Comparative analysis of four solar models for tropical sites.
ABSTRACT
This paper summarizes the results of a comparative analysis of four models (the Kasten, Muneer, Zhang and Huang, and neural networkbased models) used to predict hourly solar radiation for six tropical sites. All four models require cloud cover and nonsolar weather data to predict global, diffuse, and direct solar radiation. Predictions from the models are compared against measured solar data obtained for meteorological stations in the six tropical sites. Nonsolar weather data for the same sites were obtained from the National Climatic Data Center (NCDC). Results of the validation analysis indicate that the Zhang and Huang model is suitable for predicting hourly solar radiation in tropical climates. INTRODUCTION To perform energy analysis of new and existing buildings, detailed simulation tools are often utilized. Most detailed building energy simulation tools such DOE2 (LBL 1981) and EnergyPlus (Crawley et al. 2001) require hourly weather files to estimate energy end uses and/or indoor thermal and visual comfort. Several formats are available for weather files, including * WYEC (Weather Year for Energy Calculations), developed by the American Society for Heating, Refrigerating and AirConditioning Engineers [ASHRAE]), * TMY (Typical Meteorological Year) developed by the National Renewable Energy Laboratory [NREL]), and * TRY (Typical Reference Year), which uses an actual weather file for one year that represents the longterm statistical average (i.e., record of over 10 to 30 years). Unfortunately, hourly weather files suitable for detailed building simulation analysis are not available for several countries. The main hindrance for developing hourly weather files is the lack of measured solar data in most countries. For instance, over 90% of the hourly solar radiation data provided by the National Solar Radiation Data Base for 239 US sites is based on predictions from solar models rather than measured data (Maxwell 1998). Several solar models have been reported in the literature. Gueymard (2003) provides a review and evaluation of some solar models that use a broadband scheme and require atmospheric inputs (besides the zenith angle) such as site pressure, precipitable water, broadband aerosol optical depth, and total ozone abundance. More recently, satellite image data have been used to estimate solar radiation for various sites (Perez et al. 1996, 2004). Krarti et al. (2006) provides an extensive review of reported solar models that use readily available weather data. The comparative analysis presented in this paper is based on hourly measured solar data obtained for six sites located near the tropics. The measured hourly solar radiation data were obtained from the National Solar Radiation Data Base (NSRDB) maintained by (NREL) for the US sites (Guam and Honolulu), and from local meteorological stations for other locations (Hong Kong, Sao Paulo, Singapore, and Mexico City). Most of the measured data include global, direct, and diffuse hourly solar radiation. Nonsolar weather data for the tropical sites were obtained from the US National Climatic Data Center (NCDC). Table 1 lists the sources of the solar measured data for the six tropical sites (with latitude within [+ or ]25[degrees]) used to carry out the comparative analysis. In this paper, a brief overview of the four solar models is first presented. Then, the results of the comparative analysis are summarized and discussed. OVERVIEW OF SOLAR MODELS Four solar models have been tested with measured solar data obtained for six tropical sites. To carry out the validation analysis, nonsolar data obtained were obtained from NCDC for the same years and locations listed in Table 1. The nonsolar data include, on an hourly basis, drybulb temperature, dewpoint temperature, station pressure, sky cover, low sky cover, wind speed, wind direction, ceiling height, and relative humidity. The four solar models tested are: 1. Kasten modelthis model was used by ASHRAE to generate the International Weather for Energy Calculation (IWEC) files for selected locations around the world. 2. Muneer modelthis model is similar to the Kasten model except for the calculation of the clear sky model. 3. Zhang and Huang modelthis was recently developed and validated for developing TMY files for several Chinese locations. 4. Neural networkbased modelthe neural network approach has been shown in the literature to be suitable for predicting solar radiation. A brief overview of each model is first presented. Then, the results of the validation analysis for the solar models are discussed. Kasten Model The Kasten model is described in Davies and McKay (1989) and is based on a paper by Kasten (1983). It uses only total CA, not cloud layer information. The coefficients used in the model were derived from West German data. The global radiation [I.sub.g] is calculated from [I.sub.g] = [I'.sub.g](1  [A.sub.1]C[A.sup.[A.sub.2]]), (1) where the cloudless sky irradiance [I'.sub.g] is given by [I'.sub.g] = [I.sub.0][A.sub.3]exp([A.sub.4][T.sub.i]m), (2) with: * [T.sub.l] the Linke turbidity factor, which is linked to other indices, such as optical depths for ozone, Rayleigh and aerosol scattering, and water vapor absorption, using expressions derived in Davies and McKay (1989) * CA, the total cloud amount (fraction of celestial dome covered by clouds, which ranges from 0 to 1) * m, the relative optical air mass, which is a function of the solar zenith angle (Wong and Chow 2001) * [A.sub.1], [A.sub.2], [A.sub.3], and [A.sub.4], correlation coefficients, which are equal to 0.72, 3.2, 0.74, and 0.027 and are based on analysis of West German data. However, new coefficients are developed for each tropical site as outlined in the section on results Muneer Model The Muneer model uses the cloud amount (CA) to calculate hourly horizontal global, diffuse, and beam irradiance. This model is described by Gul et al. (1998) and Mehreen et al. (1998). A similar model was developed by Muneer et al. (1996, 1997) but requires sunshine duration rather than cloud cover. To find the total global irradiance, the irradiance under a clear sky is first estimated using the solar elevation angle, [I'.sub.g] = ([B.sub.1]sin[gamma]  [B.sub.2]) (3) [I.sub.g]/[I'.sub.g] = 1  [B.sub.3](CA/8)[.sup.[B.sub.4]], (4) where CA is the cloud amount in octas. [B.sub.1], [B.sub.2], [B.sub.3], and [B.sub.4] are coefficients that depend on the location of the measured radiation. The original coefficients derived for Hamburg, Germany, are [B.sub.1] = 910, [B.sub.2] = 30, [B.sub.3] = 0.72, and [B.sub.4] = 3.2. Zhang and Huang Model The model was originally developed using Chinese locations (Zhang and Huang 2002). It uses regression models that find the leastsquares fit between measured solar radiation data and climatic conditions, including total cloud cover, relative humidity, wind speed, and drybulb temperature. Equation 5 is the correlation for estimated hourly global solar radiation: I = {[I.sub.0]sin(h)[[c.sub.0] + [c.sub.1](CC) + [c.sub.2](CC)[.sup.2] + [c.sub.3]([T.sub.n]  [T.sub.n3]) + [c.sub.4][phi] + [c.sub.5]Vw] = d}/k when I > 0 I = 0 when I < 0 (5) where I = estimated hourly solar radiation in W/[m.sup.2] [I.sub.0] = solar constant, 1355 W/[m.sup.2] h = solar altitude angle: angle between horizontal and line to sun CC = cloud cover in tenths [phi] = relative humidity in % [T.sub.n],[T.sub.n3] = drybulb temperature at hours n and n3, respectively [V.sub.w] = wind speed in m/s [c.sub.0], [c.sub.1], [c.sub.2], [c.sub.3], [c.sub.4], [c.sub.5], d, k = regression coefficients The regression coefficients were determined from multiparameter analyses against the measured data (1993) for Beijing and Guangzhou and were found to be as follows (Zhang and Huang 2002): [c.sub.0] = 0.5598, [c.sub.1] = 0.4982, [c.sub.2] = 0.6762, [c.sub.3] = 0.02842, [c.sub.4] = 0.00317, [c.sub.5] = 0.014, d = 17.853, k = 0.843 The correlation coefficient (R) was found to be 0.93, which implies that Equation 5 provides an accurate estimation of the hourly total horizontal solar radiation in both Beijing and Guangzhou. Neural NetworkBased Model A neural network (NN) can be any model in which the output variables are computed from the input variables by compositions of basic functions or connections. Several configurations and classes of neural networks have been proposed in the literature, with specific functions and capabilities. Typically, three types of neural networks can be distinguished based on the learning model: supervised learning, unsupervised learning, and hybrid supervisedunsupervised learning (Krarti 2003). In this work, a supervised learning feedforward backpropagation neural network has been utilized. A feedforward backpropagation neural network consists of several layers of neurons that are connected to each other. In this context, a "neuron" is a simplified mathematical model of a biological neuron. A connection is a unique information transport link from one sending to one receiving neuron. Figure 1 shows a schematic diagram of the structure of a neural network. The first and last layers of neurons are called input and output layers; between them are one or more (as is the case in Figure 1) hidden layers. The neuron depicted by the small circles in Figure 1 is the fundamental building block of a network. Each element of the input set [I.sub.i] is multiplied by a weight [W.sub.i,j], and the products are summed to provide the output at the neuron, [O.sub.j]: [O.sub.j] = [summation over i][I.sub.i][W.sub.i,j] + [B.sub.j] (6) where [B.sub.j] is the bias (the activation threshold) at the neuron j. This bias avoids the tendency of a neural network to get "stuck" in a limited value area. After each [O.sub.j] is calculated, an activation function is applied to modify it. The activation function is typically a bounded monotonic function such as the standard sigmoid function, f(x) = 1/[1+exp(x)], or the hyperbolic tangent function, f(x) = tan h(x). In some cases, linear activation function is used [f(x) = x] instead of a bounded function. Typically, squashed bounded activation functions are appropriate when the target variables have to be constrained with certain limits (1, 1) or (0, 1), whereas the linear activation functions are more adequate when the target variables can take continuous values without any limits. The outputs of the activation functions for the neurons in a layer become the inputs for the layers downstream. The ultimate output of the model is the result of the activation function at the output layer. The weights [W.sub.i,j] of the neural network are adjusted iteratively so that application of a set of inputs produces the desired set of outputs. If the computed outputs do not match the known (i.e., target) values during network training, the neural network model is in error. The error E is typically calculated as the sum of squared differences between computed values [O.sub.j] and target values [T.sub.j]: E = [summation over j]([O.sub.j]  [T.sub.j])[.sup.2] (7) If E is large, then the neural network weights [W.sub.i,j] are adjusted to reduce this error, usually using the gradient descent method. Using this approach, the weight is changed from [W.sub.i,j] to [W.sub.i,j]  a dE/d[W.sub.i,j]. The parameter a is called the learning rate. Its value may be adjusted during training based on various criteria that tend to produce the best accuracy. This procedure of weight adjustment is called backpropagation and was originally discovered by Werbos in 1974 but became popular through the work of McClelland and Rumelhart (1988) and others. A simplified procedure for the learning process of neural networks is summarized below: * Provide the network with training data consisting of patterns of input variables and target outputs. [FIGURE 1 OMITTED] * Assess how closely the network output matches the target outputs. * Adapt the connection strengths (i.e., weights) between the neurons so the network produces better approximations of the desired target outputs. * Continue the process of adjusting the weights until some desired level of accuracy is achieved. If not used properly, neural networks may tend to "memorize" the noise in training data. Various techniques exist that reduce this overtraining problem. These techniques are discussed in Kreider et al. (1995). In general, however, neural networks can be very flexible models which can approximate many kinds of inputoutput mappings. It has been shown that neural networks can be used to adequately predict hourly solar radiation in Sao Paulo, Brazil (Soares et al. 2004). For this study, a neural network algorithm was developed based on a feedforward backpropagation learning approach. The input variables for the neural network include solar altitude angle, drybulb temperature, dewpoint temperature, wind speed, relative humidity, and cloud cover. The output for the neural network is global horizontal solar radiation. An iterative process was carried out to optimize neural network parameters, including the number of hidden layers and the learning rate. RESULTS The predictions for all four models are obtained for six sites using the original coefficients as well as sitefitted coefficients. The model predictions are then compared to measured hourly solar radiation data obtained for all six sites. This section summarizes the results of the comparative analysis. Kasten Model Results Results for Original Coefficients. A plot of calculated values of hourly irradiance vs. the measured values is shown in Figure 2 for Sao Paulo, Brazil. Although most of the data points are located along the diagonal, predictions from the Kasten model (with its original coefficients) are generally lower than the measured data. As indicated by Equation 1, the Kasten model uses only one variable, CA, to estimate global solar radiation. Thus, if the model's coefficients (i.e., not sitefitted coefficients) are not properly selected, the model seems to overestimate the effect of CA. [FIGURE 2 OMITTED] Over the year (including nighttime values), the mean bias error (MBE) is 53.8 W/[m.sup.2] and the root mean square error (RMSE) is 139.8 W/[m.sup.2] on an hourly basis. Excluding nighttime data (i.e., when measured irradiance is zero), MBE and RMSE values are 85.8 and 176.7 W/[m.sup.2], respectively, or 30.0% and 61.3% relative to the measured average solar radiation. Results for SiteFitted Coefficients. Figure 3 plots global horizontal radiation calculated by the Kasten model with sitefitted coefficients ([A.sub.1] = 0.68 and [A.sub.2] = 4.76) vs. measured radiation for Sao Paulo, Brazil. There is a decrease in scatter of the data along the diagonal when sitefitted coefficients are used for Kasten model. However, the Kasten model seems to estimate inaccurately for several hours a global solar radiation near 600 W/[m.sup.2]. Over the year (including nighttime values) the MBE is 1.5 W/[m.sup.2] and the RMSE is 103.4 W/[m.sup.2] on an hourly basis. Excluding nighttime data, the MBE and RMSE values are 2.5 and 132.8 W/[m.sup.2], respectively, or 0.9% and 45.3% relative to the measured average solar radiation. Muneer Model Results Results for Original Coefficients. A plot of hourly irradiance calculated using the Muneer model vs. measured values is shown in Figure 4 for Hong Kong. Similar to the Kasten model, the Muneer model relies heavily on one variable, CA, and generally underestimates global solar radiation when original model coefficients are used. [FIGURE 3 OMITTED] Over the year (including nighttime values) the MBE is 12.6 W/[m.sup.2] and the RMSE is 106.8 W/[m.sup.2] on an hourly basis. Excluding nighttime data, the values are 24.2 and 148.0 W/[m.sup.2], respectively, or 8.5% and 52.0% relative to the measured average solar radiation. Results for SiteFitted Coefficients. Figure 5 shows that predictions from the Muneer model can be improved only slightly when sitefitted coefficients ([B.sub.1] = 749.1, [B.sub.2] = 7.6, [B.sub.3] = 0.77, and [B.sub.4] = 5.2) are applied. Its heavy reliance on CA seems to limit any potential improvement of the Muneer model predictions for tropical climates. Over the year (including nighttime values) the MBE is 3.0 W/[m.sup.2] and the RMSE is 104.3 W/[m.sup.2] on an hourly basis. Excluding nighttime data, the MBE and RMSE values are 5.6 and 144.5 W/[m.sup.2], respectively, or 2.0% and 50.8% relative to the measured data. [FIGURE 4 OMITTED] [FIGURE 5 OMITTED] [FIGURE 6 OMITTED] [FIGURE 7 OMITTED] Zhang and Huang Model As indicated in the description of the Zhang and Huang model, the model coefficients ([c.sub.0], [c.sub.1], [c.sub.2], [c.sub.3], [c.sub.4], [c.sub.5], d, k) can be determined based on measured data for solar radiation for each tropical site using regression analysis. Results for Original Coefficients. Figure 6 compares hourly global solar irradiance calculated from the Zhang and Huang model using the original coefficients against measurements for Singapore. Most of the points fall along under the diagonal line. However, the model with its original coefficients underestimates hourly solar radiation. A plot of monthly average of global solar radiation is shown in Figure 7. The Zhang and Huang model consistently underestimates the solar radiation. However, the trend of the model predictions follows that of measurements, indicating a potential to improve model accuracy by adjusting the coefficients. Over the year (including nighttime values) the MBE is 62.6 W/[m.sup.2] and the RMSE is 144.7 W/[m.sup.2] on an hourly basis. Excluding nighttime data, the MBE and RMSE values are 121.0 and 201.2 W/[m.sup.2], respectively, or 34.0% and 56.5% relative to the measurements. Results for Sitefitted Coefficients. Figure 8 compares hourly global solar radiation calculated by the Zhang and Huang model with sitefitted coefficients vs. measurements. A significant improvement in model predictions is obtained as confirmed by the monthly average solar radiation plot illustrated in Figure 9. Over the year (including nighttime values) the MBE is 1.1 W/[m.sup.2] and the RMSE is 102.8 W/[m.sup.2] on an hourly basis. Excluding nighttime data, the MBE and RMSE values are respectively 2.2 and 142.9 W/[m.sup.2], or 0.9% and 40.1% relative to the measurements. Neural Network Model The training phase (i.e., the iterative learning process to estimate the weights for the neural network) was carried using hourly data for 36 days (3 days of each month). The weighting factors obtained for the NN are applied to predict hourly solar radiation for the six sites. [FIGURE 8 OMITTED] [FIGURE 9 OMITTED] [FIGURE 10 OMITTED] [FIGURE 11 OMITTED] Results for Global Irradiance. Figure 10 compares the NNpredicted hourly global solar irradiance vs. the measurements for Guam. Most of the points fall along the diagonal line, although there is some scatter. The monthly NNpredicted vs. measured global radiation is shown in Figure 11. The agreement is acceptable, but there is some noticeable difference for the months of January, March, August, and October. Over the year (including nighttime values) the MBE is 4.3 W/[m.sup.2] and the RMSE is 100.5 W/[m.sup.2] on an hourly basis. Excluding nighttime data, the MSE and RMSE values are 8.7 and 143.7 W/[m.sup.2], respectively, or 2.1% and 34.2% relative to the measurements. Predictions of Diffuse Solar Radiation Measured diffuse solar radiation data from four sitesSao Paulo, Singapore, Honolulu and Guamare obtained. Using these measured data, the predictions of the solar model (expressed as MBE and RMSE) for estimating hourly solar diffuse radiation are evaluated. A model developed by Watanabe et al (1983) is used for splitting diffuse and direct normal solar radiation from global horizontal solar radiation estimated from each of the four solar models discussed above. The original coefficients of the model were developed for sites in Japan. [K.sub.T] = I/([I.sub.0] sinh), [K.sub.TC] = 0.4368 + 0.1394 x sinh [K.sub.DS] = [K.sub.T]  (1.107 + 0.03569) x sinh + 1.681 x [sin.sup.2]h)(1  [K.sub.T])[.sup.3] when [K.sub.T] = [K.sub.TC] [K.sub.DS] = (3.996  3.862 x sinh + 1.540 x [sin.sup.2]h)[K.sub.T.sup.3] when [K.sub.T] < [K.sub.TC] [I.sub.b] = [I.sub.0] x sinh x [K.sub.DS](1  [K.sub.T])/(1  [K.sub.DS]) [I.sub.d] = [I.sub.0] x sinh([K.sub.T]  [K.sub.DS])/(1  [K.sub.DS]) (8) where I = global solar radiation on the horizontal surface, W/[m.sup.2] [I.sub.b] = direct normal (beam) solar radiation on the horizontal surface, W/[m.sup.2] [I.sub.d] = diffuse radiation, W/[m.sup.2] Table 2 summarizes the prediction errors estimated for the models using both original and sitefitted coefficients. Figure 12 illustrates the cumulative frequency curves for hourly diffuse solar radiation of Honolulu, Hawaii, obtained from various models (with original and sitefitted coefficients) and from measured data for the year 1990. The results show that generally the Zhang and Huang model (using the Watanabe diffuse solar model) has the least RMSE and provides better predictions of the diffuse hourly solar radiation. Discussion of the Results Table 3 summarizes the prediction errors of the four models with sitefitted and original coefficients to estimate global horizontal solar radiation for six tropical sites. The results show that the Zhang and Huang model is the best for prediction of solar radiation for all sites. The results also indicate that the neural networkbased model provides relatively good predictions of global solar radiation. However, the NNmodel is a "black box" model and needs training, which requires measured data to estimate the weighting factors. It is noteworthy to point out that a previous study using measured data for a nontropical site (Krarti and Seo 2006) indicated that the Zhang model with its original coefficients predicts well the hourly solar radiation, even though these coefficients were derived from a very different part of the world (China). The analysis presented in this paper, however, shows that new adjusted coefficients should be used for the Zhang and Huang model to better predict solar radiation levels in tropical climates. A companion paper (Seo et al. 2006) provides new coefficients for the Zhang and Huang model suitable for tropical sites. [FIGURE 12 OMITTED] SUMMARY Four solar models were evaluated against measured data obtained for six tropical sites. It was found that the Zhang and Huang model with sitefitted coefficients provides the best prediction of hourly solar radiation. The neural networkbased model was found also to provide good predictions. However, the NNbased model is characterized as a "black box" model (its weighing factors have no physical meanings) and is more difficult to implement than regressionbased models for typical users. Results of the comparative analysis indicated that, when sitefitted regression coefficients were used instead of the original coefficients, the Zhang and Huang model predictions were noticeably better for estimating hourly solar radiation for the tropical sites. A companion paper (Seo et al. 2006) investigates refinements of the Zhang and Huang model to extend its application to tropical climates. ACKNOWLEDGMENTS Financial support from the American Society for Heating, Refrigerating and AirConditioning Engineers (ASHRAE) is acknowledged under research project RP1309. NOMENCLATURE [A.sub.1], [A.sub.2], [A.sub.3], [A.sub.4] = correlation coefficients used in the Kasten Model [B.sub.1], [B.sub.2], [B.sub.3], [B.sub.4] = correlation coefficients used in the Muneer Model Bj = bias parameter defined for the neural network model [c.sub.0], [c.sub.1], [c.sub.2], [c.sub.3], [c.sub.4], [c.sub.5], d, k = regression coefficients used in the Zhang and Huang model CA = cloud cover in tenths E = error function defined for the neural network model h = solar altitude angle [I.sub.b] = direct normal (beam) solar radiation on the horizontal surface, W/[m.sup.2] [I.sub.d] = diffuse radiation, W/[m.sup.2] [I.sub.g] = hourly global horizontal solar radiation, W/[m.sup.2] [I.sub.0] = solar constant, 1355 W/[m.sup.2] [I.sub.j] = input variables defined for the neural network model [K.sub.DS], [K.sub.T], [K.sub.TC] = clearness indices defined by Equation 8 for Watanable model m = relative optical air mass [O.sub.j] = output variables defined for the neural network model [T.sub.i] = Linke turbidity factor [T.sub.j] = target values for the neural network model [T.sub.n], [T.sub.n3] = drybulb temperature at hours n and n3, respectively, used in Zhang and Huang model [V.sub.w] = wind speed, m/s. 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Solar radiation model. Applied Energy 69:191224. Zhang, Q.Y., and Y.J. Huang. 2002. Development of typical year weather files for Chinese locations, LBNL51436. ASHRAE Transactions 108(1):5463. Zhang, Q.Y., Y.J. Huang, H.X. Yang, and C.Z. Lou. 2003. Development of models to estimate solar radiation for Chinese locations. Journal of Asian Architecture and Building Engineering, pp. 14046. Donghyun Seo Moncef Krarti, PhD, PE Member ASHRAE Donghyun Seo is a graduate student and Moncef Krarti is a professor in the Civil, Environmental, and Agricultural Engineering Department, University of Colorado, Boulder. Table 1. Geological Information of Selected Tropical Sites and Sources of Measured Solar Radiation Data Latitude Longitude Elevation No. Site (Year) (degrees) (degrees) (m) Measured Solar Data 1 Hong Kong 22.2N 114.1E 24 Meteo. Station (2002) (King's Park) 2 Sao Paulo 23.4S 46.4W 803 Meteo. Station (2003) (Airport) 3 Singapore 1.2N 103.6E 16 Meteo. Station (1999) (Changi Arpt.) 4 Honolulu 21.2N 157.6W 5 NREL NSRDB (1990) 5 Guam (1990) 13.3N 144.5E 75 NREL NSRDB 6 Mexico City 19.3N 99.1W 2234 Meteo. Station (1993) (Airport) Table 2. Prediction Errors for Diffuse Solar Irradiance of Four Solar Models with SiteFitted and Original Coefficients for Six Tropical Sites Solar Models with Site Fitted Coefficients Kasten Model Muneer Model MBE RMSE MBE RMSE Site No.* (W/[m.sup.2]) (W/[m.sup.2]) (W/[m.sup.2]) (W/[m.sup.2]) 1 N/A N/A N/A N/A 2 1.6 100.8 2.7 98.1 3 29.7 84.9 29.5 82.9 4 23.7 67.0 14.1 61.9 5 6.4 89.7 6.0 85.0 6 N/A N/A N/A N/A Solar Models with Site Fitted Coefficients Zhang Model Neural Network MBE RMSE MBE RMSE Site No.* (W/[m.sup.2]) (W/[m.sup.2]) (W/[m.sup.2]) (W/[m.sup.2]) 1 N/A N/A N/A N/A 2 0.4 85.6 N/A N/A 3 17.4 74.2 N/A N/A 4 13.6 64.6 N/A N/A 5 10.1 86.3 N/A N/A 6 N/A N/A N/A N/A Solar Models with Original Coefficients Kasten Model Muneer Model MBE RMSE MBE RMSE Site No.* (W/[m.sup.2]) (W/[m.sup.2]) (W/[m.sup.2]) (W/[m.sup.2]) 1 N/A N/A N/A N/A 2 53.9 137.0 28.3 112.7 3 20.0 79.2 2.1 67.0 4 16.8 110.1 17.6 87.9 5 39.7 119.1 29.6 101.2 6 N/A N/A N/A N/A Solar Models with Original Coefficients Zhang Model Neural Network MBE RMSE MBE RMSE Site No.* (W/[m.sup.2]) (W/[m.sup.2]) (W/[m.sup.2]) (W/[m.sup.2]) 1 N/A N/A N/A N/A 2 20.4 98.0 N/A N/A 3 6.4 73.4 N/A N/A 4 13.2 62.9 N/A N/A 5 27.8 94.2 N/A N/A 6 N/A N/A N/A N/A * 1 = Hong Kong, 2 = Sao Paulo, 3 = Singapore, 4 = Honolulu, 5 = Guam, 6 = Mexico City. Table 3. Prediction Errors for Global Horizontal Irradiance of Four Solar Models with SiteFitted and Original Coefficients for Six Tropical Sites Solar Models with Site Fitted Coefficients Kasten Model Muneer Model MBE RMSE MBE RMSE Site No.* (W/[m.sup.2]) (W/[m.sup.2]) (W/[m.sup.2]) (W/[m.sup.2]) 1 3.2 103.2 2.9 102.7 2 1.6 100.8 2.7 98.1 3 2.9 129.3 4.1 130.9 4 3.8 82.4 2.1 80.7 5 0.7 111.0 6.0 113.2 6 0.4 115.1 2.5 112.4 Solar Models with Site Fitted Coefficients Zhang Model Neural Network MBE RMSE MBE RMSE Site No.* (W/[m.sup.2]) (W/[m.sup.2]) (W/[m.sup.2]) (W/[m.sup.2]) 1 0.8 85.9 1.0 86.5 2 0.4 85.6 3.6 86.3 3 1.2 102.4 5.5 104.2 4 0.2 71.5 1.8 69.9 5 0.2 98.4 4.2 99.1 6 1.5 108.3 5.8 106.5 Solar Models with Original Coefficients Kasten Model Muneer Model MBE RMSE MBE RMSE Site No.* (W/[m.sup.2]) (W/[m.sup.2]) (W/[m.sup.2]) (W/[m.sup.2]) 1 20.0 110.5 12.5 104.8 2 53.9 137.0 28.3 112.7 3 85.9 194.7 18.5 131.4 4 79.1 151.1 49.2 118.2 5 99.0 201.6 68.4 171.3 6 104.4 217.7 76.9 183.0 Solar Models with Original Coefficients Zhang Model Neural Network MBE RMSE MBE RMSE Site No.* (W/[m.sup.2]) (W/[m.sup.2]) (W/[m.sup.2]) (W/[m.sup.2]) 1 11.3 97.7 N/A N/A 2 20.4 98.0 N/A N/A 3 62.7 144.5 N/A N/A 4 18.1 97.5 N/A N/A 5 53.5 141.6 N/A N/A 6 33.5 147.3 N/A N/A * 1 = Hong Kong, 2 = Sao Paulo, 3 = Singapore, 4 = Honolulu, 5 = Guam, 6 = Mexico City. 

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