Temperature Mean Dry Bulb Temperature Max Dew Point Temperature Min Dew Point Temperature

1/20 2/20 1/20 1/20

Weather Element Mean Dew Point Temperature Max Wind Velocity Mean Wind Velocity Global Horizontal Irradiance Direct Normal Irradiance

Weighting Factor 2/20 1/20 1/20 5/20 5/20

In the second step, the five candidate months are ranked with respect to the closeness of the month to the long-term mean and median. The third step is to check the persistence of the mean dry bulb temperature and daily GHI so that the month that exhibits exceptional weather patterns, such as the longest run or zero runs of consecutive warm days, are excluded. The highest-ranked candidate month from the second step that survives in the third step is selected as the typical meteorological month. Finally, in the fourth step, the 12 typical meteorological months are concatenated to form a complete year. TMY data of the ten major cities in Korea were generated with 20-year (1991–2010) weather data from KMA [4]. Figure 8 shows that the yearly sums of the GHI in the TMY datasets do not change noticeably by solar radiation models, indicating their variations of not more than 2.1%. The yearly sums from all of the four models agree well with the 20-year long-term mean values within 3.0%. Accordingly, the effect of solar radiation models on the GHI is insignificant, although solar radiation models are used to obtain the DNI, which is involved in calculation of the FS statistics with the

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largest weight, as in Table 3. Note that the typical meteorological months are selected according to their ranking in the weighted sum of the FS statistics given by Equation (20). The DNI effects are likely to be addressed via the GHI because both the GHI and DNI have strongly positive correlation. Meanwhile, an important implication to investigate solar radiation models is to provide guideline to correct the current TMY data from KSES, which seriously underestimate the DNI [12]. The GHI data from KSES are generally smaller than the long-term means. However, if the city of Incheon showing 7.0% difference is excluded, the GHI data from KSES are still within the variation range by the solar radiation model.

Figure 8. Yearly sums of the GHI fromthe thelong-term long-term mean, data from Korean SolarSolar Energy Figure 8. Yearly sums of the GHI from mean,ininthe theTMY TMY data from Korean Energy Society (KSES), and in the TMY data generated using the Maxwell, the Reindl-2, and the new models. Society (KSES), and in the TMY data generated using the Maxwell, the Reindl-2, and the new models.

Figure 9 shows the yearly sums of the DNI in the TMY datasets. Compared to the GHI in Figure 8, the DNI variations by solar radiation model are increased. The Maxwell model estimates larger values by 18% on the average of the ten cities than the Lee model, while the Reindl-2 and the new models estimate larger values by 11.5% and 4.6%, respectively. The reverse order of the MBE values in Table 1 out of the Maxwell, the Reindl-2, the new, and the Lee models is consistent with the order of lines in Figure 9 (MBE is defined as measurement minus estimation). When the monthly sums of the DNI were analyzed for the nine other cities of Korea, similar seasonal variations due to solar radiation models were observed, as demonstrated in Figure 6. The consistent observations made with the nine other cities imply that the results based on the city of Daejeon should be applicable over the whole country. Meanwhile, the TMY data from KSES largely underestimate the yearly sums of the DNI such that they Figure 8. Yearly sums of GHI from the long-term in themodel. TMY dataIffrom range from 53.8% to 65.5% ofthe the counterparts frommean, the new theKorean TMY Solar dataEnergy from KSES are Society simulation, (KSES), and inthe the TMY data generated using Maxwell, thefor Reindl-2, the new models. used in energy simulation results willthe not account DNI and effects properly. Figure 9. Yearly sums of the DNI in the TMY data from KSES and in the TMY data generated using the Maxwell, the Reindl-2, and the new models.

Figure 9. Yearly sums of the DNI in the TMY data fromKSES KSESand andin inthe the TMY TMY data data generated Figure 9. Yearly sums of the DNI in the TMY data from generatedusing using the the Maxwell, the Reindl-2, and themodels. new models. Maxwell, the Reindl-2, and the new

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5. Conclusions Using solar radiation data from the city of Daejeon from 2007 to 2009, ten decomposition models that convert the GHI into the DNI were evaluated. It was demonstrated that the Orgill, Vignola, Louche, Erbs, Reindl-1, and Lam models estimated some DNI values to be unacceptably larger than 1000 W/m2 . On the contrary, the Lee and CIBSE models tend to underestimate high DNI values exceeding 750 W/m2 . The Reindl-2 and Maxwell models did not pose the extreme behaviors. According to the MBE and the RMSE, the Reindl-2 model was selected as the most suitable model for Korea. A new model based on the quasi-physical approach was developed in order to improve error statistics and remove occasional outliers. The new model resulted in significantly reduced values of MBE and RMSE compared to the Reindl-2 model, by a factor of 8.7% and by 9.8%, respectively. The largest error in the monthly sum of the DNI is also reduced from 24.3% of the Reindl-2 model to 15.7% while the yearly sums of the DNI are estimated within an error of 1.5%. When comparisons were extended to ten major cities in Korea from the standpoint of the TMY data, consistent observations in the bias trend and the seasonal variation between the models were made and, thereby, support that the evaluation results in this study are applicable throughout the nation. This study provides a guideline not only for selecting a suitable solar radiation model in Korea, but also for evaluating solar radiation models in Northeastern Asia. Weather data in energy simulation programs, such as PVsyst, TRNSYS (TRaNsient SYstem Simulation Program), and SAM, can be updated accordingly for reliable results. Furthermore, relations between irradiance components and with weather elements become more important as irradiance forecast technology advances [25]. This study will help satellite-based forecasting of solar resources in the long-term or in a broad region. Acknowledgments: This work was financially supported by grants from the Korea Institute of Energy Technology Evaluation and Planning (KETEP), the Ministry of Trade, Industry, and Energy (MOTIE) (No. 20143010071570), and the National Research Foundation of Korea (NRF), Ministry of Education (2015M3D2A1032828). Author Contributions: Hyun-Jin Lee and Chang-Yeol Yun conducted and conceived the project; Chang-Yeol Yun measured and analyzed the data; Shin-Young Kim classified the data and collected the materials; and Hyun-Jin Lee evaluated the models. Conflicts of Interest: The authors declare no conflict of interest.

Abbreviation CDF CIBSE GHI DHI DNI FS KIER KMA KSES MBE RMSE TMY

Cumulative Distribution Function Chartered Institution of Building Services Engineers Global Horizontal Irradiance Diffuse Horizontal Irradiance Direct Normal Irradiance Finkelstein-Schafer Korea Institute of Energy Research Korea Meteorological Administration Korean Solar Energy Society Mean Bias Error Root Mean Square Error Typical Meteorological Year

References 1. 2. 3.

Duffie, J.A.; Beckman, W.A. Solar Engineering of Thermal Processes, 3rd ed.; John Wiley & Sons: Somerset, NJ, USA, 2013. Lee, H.J.; Kim, J.K.; Lee, S.N.; Kang, Y.H. Numerical study on optical performances of the first central-receiver solar thermal power plant in Korea. J. Mech. Sci. Technol. 2016, 30, 1911–1921. [CrossRef] Vu, N.H.; Shin, S. A concentrator photovoltaic system based on a combination of prism-compound parabolic concentrators. Energies 2016, 9, 645. [CrossRef]

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Korea Meteorological Administration (KMA). Available online: https://data.kma.go.kr/cmmn/main.do (accessed on 8 February 2017). Wong, L.; Chow, W. Solar radiation model. Appl. Energy 2001, 69, 191–224. [CrossRef] Mousavi Maleki, S.A.; Hizam, H.; Gomes, C. Estimation of hourly, daily and monthly global solar radiation on inclined surfaces: Models re-visited. Energies 2017, 10, 134. [CrossRef] Lee, K.; Yoo, H.; Levermore, G.J. Generation of typical weather data using the ISO test reference year (TRY) method for major cities of South Korea. Energy Build. 2010, 45, 956–963. [CrossRef] Lee, K.; Yoo, H.; Levermore, G.J. Quality control and estimation hourly solar irradiation on inclined surfaces in South Korea. Renew. Energy 2013, 57, 190–199. [CrossRef] Sengupta, M.; Habte, A.; Kurtz, S.; Dobos, A.; Wilbert, S.; Lorenz, E.; Stoffel, T.; Renné, D.; Gueymard, C.A.; Myers, D. Best Practices Handbook for the Collection and Use of Solar Resource Data for Solar Energy Applications; National Renewable Energy Laboratory: Golden, CO, USA, 2015. Wilcox, S.; Marion, W. Users Manual for TMY3 Data Sets; National Renewable Energy Laboratory: Golden, CO, USA, 2008. Zang, H.; Wang, M.; Huang, J.; Wei, Z.; Sun, G. A hybrid method for generation of typical meteorological years for different climates of China. Energies 2016, 9, 1094. [CrossRef] Korean Solar Energy Society (KSES). Available online: http://www.kses.re.kr/data_06/list_hi.php (accessed on 8 February 2017). Maxwell, E.L. A Quasi-Physical Model for Converting Hourly Global Horizontal to Direct Normal Insolation; Solar Energy Research Institute: Golden, CO, USA, 1987. Myers, D.R. Solar radiation modeling and measurements for renewable energy applications: Data and model quality. Energy 2005, 30, 1517–1531. [CrossRef] Habte, A.; Sengupta, M.; Andreas, A.; Wilcox, S.; Stoffel, T. Intercomparison of 51 radiometers for determining global horizontal irradiance and direct normal irradiance measurements. Sol. Energy 2016, 133, 372–393. [CrossRef] Orgill, J.; Hollands, K. Correlation equation for hourly diffuse radiation on a horizontal surface. Sol. Energy 1977, 19, 357–359. [CrossRef] Vignola, F.; McDaniels, D. Beam-global correlations in the Pacific Northwest. Sol. Energy 1986, 36, 409–418. [CrossRef] Louche, A.; Notton, G.; Poggi, P.; Simonnot, G. Correlations for direct normal and global horizontal irradiation on a French Mediterranean site. Sol. Energy 1991, 46, 261–266. [CrossRef] Lam, J.C.; Li, D.H. Correlation between global solar radiation and its direct and diffuse components. Build. Environ. 1996, 31, 527–535. [CrossRef] Guide, J. Weather, Solar and Illuminance Data; Chartered Institution of Building Services Engineers (CIBSE): London, UK, 2002. Erbs, D.; Klein, S.; Duffie, J. Estimation of the diffuse radiation fraction for hourly, daily and monthly-average global radiation. Sol. Energy 1982, 28, 293–302. [CrossRef] Reindl, D.T.; Beckman, W.A.; Duffie, J.A. Diffuse fraction correlations. Sol. Energy 1990, 45, 1–7. [CrossRef] Lave, M.; Hayes, W.; Pohl, A.; Hansen, C.W. Evaluation of global horizontal irradiance to plane-of-array irradiance models at locations across the United States. IEEE J. Photovolt. 2015, 5, 597–606. [CrossRef] Finkelstein, J.M.; Schafer, R.E. Improved goodness-of-fit tests. Biometrika 1971, 58, 641–645. [CrossRef] Kleissl, J. Solar Energy Forecasting and Resource Assessment; Academic Press: Cambridge, MA, USA, 2013. © 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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Comparison of Solar Radiation Models to Estimate Direct Normal Irradiance for Korea Hyun-Jin Lee 1, *, Shin-Young Kim 2 and Chang-Yeol Yun 2, * 1 2

*

School of Mechanical Engineering, Kookmin University, 77 Jeongneung-ro, Seongbuk-gu, Seoul 02727, Korea New and Renewable Energy Resource Center, Korea Institute of Energy Research, 152 Gajeong-ro, Yuseong-gu, Daejeon 34129, Korea; [email protected] Correspondence: [email protected] (H.-J.L.); [email protected] (C.-Y.Y.); Tel.: +82-2-910-5466 (H.-J.L.); +82-42-860-3746 (C.-Y.Y.)

Academic Editor: Francesco Calise Received: 10 February 2017; Accepted: 20 April 2017; Published: 30 April 2017

Abstract: Reliable solar radiation data are important for energy simulations in buildings and solar energy systems. Although direct normal irradiance (DNI) is required for simulations, in addition to global horizontal irradiance (GHI), a lack of DNI measurement data is quite often due to high cost and maintenance. Solar radiation models are widely used in order to overcome the limitation, but only a few studies have been devoted to solar radiation data and modeling in Korea. This study investigates the most suitable solar radiation model that converts GHI into DNI for Korea, using measurement data of the city of Daejeon from 2007 to 2009. After ten existing models were evaluated, the Reindl-2 model was selected as the best. A new model was developed for further improvement, and it substantially decreased estimation errors compared to the ten investigated models. The new model was also evaluated for nine major cities other than Daejeon from the standpoint of typical meteorological year (TMY) data, and consistent evaluation results confirmed that the new model is reliably applicable across Korea. Keywords: solar radiation model; global horizontal irradiance (GHI); direct normal irradiance (DNI); typical meteorological year (TMY); decomposition model

1. Introduction Continuous global energy issues, such as climate change and energy shortages, have increased the interest in energy-efficient buildings and solar energy systems. The energy simulation of such systems is critical for accurate performance evaluation and, ultimately, for optimal design. As the most important input to the energy simulation, reliable solar radiation data must be given in advance. The most useful solar radiation data are global horizontal irradiance (GHI), but direct normal irradiance (DNI) or diffuse horizontal irradiance (DHI) are also important. For example, irradiance on the surface of a solar collector or solar cell is determined when either DNI or DHI is given in addition to the GHI [1]. Note that the GHI, DNI, and DHI are interdependent and, thus, knowing two irradiances out of three is sufficient. Solar radiation measurements are often limited to a few locations or short-term periods in some countries. Furthermore, in general, availability of DNI (or DHI) data is much lower than that of GHI data because DNI measurement using a sun tracker costs more and needs more careful maintenance. Along with research efforts for energy-efficient buildings and solar energy systems, the demand for DNI data has increased significantly in Korea [2,3]. Even though the Korea Meteorological Administration (KMA) provides GHI data, as well as other meteorological data, such as dry bulb temperature and wind speed, DNI is not included [4]. When DNI data are not available, it is necessary to rely on a solar radiation model that accounts for regional climate characteristics. Many solar radiation models to Energies 2017, 10, 594; doi:10.3390/en10050594

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estimate DNI with GHI have been developed [5,6]. However, only a few studies have been devoted to solar radiation data and modeling for Korea, and even fewer studies to hourly DNI data. A lack of DNI measurement data has been a major obstacle to meaningful studies. Recently, Lee et al. [7] modeled GHI with cloud cover data for major cities in Korea and, successively, Lee et al. [8] reported a solar radiation model developed for estimating DNI with GHI. However, the model of Lee et al. [8] tends to underestimate DNI data such that DNI values exceeding 750 W/m2 seldom occur. Long-term, 20 or 30 years, solar radiation data collected on an hourly basis are desirable to reflect climatic characteristics at a specific location and obtain reliable simulation results [9]. Since a direct handling of massive data is burdensome, representative datasets generated from raw, long-term data are often used. The representative data, usually referred to as the typical meteorological year (TMY) data, contain 8760 hourly values of meteorological elements for the one-year duration [10,11]. The Korean Solar Energy Society (KSES) has shared the TMY data of seven cities in Korea but, unfortunately, its TMY datasets reveal unreasonably low DNI values [12]. As a result, the users relying on the TMY data from KSES have a risk of underestimating DNI effects in their energy simulations. This study aims at investigating solar radiation models, including a newly developed model, for the estimation of DNI from GHI in Korea and providing a guideline for the selection of solar radiation models in energy simulations. In the beginning, ten well-known solar radiation models are evaluated with three years of data from the city of Daejeon in Korea. Then, a new model based upon the quasi-physical approach proposed by Maxwell [13] is presented. Finally, from the standpoint of the TMY data, variations of solar irradiance due to the solar radiation model are analyzed, and the nationwide extension of the new model is investigated. 2. Evaluation of Existing Solar Radiation Models KMA as a national representative provides meteorological data over 100 locations [4]. GHI is also available at some locations, but DNI is not available at all. Meanwhile, Korea Institute of Energy Research (KIER) measured both GHI and DNI in the city of Daejeon for research purpose. In this study, KIER measurement data from 2007 to 2009 were used for evaluation of solar radiation models and for development of a new model. The city of Daejeon is located approximately at the center of Republic of Korea, and its latitude, longitude, and altitude are 36.18◦ , 127.24◦ , and 77.1 m, respectively. The pyranometer for GHI measurement was a CMP 11 model from Kipp & Zonen Company in Delft, the Netherlands whereas the pyrheliometer for DNI measurement was a CHP 1 model with a SOLYS 2 sun tracker from the same company. Both GHI and DNI were measured every minute and averaged over 60 min to get hourly data. The uncertainties originated from both of the sensors are less than 1%. Based on references [14,15] and experiences, the estimated measurement uncertainties of GHI and DNI are generally 3% on the average and 5% at most. For the three years, the average percentages of missing GHI and DNI data were 2.3% and 1.9%, respectively. The data pair that misses either GHI or DNI and was measured when the zenith angle of the sun was larger than 85◦ were excluded. The number of the remained pairs of GHI and DNI measurements totals 11,928. Solar radiation models to estimate DNI can be classified into two categories, parametric and decomposition models [5]. In parametric models, solar radiation is obtained from other meteorological parameters, such as cloud cover, atmospheric turbidity, pressure, and water content. On the other hand, decomposition models rely on correlations between global, direct, and diffuse components of solar radiation. Whereas parametric models require detailed information of the atmospheric conditions, decomposition models are relatively easy to use once the GHI is known. Out of many decomposition models, ten were selected in this study: Orgill and Hollands [16], Vignola and McDaniels [17], Louche et al. [18], Lee et al. [8], Lam and Li [19], CIBSE [20], Erbs et al. [21], Maxwell [13], and two from Reindl et al. [22]. These models are widely used for modeling solar radiation, e.g., in relevant textbooks [1] and in model comparison studies [5,8,23]. The selected models were evaluated by comparing the modeled and the measured DNI data.

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Most of the decomposition models use correlations between global, direct, and diffuse solar radiation. Once the global, direct, diffuse, and extraterrestrial irradiances on a horizontal surface are given by It , In , Id , and I0 , respectively, three non-dimensional parameters—the clearness index, k t = It /I0 , the direct beam transmittance, k n = In /I0 , and the diffuse fraction, k d = Id /It —can be defined. Note that It represents GHI. DNI is denoted by Gn and is related to In by the equation of In = Gn cos θz , in which θz is the zenith angle of the sun. Usually, the correlations of a solar radiation model render k n or k d as a function of k t in separate intervals divided by k t values. If a model yields k d rather than k n , DNI is calculated via Gn = It (1 − k d )/ cos θz after k d is obtained. Only three models among the ten models evaluated in this study are presented in the following for brevity. The Lee model [8] is selected because it was recently developed with measurement data from the city of Daejeon in 2009. The Reindl-2 [22] and Maxwell [13] models are selected because they show good performances compared to the others, which will be demonstrated later. The Reindl-2 model was developed with measurement data at five European and North American locations, and the term ofcos θz is added as the second input parameter besides k t . The Maxwell model will be explained in the next section. The rest of the other models can be found in the corresponding articles. (1) Lee model [8]: k d = 0.92 for k t ≤ 0.2

(1)

k d = 0.691 + 2.4306k t − 7.3371k2t + 4.7002k3t for 0.2 < k t

(2)

(2) Reindl-2 model [22]: k d = 1.02 − 0.254k t + 0.0123 cos θz for k t ≤ 0.3

(3)

k d = 1.4 − 1.749k t + 0.177 cos θz for 0.3 < k t < 0.78

(4)

k d = 0.486k t − 0.182 cos θz for k t ≥ 0.78

(5)

k n = k nc − ∆k n

(6)

k nc = 0.866 − 0.122m a + 0.0121m2a − 0.000653m3a + 0.000014m4a

(7)

∆k n = A + B exp(Cm a )

(8)

A = 0.512 − 1.56k t + 2.286k2t − 2.222k3t for k t ≤ 0.6

(9)

A = −5.743 + 21.77k t − 27.49k2t + 11.56k3t for k t > 0.6

(10)

B = 0.37 + 0.962k t for k t ≤ 0.6

(11)

B = 41.4 − 118.5k t + 66.05k2t + 31.9k3t for k t > 0.6

(12)

C = −0.28 + 0.923k t − 2.048k2t for k t ≤ 0.6

(13)

C = −47.01 + 184.2k t − 222.0k2t + 73.81k3t for k t > 0.6

(14)

(3) Maxwell model [13]:

Except for the Maxwell model, the correlations for k n or k d are a polynomial function of k t . The only difference lies in the coefficients that account for climate characteristics where the solar irradiance data were measured and used for developing each model. Meanwhile, the Maxwell model possesses a different functional form because the quasi-physical approach is applied; that is, it, in part, reflects the physics involved in the atmospheric transmission of solar radiation. In order to identify proper models for Korea, DNI values were calculated with the ten candidate models using the selected GHI data as the input and compared with actual DNI measurement data. Linear regression analysis between measured and modeled DNI data was conducted for each model. Based on the regression analyses, the ten models were divided into three groups from the standpoint

possesses a different functional form because the quasi-physical approach is applied; that is, it, in part, reflects the physics involved in the atmospheric transmission of solar radiation. In order to identify proper models for Korea, DNI values were calculated with the ten candidate models using the selected GHI data as the input and compared with actual DNI measurement 4data. Energies 2017, 10, 594 of 12 Linear regression analysis between measured and modeled DNI data was conducted for each model. Based on the regression analyses, the ten models were divided into three groups from the standpoint that the estimation estimation is is larger larger than than the the measurement measurementand andhigh highDNI DNIvalues valuesof ofmore morethan than750 750W/m W/m22 are properly estimated. estimated. The Thefirst firstgroup, group,which whichincludes includes Orgill, Vignola, Louche, Erbs, thethe Orgill, thethe Vignola, thethe Louche, the the Erbs, the the Reindl-1, Lam models, estimates DNIvalues valueslarger largerthan thanthe the measurement measurement and yields Reindl-1, andand thethe Lam models, estimates DNI yields unacceptably time to time. Figure 1a shows the scatterplot obtained with thewith Vignola unacceptably high highvalues valuesfrom from time to time. Figure 1a shows the scatterplot obtained the 2 , but such model as a representative of the first group. Some estimated DNI values exceeded 1000 W/m Vignola model as a representative of the first group. Some estimated DNI values exceeded 1000 2, butvalues high are very Korea. W/mDNI such high DNI rarely valuesobserved are very in rarely observed in Korea.

Figure 1. Scatterplots of direct normal irradiance (DNI) measurements and estimations, in which the Figure 1. Scatterplots of direct normal irradiance (DNI) measurements and estimations, in which the red curves curves represent represent the the linear linear regression regressionfits: fits: (a) (a) Vignola; Vignola;(b) (b)Lee; Lee;(c) (c)Reindl-2; Reindl-2;and and(d) (d)Maxwell. Maxwell. red

On the contrary, the Lee and the CIBSE models, which belong to the second group, seldom yield thevalues, contrary, the Leeinand the CIBSE models, which belong the to the second group, seldom yield high On DNI resulting underestimation. Figure 1b shows scatter plot obtained with the high DNI values, resulting in underestimation. Figure 1b shows the scatter plot obtained with the Lee model as a representative of the second group. The linear regression trend beyond DNI values Lee model 750 as aW/m representative of the second group.that Theadditional linear regression trend values 2 slopes downward, exceeding suggesting correlation atbeyond large ktDNI should be 2 exceeding 750 W/m slopes downward, suggesting that additional correlation at large k t should be introduced. Even though the Lee model was developed with the DNI measurement data in Korea [8], introduced. Even though the Lee model was developed with the DNI measurement data in Korea [8], its underestimation implies that one-year data used for the model development are not enough for its underestimation implies that one-year data used for the model development are not enough for proper estimation. proper estimation. The third group includes the Reindl-2 and the Maxwell models. These models estimate DNI The third group includes the Reindl-2 and the Maxwell models. These models estimate DNI values values larger than those measured, but they do not pose extreme behaviors, in contrast to those in larger than those measured, but they do not pose extreme behaviors, in contrast to those in the first and the first and the second groups. Lave et al. [23] demonstrated that the Reindl-2 and the Maxwell the second groups. Lave et al. [23] demonstrated that the Reindl-2 and the Maxwell models use the term models use the term of cos θz (via air mass in the Maxwell model) in addition to kt and, thus, they of cos θz (via air mass in the Maxwell model) in addition to k t and, thus, they outperform other models kt this outperform other models that use only, which consistent frequency, with this study. The observation that use k t only, which is consistent with study. Theisobservation normalized by dividing frequency, normalized bydata dividing with thecalculated total number of datatopoints, wasofcalculated to with the total number of points, was according the level the DNI. according Histograms the level of the DNI. Histograms of the normalized data shown in Figure 2 indicate that the Reindl-2 of the normalized data shown in Figure 2 indicate that the Reindl-2 model suitably estimates the model suitably estimates the observation frequency DNI in each bin. On the tends other observation frequency of DNI values in each DNI bin.ofOn thevalues other hand, theDNI Maxwell model 2 hand, the Maxwell overestimate values . in the bins when from the 750scatterplots to 950 W/min2. to overestimate DNImodel valuestends in theto bins from 750 toDNI 950 W/m However, However, the scatterplots in Figure 1c,dthe areReindl-2 compared, it is clear thatiswith the Reindl-2 model Figure 2c,dwhen are compared, it is clear that with model the DNI occasionally estimated the DNI is occasionally estimated too high. too high.

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Figure 2. Histograms of DNI measurement and estimation, in which frequency is normalized by

Figure 2. Histograms of DNI measurement and estimation, in which frequency is normalized by dividing by the total number of data points. dividing by the total number of data points.

The coefficient of determination (R2), the mean bias error (MBE), the root mean square error

The coefficient of determination (R2 ), the mean biascalculated error (MBE), the rootthe mean square error (RMSE), and the median absolute deviation (MAD) were to distinguish goodness-of2 values remarkably, roughly fit of each model and summarized in Table 1. The models do not alter R (RMSE), and the median absolute deviation (MAD) were calculated to distinguish the goodness-of-fit in the variation range of 2%, and the R21. ofThe the Reindl-2 the best. The MBE is defined roughly as the of each model and summarized in Table models model do notisalter R2 values remarkably, in sum of the measurements minus the estimation. Therefore, a negative value of MBE means 2 the variation range of 2%, and the R of the Reindl-2 model is the best. The MBE is defined as the sum of overestimation of a model, and a positive value means the opposite. Whereas the MBE is a good the measurements minus the estimation. Therefore, a negative value of MBE means overestimation of a measure for yearly-based estimation, the RMSE is for hourly-based estimation. If the best model was model, and a positive value means the opposite. Whereas the MBE is a good measure for yearly-based to be selected from the ten investigated models, the first selection criterion is to exclude extreme estimation, the Accordingly, RMSE is for out hourly-based If the statistics best model was and to be selected from the ten behaviors. of the third estimation. group, with better of MBE RMSE the Reindl-2 investigated models, the first selection criterion is to exclude extreme behaviors. Accordingly, model becomes the most suitable for estimating the DNI in Korea. The unacceptable underestimationout of the third with betterthe statistics of MBE and RMSE thethough Reindl-2 model the Note mostthat suitable of thegroup, DNI disqualifies CIBSE and the Lee models even their MBEsbecomes are smaller. the RMSEthe values roughly range fromunderestimation 25% to 35% of of thethe mean value and for estimating DNIininTable Korea.1 The unacceptable DNIDNI disqualifies theare CIBSE significantly larger than the measurement approximately 3%. values in Table 1 roughly and the Lee models even though their MBEsuncertainty are smaller.byNote that the RMSE range from 25% to 35% of the mean DNI value and are significantly larger than the measurement Table 1. Regression analysis of each model with a linear polynomial of y = C1 x + C0 : the coefficient uncertainty by approximately 3%. of determination (R2), the mean bias error (MBE) as the measurement minus estimation, the root mean square error (RMSE), and the median absolute deviation (MAD).

Table 1. Regression analysis of each model with a linear polynomial of y = C1 x + C0 : the coefficient of Group Modelbias error (MBE) Cas 1 the measurement C0 R2minus MBE RMSE determination (R2 ), the mean estimation, the rootMAD mean and Hollands −11.91 75.16 28.14 square errorOrgill (RMSE), and the [16] median absolute0.99 deviation15.70 (MAD). 92.8% 92.3% 93.0% 2 R92.9% 92.8% 92.9% 92.3% 91.8% 93.0% 92.7% 92.9% 92.9% 92.9% 93.7% 91.8% 93.2% 92.7% 94.5%

−19.95 −45.45 MBE −20.92 −11.91 −23.17 −19.95 −39.16 −45.45 16.23 −20.92 6.93 −23.17 −17.82 −39.16 −37.05 16.23 −2.04

Reindl of et al. [22]—2 3. III Development a New Solar Radiation0.97 Model 26.24 1.04 26.78 Maxwell [13]

93.7% 93.2%

3.1. Methodology New

94.5%

I Group

I II III

II

Vignola and McDaniels [17] Louche et al. [18] Model Erbs et al. [21] Reindl et al. [22]—1 [16] Orgill and Hollands Vignola and McDaniels [17] Lam and Li [19] Louche et [8] al. [18] Lee et al. Erbs et al. CIBSE [20][21] Reindl et al. [22]—1 Reindl et al. [22]—2 Lam and Li [19] Maxwell [13] Lee et al. New [8]

CIBSE [20]

0.96 1.06 C1 1.03 0.990.98 0.960.92 1.060.78 1.030.89 0.98 0.97 0.92 1.04 0.780.96

0.89

0.96

30.16 29.33 C012.08 15.70 27.22 30.16 60.77 29.33 42.12 12.08 20.77 27.22 26.24 60.77 26.78 42.12 12.16

20.77

12.16

92.9%

78.11 92.02 RMSE 80.07 75.16 77.10 78.11 86.37 92.02 85.43 80.07 72.63 77.10 70.29 86.37 84.95 85.43 63.37

32.58 31.70 MAD 29.62 28.14 30.20 32.58 37.98 31.70 37.80 29.62 30.07 30.20 26.84 37.98 30.38 37.80 26.57

−17.82 −37.05

70.29 84.95

26.84 30.38

−2.04

63.37

26.57

6.93

72.63

30.07

In the previous section, the Reindl-2 model turned out to be the most suitable model. However,

3. Development ofsatisfactory. a New Solar Radiation Model it is not entirely Above all, the MBE is still large, and some outliers can occur, as indicated

in Figure 1c. An effort to improve the solar radiation model was made. The Reindl-2 and the Maxwell

3.1. Methodology models naturally became good candidates due to the aforementioned comparison results. The ReindlIn the previous section, the Reindl-2 model turned out to be the most suitable model. However, it is not entirely satisfactory. Above all, the MBE is still large, and some outliers can occur, as indicated in Figure 1c. An effort to improve the solar radiation model was made. The Reindl-2 and the Maxwell models naturally became good candidates due to the aforementioned comparison results. The Reindl-2

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model uses a simple curve-fitting approach. Hence, the modification based on the quasi-physical 2 model uses a simple curve-fitting approach. Hence, the modification based on the quasi-physical approach of the Maxwell model [13] is likely to be a better estimate for Korea. approach of the Maxwell model [13] is likely to be a better estimate for Korea. Maxwell’s quasi-physical approach was established upon the three following assumptions: First, Maxwell’s quasi-physical approach was established upon the three following assumptions: First, air mass, m a , ismthe, is dominant parameter affecting the relationship between k and k . Second, using a air mass, the dominant parameter affecting the relationship betweenn kn andt kt . Second, a physical model to calculate clear-sky k n will provide a physically-based reference, from which changes using a physical model to calculate clear-sky kn will provide a physically-based reference, from in k n can be calculated. Third, the seasonal, annual, and climate variations in the relationship between which changes in kn can be calculated. Third, the seasonal, annual, and climate variations in the k n and k t are entirely accounted for by parametric functions in k t that relate ∆k n to m a , cloud cover, relationship between kn and kt are entirely accounted for by parametric functions in kt that relate and precipitable water vapor. The second hypothesis explains Equation (6) above. If clear-sky k n is Δ k to m a , cloud cover, and precipitable water vapor. The second hypothesis explains Equation (6) defined nas k nc for limiting values, ∆k n represents the deviation from it. Maxwell [13] adopted the above. If clear-sky kn is defined as knc for limiting values, Δ k n represents the deviation from it. Bird clear-sky model for k nc , which corresponds to Equation (7). According to the first and third Maxwell [13] adopted the Bird clear-sky model for knc , which corresponds to Equation (7). According hypotheses that were obtained from statistical analyses, ∆k n has a functional form, as in Equation (8). to the first and third hypotheses that were obtained from statistical analyses, Δ k has a functional The coefficients A, B, and C in Equation (8) were determined by fitting solarn radiation data from form, as in Equation (8). The coefficients A, B, and C in Equation (8) were determined by fitting solar Atlanta, Georgia, USA in 1981. radiation data from Atlanta, Georgia, USA in 1981. Development of a new solar radiation model based on the quasi-physical approach starts from Development of a new solar radiation model based on the quasi-physical approach starts from accepting Equations (6)–(8). theremaining remaining to determine the coefficients A,using B, and C accepting Equations (6)–(8).Then, Then, the tasktask is tois determine the coefficients A, B, and C usingsolar solarradiation radiation data from Korea, which willrise givetorise to correlations similar to Equations (9)–(14). data from Korea, which will give correlations similar to Equations (9)–(14). The The first todivide dividektk tinto intothe the intervals whose median values increase 0.05, at starting at 0.25, first step step isisto intervals whose median values increase by 0.05,by starting 0.25, and and in each interval of k the regression analysis between ∆k and m is carried out. For example, n m a isa carried out. For example, in each interval of kt t the regression analysis between Δ k n and Figure 3 shows the scatterplot fromfrom the regression analysis at k t at= 0.25, in which the results represent kt = 0.25 , in which the results Figure 3 shows the scatterplot the regression analysis very represent cloudy conditions, as indicated by the range of 0.225 ≤ k < 0.275. Therefore, they correspond to t 0.225 ≤ kt < 0.275 . Therefore, they very cloudy conditions, as indicated by the range of the limiting caseto of the ∆k nlimiting = k nc , implying the extraterrestrial solar radiation is completely absorbed correspond case of Δkthat n = knc , implying that the extraterrestrial solar radiation is or scattered by the atmosphere and, thus, the direct normal component solar component radiation isofessentially completely absorbed or scattered by the atmosphere and, thus, the directof normal solar zero. radiation The black thin curve in Figure 3 represents k in Equation (7), and the fact that all thethe data in is essentially zero. The black thin curve nc in Figure 3 represents knc in Equation (7),ofand Figure 3 are curve supports this statement. fact that located all of thebelow data inthis Figure 3 are located below this curve supports this statement.

k and the air mass at k = 0.25 , in which the solid red curve represents Figure 3. Scatterplot of Δand Figure 3. Scatterplot of ∆k n n the air mass at k t =t 0.25, in which the solid red curve represents the the regression fit and thethin blackcurve thin curve represents in Equation regression fit and the black represents k nc ink ncEquation (7). (7).

The similar regression analyses (not presented) to determine the coefficients A, B, and C were

The similar regression analyses (not presented) to determine the coefficients A, B, and C were repeated at each interval of kt until kt = 0.70 . Note that Maxwell [13] presented the regression repeated at each interval of k t until kkt = 0.70. Note that Maxwell [13] presented the regression analyses analyses up to kt = 0.8 because t > 0.81 was not available in the solar radiation data from Atlanta. up to k t = 0.8 because k t > 0.81 was not available in the solar radiation data from Atlanta. For Daejeon, For Daejeon, Korea, there are some solar radiation data even when kt > 0.75 , but they are not enough Korea, there are some solar radiation data even when k t > 0.75, but they are not enough to derive to derive statistically meaningful fits. The reason is that the solar radiation in the southeastern region statistically meaningful fits. The reason is that the solar radiation in the southeastern region of the US of the US is more abundant than in Northeastern Asia. After the coefficients A, B, and C were is more abundant thaninterval in Northeastern Asia. After the coefficients A, B, C were at determined at each of kt , another regression analysis was carried outand in order to fitdetermined A, B, and each C interval of k , another regression analysis was carried out in order to fit A, B, and C as a function t as a function of kt . The development procedure can be summarized as follows: of k t . The development procedure can be summarized as follows: • Calculate cos θz , m a , and I0 on an hourly basis ( I0 is calculated based on [13]). kn with • Calculate cos θkzt, m I0 onthe anmeasured hourly basis calculated on [13]). • Calculate and GHI,(II0t isand DNI, G nbased . a , and • Calculate k t and k n with the measured GHI, It and DNI, Gn . • Divide the intervals of k t and the group data by the interval.

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Conduct a regression analysis between ∆k n and m a and determine the coefficients A, B, and C in each group. Energies 2017, 10, 594intervals of k and the group data by the interval. 7 of 12 • Divide the t Conduct another regression analysis to fit the coefficients A, B, and C as a function of k • Conduct a regression analysis between Δ k n and m a and determine the coefficients A, B,t .and •

Divide the intervals of kt and the group data by the interval. C in each group. m a and determine • Conduct a regression analysis between thefunction coefficients B, and Figure 4 shows that the coefficients A and BΔ kcan be expressed as a linear andA, a third-order n and • Conduct another regression analysis to fit the coefficients A, B, and C as a function of kt . polynomial function over the entire range, respectively. Meanwhile, the fitting of the coefficient C with C in each group. Figure 4 shows the coefficients Atoand B separating can be expressed asthe aC linear function 0.25 andkta≤ • Conduct anotherthat regression analysis fitfor the coefficients B, and as a function of . thirda third-order polynomial function is required kA, two ranges, k t ≤ 0.50 t into order polynomial function over the entire range, respectively. Meanwhile, the fitting of the coefficient and 0.50 ≤ k ≤ 0.70. Similarly to the derivation of the original model by Maxwell [13], extrapolation t Figure 4 shows that the coefficients A and can be expressed as a linear function and aranges, thirdk into C with a third-order polynomial function is Brequired for separating the two is also applied in thisfunction study over when > 0.75. Finally, the correlations int the solar radiation model order polynomial thek tentire range, respectively. Meanwhile, the fitting of the coefficient 0.25 ≤ kt ≤ 0.50 and 0.50 ≤ kt ≤ 0.70 . Similarly to the derivation of the original model by Maxwell developed the data from Daejeon, Korea can be written follows: kt into the two ranges, C withwith a third-order polynomial function is required for as separating [13], extrapolation is also applied in this study when kt > 0.75 . Finally, the correlations in the solar 0.25 ≤ kt ≤ 0.50 and 0.50 ≤ kt ≤ 0.70 . Similarly to the derivation of the original model by Maxwell radiation model developed with the data Daejeon, Koreat can be written as follows: A =from 0.3452 − 0.3782k (15) [13], extrapolation is also applied in this study when kt > 0.75 . Finally, the correlations in the solar A = 0.3452 − 0.3782 kt can be written as follows: (15) radiation model developed with the data from Daejeon, Korea 2 3

B = 0.5329 + 0.2676k t − 0.0216k t + 0.1584k t

(16)

2 3 B = 0.5329 +A0.2676 kt −−0.0216 = 0.3452 0.3782kkt t + 0.1584 kt

(16) (15)

C = −0.2117 − 0.0513 kt + 1.2976 kt2 − 3.3222 kt3 forkt3kt ≤ 0.5 B = 0.5329 + 0.2676 kt − 0.0216 kt2 + 0.1584

(17) (16)

CC==0.7221 − 10.2801 ktk+t +30.3285 27.9766 forkt 0.5 < kt −0.2117 − 0.0513 1.2976 ktt2 − 3.3222 kt3ktfor ≤ 0.5

(18) (17)

C = 0.7221 − 10.2801kt + 30.3285 kt2 − 27.9766 kt3 for 0.5 < kt

(18)

C = −0.2117 − 0.0513k t + 1.2976k2t − 3.3222k3t for k t ≤ 0.5 C = 0.7221 − 10.2801k t + 30.3285k2t2 − 27.9766k3 3t for 0.5 < k t

(17) (18)

Figure 4. Regression analysis to fit the coefficients A, B, and C according to Equation (8).

Figure 4. Regression analysis to fit the coefficients A, B, and C according to Equation (8).

3.2. ResultsFigure 4. Regression analysis to fit the coefficients A, B, and C according to Equation (8).

3.2. Results

The scatterplot with the new model in Figure 5 is qualitatively similar to the counterpart in 3.2. Results

The scatterplot with thereduces new model in Figure 5 is estimated qualitatively tomodel the counterpart Figure 1d. It significantly the occasional outliers by thesimilar Reindl-2 shown in in The scatterplot with the new model in Figure 5 is qualitatively similar to the counterpart in in Figure 1c. The goodness-of-fit is greatly improved with the new model. Table 1 shows the MBEshown and Figure 1d. It significantly reduces the occasional outliers estimated by the Reindl-2 model Figure 1d. It significantly reduces the occasional outliers estimated by the Reindl-2 model shown in the1c. RMSE significantly improved respect to thethe Reindl-2 model, by a factor of 8.7 and by Figure The are goodness-of-fit is greatlywith improved with new model. Table 1 shows the MBE and Figure 1c. The goodness-of-fit is greatly improved with in theFigure new model. Table 1 shows the MBE and 9.8%, respectively. However, the normalized frequency 2 demonstrates that the new model the RMSE are significantly improved with respect to the Reindl-2 model, by a factor of 8.7 and by the RMSE poorer are significantly improved withinrespect to the Reindl-2 model, by750 a factor is slightly than the Reindl-2 model estimating high DNI exceeding W/m2.of 8.7 and by 9.8%,9.8%, respectively. However, thethe normalized frequency demonstrates thatthe the new model is respectively. However, normalized frequencyin inFigure Figure 22 demonstrates that new model 2. slightly poorer than the Reindl-2 model in estimating high DNI exceeding 750 W/m 2 is slightly poorer than the Reindl-2 model in estimating high DNI exceeding 750 W/m .

Figure 5. Scatterplot of the DNI measurement and estimation from the new model. Figure 5. Scatterplot of the DNI measurement and estimation from the new model.

Figure 5. Scatterplot of the DNI measurement and estimation from the new model.

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As summarized in Table 2, the error of the yearly sum of DNI supports that the new model remarkably error in Energies improves 2017, 10, 594 estimation performance. For all three years—2007, 2008, and 2009—the 8 of 12 Energies 2017, 10, 594 8 of 12 estimation does not exceed 1.5%. Figure 6 shows variations of errors for estimating the monthly sum. Tableto 2, smaller the errorerrors of thecompared yearly sumto ofthe DNIMaxwell supportsand thatthe the Reindl-2 new model In general,As thesummarized new modelinleads models. As summarized in Table 2, the error of the yearly sumyears—2007, of DNI supports that 2009—the the new model remarkably improves estimation performance. For all three 2008, and errormodel The largest error for the Maxwell model is 35.5% in February 2009 and 24.3% for the Reindl-2 remarkably performance. For all three years—2007, 2008, and 2009—the error in estimationimproves does notestimation exceed 1.5%. Figure 6 shows variations of errors for estimating the monthly in Julyin2007. However, theexceed largest error for 6the newvariations model isofonly 15.7%estimating in Februarymonthly 2009. Errors estimation does 1.5%. Figure shows errors sum. In general, thenot new model leads to smaller errors compared to thefor Maxwell and the the Reindl-2 are remarkably decreased around the winter and the spring seasons, they are increased in sum. In general, theerror new model leads to smaller to 2009 thewhereas Maxwell and models. The largest for the Maxwell model iserrors 35.5%compared in February and 24.3% forthe theReindl-2 Reindllargest error forand the the Maxwell model is in February and 24.3% for the Reindlseveralmodels. the2007. summer the autumn seasons. new model toinestimate DNI 2 months model The inof July However, largest error for35.5% the The new model is2009 onlytends 15.7% February 2009.values 2Errors model in July 2007. However, thearound largestmodels error forregardless theand newthe model is only 15.7% in February smaller thanare the Maxwell and the Reindl-2 ofspring the month, thereby resulting remarkably decreased the winter seasons, whereas they2009. are in the areinremarkably aroundin the winter the spring seasons, increased several of the summer and the autumn new modelwhereas tends to they estimate overallErrors downward shiftsmonths of decreased monthly errors Figure 6. and In seasons. order toThe investigate the effects of are seasonal increased in several months of Maxwell the summer and the autumnmodels seasons.regardless The new model tends to estimate DNI values smaller than the and the Reindl-2 of the month, thereby positions of the sun, the hourly errors in June 2009 and December 2009 against the solar zenith angle, DNI values smaller than the Maxwell and Reindl-2 models regardless of thetomonth, thereby in the overall downward shifts of the monthly errors in Figure 6. In investigate the θz , areresulting presented in Figure 7. The smaller estimation by the new model is order essentially independent of resulting in the overall downward shifts of monthly errors in Figure 6. In order to investigate the effects of seasonal positions of the sun, the hourly errors in June 2009 and December 2009 against the θz , which is generally true for the non-presented months as well. Consequently, it can be concluded effects of seasonal the sun, the June 2009 and December θz , are ofpresented solar zenith angle,positions in hourly Figure errors 7. Theinsmaller estimation by the2009 newagainst modelthe is that the underestimation thepresented new model consistently occurs estimation throughout year and contributes solar zenith angle, θz of , are in Figure 7. The smaller by athe new model is θ essentially independent of z , which is generally true for the non-presented months as well. to decreasing largest monthly error. isSince the months errors are months decreased dominate generally true for where the non-presented as well. essentiallythe independent of θz , which Consequently, it can be concluded that the underestimation of the new model consistently occurs those where errors are increased, performance in the yearly estimation is improved. In spite of better Consequently, it can concluded that the underestimation of the new model throughout a year andbecontributes to decreasing the largest monthly error. Sinceconsistently the months occurs where estimation in yearly irradiances, caution must be paid when the new model is applied forwhere estimating throughout a year and contributes to decreasing the largest monthly error. Since the months errors are decreased dominate those where errors are increased, performance in the yearly estimation errors are decreased dominate those where errors are increased, performance in the yearly estimation monthly irradiances around the summer and the autumn seasons. is improved. In spite of better estimation in yearly irradiances, caution must be paid when the new is improved. In spite of better estimation in yearly irradiances, be paid when the new model is applied for estimating monthly irradiances around thecaution summermust and the autumn seasons. model is appliedTable for estimating monthly around the summer autumn seasons. 2. Relative errors irradiances for estimating the yearly sum ofand the the DNI. Table 2. Relative errors for estimating the yearly sum of the DNI. Table 2. Relative errors for2007 estimating2008 the yearly2009 sum of the DNI. Model Model 2007 2008 2009 Model 2007 2008 12.3% 12.3% 2009 Maxwell 15.3% 13.2% Maxwell 15.3% 13.2% Reindl-2 7.7% 5.7% Maxwell 15.3% 13.2% Reindl-2 7.7% 5.7% 5.5% 12.3% 5.5% New 1.5% − 0.1% Reindl-2 7.7% 5.7% 5.5% New 1.5% −0.1% −0.5% −0.5% New 1.5% −0.1% −0.5%

Figure 6. Relative errors for estimating the monthly sum of the DNI.

Figure 6. Relative errors themonthly monthly sum of the DNI. Figure 6. Relative errorsfor forestimating estimating the sum of the DNI.

Figure 7. Effects of the solar zenith angle on errors for estimating the hourly DNI, in which the dotted Figure 7. Effects of the solar zenith angle on errors for estimating the hourly DNI, in which the dotted

the solidoflines the averages of the Reindl-2 and the newthe models in each interval Figureand 7. Effects the indicate solar zenith angle on errors for estimating hourly DNI,separate in which the dotted and the solid lines angle, indicate the averages the Reindl-2 new models of the solar zenith respectively: (a)ofJune 2009; andand (b) the December 2009. in each separate interval and the lines indicate averages (a) of June the Reindl-2 the new models in each separate interval of solid the solar zenith angle,the respectively: 2009; andand (b) December 2009. of the solar zenith angle, respectively: (a) June 2009; and (b) December 2009.

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4. Variations of Solar Irradiance in TMY Data Since the DNI measurement data of other cities in Korea were not accumulated systematically and sufficiently, the previous results obtained with data from the city of Daejeon cannot be directly validated for the nationwide extension. The ten major cities in Korea, including Daejeon, are considered: Busan, Cheongju, Daegu, Gangneung, Gwangju, Incheon, Jeju, Jeonju, and Seoul. The solar radiation models were compared in terms of the TMY data instead of the data of a specifically-selected year. Since the TMY data represent regional climatic characteristics for the one-year duration, they are generally used for energy simulations and facilitate evaluation [10]. In the following, TMY data of each city were generated after DNI was estimated with solar radiation models. Then, variations of solar irradiance were investigated to identify similar trends across the ten major cities. A TMY dataset consists of the months selected from individual years and concatenated to form a complete year. In this study, the method of National Renewable Energy Lab (NREL) in the US was adopted for generation of TMY data [10]. The first step is to select five candidate months close to the long-term weather characteristics for each month. For the selection, monthly cumulative distribution functions (CDFs) for the daily data of a weather element are compared with the long-term CDF. According to the Finkelstein-Schafer (FS) statistics in Equation (19) [24], the deviation of the CDF of a specific month from the long-term CDF is calculated for the j-th weather element, in which the subscript n indicates the number of days in a month and xi denotes daily data on i-th day: FS j =

1 n CDFlong−term ( xi ) − CDFmonthly ( xi ) ∑ n i =1

(19)

The ten weather elements in Table 3 are judged to be more important than the others, and a weighted sum of the FS statistics is used to select the five candidate months. 10

WFS j =

∑ w j FSj

(20)

j =1

Table 3. Weighting factors for the Finkelstein-Schafer (FS) statistics. Weather Element

Weighting Factor

Max Dry Bulb Temperature

1/20

Min Dry Bulb Temperature Mean Dry Bulb Temperature Max Dew Point Temperature Min Dew Point Temperature

1/20 2/20 1/20 1/20

Weather Element Mean Dew Point Temperature Max Wind Velocity Mean Wind Velocity Global Horizontal Irradiance Direct Normal Irradiance

Weighting Factor 2/20 1/20 1/20 5/20 5/20

In the second step, the five candidate months are ranked with respect to the closeness of the month to the long-term mean and median. The third step is to check the persistence of the mean dry bulb temperature and daily GHI so that the month that exhibits exceptional weather patterns, such as the longest run or zero runs of consecutive warm days, are excluded. The highest-ranked candidate month from the second step that survives in the third step is selected as the typical meteorological month. Finally, in the fourth step, the 12 typical meteorological months are concatenated to form a complete year. TMY data of the ten major cities in Korea were generated with 20-year (1991–2010) weather data from KMA [4]. Figure 8 shows that the yearly sums of the GHI in the TMY datasets do not change noticeably by solar radiation models, indicating their variations of not more than 2.1%. The yearly sums from all of the four models agree well with the 20-year long-term mean values within 3.0%. Accordingly, the effect of solar radiation models on the GHI is insignificant, although solar radiation models are used to obtain the DNI, which is involved in calculation of the FS statistics with the

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largest weight, as in Table 3. Note that the typical meteorological months are selected according to their ranking in the weighted sum of the FS statistics given by Equation (20). The DNI effects are likely to be addressed via the GHI because both the GHI and DNI have strongly positive correlation. Meanwhile, an important implication to investigate solar radiation models is to provide guideline to correct the current TMY data from KSES, which seriously underestimate the DNI [12]. The GHI data from KSES are generally smaller than the long-term means. However, if the city of Incheon showing 7.0% difference is excluded, the GHI data from KSES are still within the variation range by the solar radiation model.

Figure 8. Yearly sums of the GHI fromthe thelong-term long-term mean, data from Korean SolarSolar Energy Figure 8. Yearly sums of the GHI from mean,ininthe theTMY TMY data from Korean Energy Society (KSES), and in the TMY data generated using the Maxwell, the Reindl-2, and the new models. Society (KSES), and in the TMY data generated using the Maxwell, the Reindl-2, and the new models.

Figure 9 shows the yearly sums of the DNI in the TMY datasets. Compared to the GHI in Figure 8, the DNI variations by solar radiation model are increased. The Maxwell model estimates larger values by 18% on the average of the ten cities than the Lee model, while the Reindl-2 and the new models estimate larger values by 11.5% and 4.6%, respectively. The reverse order of the MBE values in Table 1 out of the Maxwell, the Reindl-2, the new, and the Lee models is consistent with the order of lines in Figure 9 (MBE is defined as measurement minus estimation). When the monthly sums of the DNI were analyzed for the nine other cities of Korea, similar seasonal variations due to solar radiation models were observed, as demonstrated in Figure 6. The consistent observations made with the nine other cities imply that the results based on the city of Daejeon should be applicable over the whole country. Meanwhile, the TMY data from KSES largely underestimate the yearly sums of the DNI such that they Figure 8. Yearly sums of GHI from the long-term in themodel. TMY dataIffrom range from 53.8% to 65.5% ofthe the counterparts frommean, the new theKorean TMY Solar dataEnergy from KSES are Society simulation, (KSES), and inthe the TMY data generated using Maxwell, thefor Reindl-2, the new models. used in energy simulation results willthe not account DNI and effects properly. Figure 9. Yearly sums of the DNI in the TMY data from KSES and in the TMY data generated using the Maxwell, the Reindl-2, and the new models.

Figure 9. Yearly sums of the DNI in the TMY data fromKSES KSESand andin inthe the TMY TMY data data generated Figure 9. Yearly sums of the DNI in the TMY data from generatedusing using the the Maxwell, the Reindl-2, and themodels. new models. Maxwell, the Reindl-2, and the new

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5. Conclusions Using solar radiation data from the city of Daejeon from 2007 to 2009, ten decomposition models that convert the GHI into the DNI were evaluated. It was demonstrated that the Orgill, Vignola, Louche, Erbs, Reindl-1, and Lam models estimated some DNI values to be unacceptably larger than 1000 W/m2 . On the contrary, the Lee and CIBSE models tend to underestimate high DNI values exceeding 750 W/m2 . The Reindl-2 and Maxwell models did not pose the extreme behaviors. According to the MBE and the RMSE, the Reindl-2 model was selected as the most suitable model for Korea. A new model based on the quasi-physical approach was developed in order to improve error statistics and remove occasional outliers. The new model resulted in significantly reduced values of MBE and RMSE compared to the Reindl-2 model, by a factor of 8.7% and by 9.8%, respectively. The largest error in the monthly sum of the DNI is also reduced from 24.3% of the Reindl-2 model to 15.7% while the yearly sums of the DNI are estimated within an error of 1.5%. When comparisons were extended to ten major cities in Korea from the standpoint of the TMY data, consistent observations in the bias trend and the seasonal variation between the models were made and, thereby, support that the evaluation results in this study are applicable throughout the nation. This study provides a guideline not only for selecting a suitable solar radiation model in Korea, but also for evaluating solar radiation models in Northeastern Asia. Weather data in energy simulation programs, such as PVsyst, TRNSYS (TRaNsient SYstem Simulation Program), and SAM, can be updated accordingly for reliable results. Furthermore, relations between irradiance components and with weather elements become more important as irradiance forecast technology advances [25]. This study will help satellite-based forecasting of solar resources in the long-term or in a broad region. Acknowledgments: This work was financially supported by grants from the Korea Institute of Energy Technology Evaluation and Planning (KETEP), the Ministry of Trade, Industry, and Energy (MOTIE) (No. 20143010071570), and the National Research Foundation of Korea (NRF), Ministry of Education (2015M3D2A1032828). Author Contributions: Hyun-Jin Lee and Chang-Yeol Yun conducted and conceived the project; Chang-Yeol Yun measured and analyzed the data; Shin-Young Kim classified the data and collected the materials; and Hyun-Jin Lee evaluated the models. Conflicts of Interest: The authors declare no conflict of interest.

Abbreviation CDF CIBSE GHI DHI DNI FS KIER KMA KSES MBE RMSE TMY

Cumulative Distribution Function Chartered Institution of Building Services Engineers Global Horizontal Irradiance Diffuse Horizontal Irradiance Direct Normal Irradiance Finkelstein-Schafer Korea Institute of Energy Research Korea Meteorological Administration Korean Solar Energy Society Mean Bias Error Root Mean Square Error Typical Meteorological Year

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