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 solar radiation, n the emission and diffusion of actinic rays from the sun. Overexposure may result in sunburn, keratosis, skin cancer, or lesions associated with photosensitivity. for six tropical sites. All four models require cloud cover and nonsolar weather data to predict global, diffuse diffuse /dif·fuse/ 1. (difus´) not definitely limited or localized. 2. (difuz´) to pass through or to spread widely through a tissue or substance. dif·fuse adj. , and direct solar radiation. Predictions from the models are compared against measured solar data obtained for meteorological me·te·or·ol·o·gy n. The science that deals with the phenomena of the atmosphere, especially weather and weather conditions. [French météorologie, from Greek 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 A tropical climate is a type of climate typical in the tropics. Köppen's widelyrecognized scheme of climate classification defines it as a nonarid climate in which all twelve months have mean temperatures above 18°C (64.4 °F). . 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 LBL  Lawrence Berkeley Laboratory, Berkeley, CA, USA. 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 re·frig·er·ate tr.v. re·frig·er·at·ed, re·frig·er·at·ing, re·frig·er·ates 1. To cool or chill (a substance). 2. To preserve (food) by chilling. and AirConditioning Engineers [ASHRAE ASHRAE American Society of Heating, Refrigerating & Air Conditioning Engineers ]), * TMY TMY The Midnight Youth (band) (Typical Meteorological Year A typical meteorological year (TMY) is a collation of selected weather data for a specific location, generated from a data bank much longer than a year in duration. It is specially selected so that it 'showcases' the range of weather phenomena for the location in question: the ) developed by the National Renewable Energy Laboratory The National Renewable Energy Laboratory (NREL), located in Golden, Colorado, as part of the U.S. Department of Energy, is the United States' primary laboratory for renewable energy and energy efficiency research and development. [NREL NREL National Renewable Energy Laboratory NREL Natural Resource Ecology Laboratory (Colorado State University, Fort Collins, CO) ]), 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 (language, simulation) SIMulation ANalysis  (SIMAN) A simulation language, especially for manufacturing systems, developed by C. Dennis Pegden in 1983. ["Introduction to Simulation using SIMAN", C.D. Pegden et al, McGrawHill 1990]. are not available for several countries. The main hindrance hin·drance n. 1. a. The act of hindering. b. The condition of being hindered. 2. One that hinders; an impediment. See Synonyms at obstacle. 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 Zenith Angle can refer to:
adj. Capable of being precipitated. water, broadband aerosol aerosol (âr`əsōl,–sŏl): see colloid. aerosol System of tiny liquid or solid particles evenly distributed in a finely divided state through a gas, usually air. 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 tropics, also called tropical zone or torrid zone, all the land and water of the earth situated between the Tropic of Cancer at lat. 23 12°N and the Tropic of Capricorn at lat. 23 12°S. . The measured hourly solar radiation data were obtained from the National Solar Radiation Data Base (NSRDB NSRDB National SIGINT Requirements Database ) maintained by (NREL) for the US sites (Guam and Honolulu), and from local meteorological stations for other locations (Hong Kong Hong Kong (hŏng kŏng), Mandarin Xianggang, special administrative region of China, formerly a British crown colony (2005 est. pop. 6,899,000), land area 422 sq mi (1,092 sq km), adjacent to Guangdong prov. , Sao Paulo, Singapore, and Mexico City Mexico City Spanish Ciudad de México City (pop., 2000: city, 8,605,239; 2003 metro. area est., 18,660,000), capital of Mexico. Located at an elevation of 7,350 ft (2,240 m), it is officially coterminous with the Federal District, which occupies 571 sq mi ). 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 latitude, angular distance of any point on the surface of the earth north or south of the equator. The equator is latitude 0°, and the North Pole and South Pole are latitudes 90°N and 90°S, respectively. 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 The drybulb temperature is the temperature of air measured by a thermometer freely exposed to the air but shielded from radiation and moisture. In construction, it is an important consideration when designing a building for a certain climate. , dewpoint temperature, station pressure, sky cover, low sky cover, wind speed, wind direction, ceiling height, and relative humidity relative humidity n. The ratio of the amount of water vapor in the air at a specific temperature to the maximum amount that the air could hold at that temperature, expressed as a percentage. . 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 neural network or neural computing, computer architecture modeled upon the human brain's interconnected system of neurons. Neural networks imitate the brain's ability to sort out patterns and learn from trial and error, discerning and extracting 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 ir·ra·di·ant adj. Sending forth radiant light. [Latin irradi [I'.sub.g] is given by [I'.sub.g] = [I.sub.0][A.sub.3]exp exp abbr. 1. exponent 2. exponential ([A.sub.4][T.sub.i]m), (2) with: * [T.sub.l] the Linke turbidity turbidity /tur·bid·i·ty/ (terbid´ite) cloudiness; disturbance of solids (sediment) in a solution, so that it is not clear.tur´bid Turbidity The cloudiness or lack of transparency of a solution. factor, which is linked to other indices, such as optical depths for ozone, Rayleigh and aerosol scattering scattering In physics, the change in direction of motion of a particle because of a collision with another particle. The collision can occur between two charged particles; it need not involve direct physical contact. , and water vapor absorption, using expressions derived in Davies and McKay (1989) * CA, the total cloud amount The proportion of sky obscured by cloud, expressed as a fraction of sky covered. (fraction of celestial ce·les·tial adj. 1. Of or relating to the sky or the heavens: Planets are celestial bodies. 2. Of or relating to heaven; divine: celestial beings. 3. 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 Correlation Coefficient A measure that determines the degree to which two variable's movements are associated. The correlation coefficient is calculated as: , 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 gul n. A stylized octagonal motif in Oriental rugs. [Persian, rose; see julep.] 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 The solar elevation angle is the elevation angle of the sun. That is, the angle between the direction of the sun and the (idealized) horizon. It can be calculated, to a good approximation, using the following formula: [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 Hamburg, city, Germany Hamburg (häm`brkh), officially Freie und Hansestadt Hamburg (Free and Hanseatic City of Hamburg), city (1994 pop. , 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 solar constant, the average amount of radiant energy received by the earth's atmosphere from the sun; its value is about 2 calories per min incident on each square centimeter of the upper atmosphere. , 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 Regression coefficient Term yielded by regression analysis that indicates the sensitivity of the dependent variable to a particular independent variable. See: Parameter. regression coefficient 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 Supervised learning is a machine learning technique for creating a function from training data. The training data consist of pairs of input objects (typically vectors), and desired outputs. , unsupervised learning Unsupervised learning is a method of machine learning where a model is fit to observations. It is distinguished from supervised learning by the fact that there is no a priori output. In unsupervised learning, a data set of input objects is gathered. , 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 Neurons Nerve cells in the brain, brain stem, and spinal cord that connect the nervous system and the muscles. Mentioned in: Speech Disorders that are connected to each other. In this context, a "neuron neuron, specialized cell in animals that, as a unit of the nervous system, carries information by receiving and transmitting electrical impulses. neuron or nerve cell Any of the cells of the nervous system. " is a simplified mathematical model
The neuron depicted de·pict tr.v. de·pict·ed, de·pict·ing, de·picts 1. To represent in a picture or sculpture. 2. To represent in words; describe. See Synonyms at represent. 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 summation n. the final argument of an attorney at the close of a trial in which he/she attempts to convince the judge and/or jury of the virtues of the client's case. (See: closing argument) 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 “Monotonic” redirects here. For other uses, see Monotone. In mathematics, a monotonic function (or monotone function) is a function which preserves the given order. such as the standard sigmoid function “Scurve” redirects here. For the recording company, see SCurve Records. A sigmoid function is a mathematical function that produces a sigmoid curve — a curve having an "S" shape. , f(x) = 1/[1+exp(x)], or the hyperbolic hy·per·bol·ic also hy·per·bol·i·cal adj. 1. Of, relating to, or employing hyperbole. 2. Mathematics a. Of, relating to, or having the form of a hyperbola. b. tangent tangent, in mathematics. 1 In geometry, the tangent to a circle or sphere is a straight line that intersects the circle or sphere in one and only one point. function, f(x) = tan h(x). In some cases, linear activation function is used [f(x) = x] instead of a bounded function In mathematics, a function f defined on some set X with real or complex values is called bounded, if the set of its values is bounded. In other words, there exists a number M>0 such that n. 1. Any of various tendrilbearing plants of the genus Cucurbita, having fleshy edible fruit with a leathery rind and unisexual flowers. 2. The fruit of any of these plants, eaten as a vegetable. bounded activation functions are appropriate when the target variables have to be constrained con·strain tr.v. con·strained, con·strain·ing, con·strains 1. To compel by physical, moral, or circumstantial force; oblige: felt constrained to object. See Synonyms at force. 2. 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 Gradient descent is an optimization algorithm. To find a local minimum of a function using gradient descent, one takes steps proportional to the negative of the gradient (or the approximate gradient) of the function at the current point. 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 mem·o·rize tr.v. mem·o·rized, mem·o·riz·ing, mem·o·riz·es 1. To commit to memory; learn by heart. 2. Computer Science To store in memory: " the noise in training data. Various techniques exist that reduce this overtraining overtraining training horses or dogs too hard so that they lose spirit. overtraining Sports medicine A general term for any practice of, or training for, a particular sport which is in excess of that necessary to participate in the sport , which 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 it·er·a·tive adj. 1. Characterized by or involving repetition, recurrence, reiteration, or repetitiousness. 2. Grammar Frequentative. Noun 1. 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 o·ver·es·ti·mate tr.v. o·ver·es·ti·mat·ed, o·ver·es·ti·mat·ing, o·ver·es·ti·mates 1. To estimate too highly. 2. To esteem too greatly. the effect of CA. [FIGURE 2 OMITTED] Over the year (including nighttime values), the mean bias error (MBE MBE (in Britain) Member of the Order of the British Empire MBE n abbr (BRIT) (= Member of the Order of the British Empire) → título ceremonial MBE n abbr (Brit) (= ) is 53.8 W/[m.sup.2] and the root mean square error (RMSE RMSE Root Mean Square Error RMSE Root Mean Squared Error ) 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 scat·ter v. 1. To cause to separate and go in different directions. 2. To separate and go in different directions; disperse. 3. To deflect radiation or particles. n. 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 In statistics, a mathematical method of modeling the relationships among three or more variables. It is used to predict the value of one variable given the values of the others. For example, a model might estimate sales based on age and gender. . 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 MSE Mouse (computer) MSE Materials Science & Engineering MSE Mean Squared Error MSE Mean Square Error MSE Master of Science in Engineering MSE Manufacturing Systems Engineering MSE Mechanically Stabilized Earth 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 sinh abbr. hyperbolic sine sinh Abbreviation of hyperbolic sine ), [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 Noun 1. horizontal surface  a flat surface at right angles to a plumb line; "park the car on the level" level floor, flooring  the inside lower horizontal surface (as of a room, hallway, tent, or other structure); "they needed rugs to cover the bare , 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 For the city and county of Honolulu, see City & County of Honolulu. “Honolulu” redirects here. For other uses, see Honolulu (disambiguation). Honolulu is the capital as well as the most populous community of the State of Hawaii, United States. , 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 nomenclature /no·men·cla·ture/ (no´menkla?cher) a classified system of names, as of anatomical structures, organisms, etc. binomial 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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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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