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Verification of some very simple clear and cloudy sky models to evaluate global solar irradiance
PII: S0038-092X(97)00057-1
Solar Energy Vol. 61, No. 4, pp. 251–264, 1997 © 1997 Elsevier Science Ltd All rights reserved. Printed in Great Britain 0038-092X/97 $17.00+0.00
VERIFICATION OF SOME VERY SIMPLE CLEAR AND CLOUDY SKY MODELS TO EVALUATE GLOBAL SOLAR IRRADIANCE VIOREL BADESCU† Candida Oancea Institute of Solar Energy, Faculty of Mechanical Engineering, Polytechnic University of Bucharest, Spl Independentei 313, Bucharest 79590, Romania ( E-mail: [email protected]) Received 14 June 1996; revised version accepted 25 May 1997 Communicated by AMOS ZEMEL Abstract—Several very simple clear sky and cloudy sky global irradiance models were tested under the climate and latitudes of Romania (Eastern Europe). The very simple clear sky models do not require meteorological data while the very simple cloudy sky models need only data on the total cloud amount. Three slightly more complex cloudy sky models which use additional meteorological inputs were also tested. The mean absolute error of the very simple clear sky models varies between 7 and 14% in July and between 12 and 19% in January. The best model originates from Western Europe. The performance of the very simple cloudy sky models is comparable to that of the more complicated ones. This is the main conclusion of the paper. Generally, their root mean square error varies between 35 and 45% in close agreement with results from the literature. Twelve classes of daily average cloudiness were defined. The model accuracy is better for the first six classes (with smaller cloud amount) and worse for the other six classes. © 1997 Elsevier Science Ltd.
1. INTRODUCTION
With an increasing number of solar energy applications the need for solar radiation data becomes more and more important. The kind of data required depends on application and user. For example, average monthly or daily data are required for climatological studies or to conduct feasibility studies for solar energy systems. Data for hourly (or shorter) periods are needed to simulate the performance of solar devices or during collector testing and other activities. In most countries, however, the spatial density of the actinometric station is inadequate (Davies et al., 1988). This situation prompted the development of calculation procedures to provide radiation estimates for places where measurements are not carried out and for places where there are gaps in the measurement records. There are numerous models for solar radiation computation, ranging from very complicated computer codes to empirical relations (for reviews see e.g. the reports of the Task IX of the International Energy Agency (IEA): May et al., 1984; Bener, 1984; Davies et al., 1988; Festa and Ratto, 1993). Choosing among these models usually takes into account two features: (1) the availability of meteorological and other kind of data required as the model input and † ISES member. 251
(2) the model accuracy. For most practical purposes and users the first criterion renders the sophisticated programs based on the solution of the radiative transfer equation unusable. As a consequence, the other models (which we call here ‘‘simple models’’) were widely tested. This paper deals with very simple models for computing global solar irradiance. Here a ‘‘very simple model’’ is defined as follows: (i) very simple clear sky models do not require any meteorological parameter as input and (ii) very simple cloudy sky models require as input a single meteorological parameter associated to the cloudiness degree (e.g. the cloud amount or the relative sunshine). The very simple models are important because the majority of the people involved in practical solar energy applications have access (for various reasons) to this kind of models only. The paper has two main purposes. First, one intends to prove that very simple models with sufficient accuracy can be identified for a given location. These models have to obey the current users exigencies (i.e. to evaluate ‘‘instantaneous’’ global irradiance). Second, one studies the dependence of their accuracy on different procedures of averaging the input data. Indeed, many users have access to averaged meteorological data only. The models we test in this work belong to two categories. First, there are the very simple
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models, which are our main interest. Second, there are some simple models. Their testing is important from two points of view. First, because the literature is surprisingly silent concerning testing of solar radiation models in Eastern Europe. Second, these models can be used as a reference level for the first category of procedures. Actinometric and meteorological data from Romania are used here. However, both the subject and the results are of general significance. Indeed, Romania serves here as an arbitrarily chosen environment for model application, and it is expected that very simple (but relatively accurate) models can be identified for any other location. In Section 2 we give details about the meteorological and actinometric data bases used in this work. Also, we present the statistical indicators of accuracy used to evaluate the models performance. Section 3 deals with the evaluation of five very simple cloudless sky models. First, they are used to compute global irradiance for clear sky. Then, their accuracy to predict the global irradiance for sky with a small cloud amount is tested. Five simple and two very simple cloudy sky models are verified in Section 4. Three types of model evaluations are performed. They differ in the way the input data (cloud amount and atmospheric air temperature) are averaged. Section 5 contains the conclusion of this work. 2. DATA AND INDICATORS OF ACCURACY
Meteorological and actinometric data measured in the following two Romanian localities were used in this work: Bucharest ( latitude 44.5°N, longitude 26.2°E, altitude 131 m asl ) and Jassy (47.2°N, 27.6°E, 130 m asl ). The climate of both localities is temperate-continental. The climatic index of continentality (Ivanov) is 131.9% at Bucharest and 129.9% at Jassy (Badescu, 1991). Data measured on about 1200 days during January and July were used in the analysis. In Bucharest we used data collected in the years 1960–1969 while in Jassy the data we used were collected during 1964–1973 ( RMHI, 1974). The data consist of global solar radiation, total amount of cloud cover and ambient temperature. They were measured at 6.00, 9.00, 12.00, 15.00 and 18.00 local standard time (LST ) in July and at 9.00, 12.00 and 15.00 LST in January. Mean monthly values of atmospheric
air pressure and relative humidity were also used. The readings of global solar irradiance were performed on Robitzsch actinographs whose maximum relative error was evaluated at 5% (Neacsa and Susan, 1984; Ciocoiu et al., 1974; Creteanu, 1984). The total cloud amount was estimated by eye by trained weather observers (in tenths of the celestial vault). Here we denote by PC (point cloudiness) the fractional total cloud amount. The daily average of point cloudiness ( PC ) was simply computed as the day arithmetic mean of the three or five ‘‘instantaneous’’ estimations. We used several statistical indicators for testing the models. They include the usual root mean square and mean bias errors (RMSE and MBE, respectively) and the mean absolute error (MAE ). To obtain dimensionless statistical indicators we expressed MBE, RMSE and MAE as fractions of the mean solar global irradiance during the respective time interval. This was the current practice during the major testing activities performed under the IEA Task IX (Davies et al., 1988). Thus, MBE, MAE and RMSE are given by: n ∑ (G −G ) i,comp i,meas MBE= i=1 , n ∑ G i,meas i=1 n ∑ |G −G | i,comp i,meas MAE= i=1 , n ∑ G i,meas i=1 1 n 1/2 ∑ (G −G )2 i,comp i,meas n i=1 RMSE= 1 n ∑ G (1) i,meas n i=1 where G and G are the i-th measured i,meas i,comp and computed values of global solar irradiance, respectively, while n is the number of measurements taken into account.
C
D
3. VERY SIMPLE CLOUDLESS SKY MODELS
Numerous very simple cloudless sky solar radiation models have been previously proposed. Seven models were selected in a first stage. After a short testing procedure we decided not to use the models of Lumb (1964) and
Clear and cloudy sky models to evaluate global solar irradiance
Suehrcke and McCormick (1988), which perform poorly under the temperate-continental climatic conditions of Romania. Finally, five very simple models were tested. They are: (1) Daneshyar–Paltridge–Proctor model (DPP) (Daneshyar, 1978; Paltridge and Proctor, 1976); (2) Kasten–Czeplak model ( KC ) ( Kasten and Czeplak, 1980); (3) Hourwitz model (H ) (Hourwitz, 1945; Hourwitz, 1946); (4) Berger–Duffie model (BD) (Berger, 1979); and (5) Adnot–Bourges–Campana–Gicquel model (ABCG) (Adnot et al., 1979). A short description of these models is given in Appendix B. We performed two types of model evaluations, described in Sections 3.1 and 3.2. These evaluations refer to skies with different degree of cloudiness. The purpose of the two evaluations was to outline the models performance and to study the possibility of using the models beyond their ‘‘design’’ limit (i.e. for cloudy skies), respectively. In Section 3.1 the models are used to compute global irradiance on clear skies. Their accuracy in predicting the global irradiance for skies with (a small ) cloud amount is tested in Section 3.2. 3.1. Very simple clear sky models applied for clear skies As a first step, we selected from the data set those readings of global solar irradiance associated with clear sky situations (PC=0). The statistics of these readings are summarized in Table 1. They reflect the dependence of the measured global irradiance on both month and latitude. No official report is available concerning the dependence of the measurement accuracy on the level of solar irradiance. We applied the five very simple models to all clear sky readings (for general computing relationships see Appendix A). The models show a comparable accuracy ( Table 2). A ranking procedure similar to that of Davies et al. (1988) Table 1. Statistics of measured global solar irradiance associated with clear sky situations (PC=0). Data from Bucharest and Jassy during January and July were used Bucharest Month Number of measurements Minimum ( W m−2) Maximum ( W m−2) Average ( W m−2) Standard deviation ( W m−2)
Jassy
July 258
January 147
July 175
January 94
36.6 975.7 485.0 292.7
27.9 692.0 253.3 131.0
39.1 961.7 441.3 277.3
27.18 719.95 213.7 123.2
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was used. The models were ranked for each performance statistic (MBE, MAE and RMSE ). For each statistic a weight of 5 was assigned to the model with the best performance, 4 to the one with the second best, and so on. Then, the ranking counts were pooled for both months and both localities and the final hierarchy was established. The ABCG and H models perform the best. The accuracy is higher in the summer and lower in the winter, as other authors already observed (from the recent literature see e.g. Degaetano et al., 1995). There is a slight dependence on locality. The result obtained at Jassy are worse than those obtained at Bucharest, where the standard deviation of the observed data is also higher (see Table 1). It is interesting to compare our results with those obtained in Canada by Gueymard (1993), who tested eleven simple clear sky models which use meteorological data as inputs to compute hourly average global irradiance (among others). Generally, for the eleven models he obtained a MBE between 4.6 and 7.8% while the RMSE varied between 4.4 and 10.5%. However, for Montreal, which is closer in latitude to Romania, the MBE and RMSE varied between 0.3 and 13.6% and between 6.6 and 16.6%, respectively. Thus, the very simple models we test here are (relatively) not far away from the more complicated models. Figure 1 shows the correlation between the measured data and the global irradiance computed by using the ABCG model. There is a systematic underestimation of the measured data. The separate ‘‘clouds’’ of filled squares can be related to the finite (and small ) number of readings per day. The first left cloud corresponds to the minimum values measured in July and January. It also contains the values below the mean from January. This first cloud is relatively free of systematic errors. The second cloud comprises the values below the mean from July, while the other two clouds correspond to the maximum values from January and July, respectively. All these three clouds have systematic underestimation, especially in July. A more complete description of the ABCG model accuracy is given in Fig. 2, where an ad hoc ‘‘average clear month’’ July was built by using those readings of global irradiance collected in Bucharest during the months of July of 1960–1969 associated with a null point cloudiness. As we see, the accuracy of the model is generally better in the middle of the day. This
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Model DPP KC H BD ABCG
July Bucharest MAE MBE RMSE MAE 0.14 0.11 0.09 0.09 0.07
−0.09 −0.06 0.08 0.03 −0.03
0.17 0.15 0.12 0.12 0.11
0.14 0.11 0.14 0.12 0.09
Jassy MBE
RMSE
−0.06 −0.02 0.13 0.08 0.01
0.17 0.14 0.16 0.15 0.11
January Bucharest MAE MBE RMSE MAE 0.17 0.14 0.12 0.14 0.14
−0.09 −0.06 0.05 0.09 −0.07
0.22 0.18 0.16 0.19 0.18
0.19 0.16 0.12 0.15 0.16
Jassy MBE
RMSE
−0.09 −0.10 0.00 0.08 −0.11
0.24 0.20 0.15 0.19 0.21
The models were used to compute global solar irradiance for clear skies (PC=0) at Bucharest and Jassy during January and July. RMSE is the root mean square error, MBE is the mean bias error and MAE is the mean absolute error. The models are: DPP: Daneshyar–Paltridge–Proctor (Daneshyar, 1978; Paltridge and Proctor, 1976); KC: Kasten–Czeplak ( Kasten and Czeplak, 1980); H: Hourwitz (Hourwitz, 1945, Hourwitz, 1946); BD: Berger–Duffie (Berger, 1979); ABCG: Adnot–Bourges–Campana–Gicquel (Adnot et al., 1979).
model intrinsic limitations but also of the decrease in the measurement accuracy, which usually occurs at a lower level of global irradiance. It is interesting to note, however, that recent studies showed that a possible cause of such a systematic error can be simply a few minutes error in the reading times (Suehrcke, 1994).
Fig. 1. Scatter plot of observed and computed values of global irradiance on clear sky. The very simple model ABCG was used. It was applied to all clear sky situations (PC=0) from Bucharest and Jassy during all the months of January and July considered in this study.
is, of course, a good recommendation for the potential users of the model engaged in solar energy applications. Indeed, the model has a better accuracy during that period of the day when usually the solar devices provide the most part of their useful energy. The larger differences between the observed and computed values near sunrise and (especially) near sunset seem to be systematic. This could be a consequence of the
3.2. Very simple clear sky models applied for cloudy skies It is known that the procedures to evaluate solar energy for clear skies are generally more precise and easy to use than those dealing with solar radiation for cloudy skies. Thus, a natural question is: what is the accuracy of the very simple clear sky models during days characterized by a low degree of cloudiness? As a first step we classified the 1200 days with observed data in twelve classes, according to the values of their daily average point cloudiness PC . As an example, Table 3 gives day details on the cloudiness classes from July at Bucharest. ‘‘The average’’ PC in this table day is the mean of all the average daily values
Fig. 2. Relative errors for global irradiance values computed by using the very simple model ABCG. An average ‘‘clear month’’ July was built by using only those readings of global irradiance collected in Bucharest during the months of July in 1960–1969 associated to a null point cloudiness. The relative error is defined as: (computed value/measured value)−1.
Clear and cloudy sky models to evaluate global solar irradiance Table 3. Features of the cloudiness classes in July at Bucharest (1960–69)
Class of cloudiness
Range of
PC day
1 2 3 4 5 6 7 8 9 10 11 12
0.00 0.01–0.10 0.11–0.20 0.21–0.30 0.31–0.40 0.41–0.50 0.51–0.60 0.61–0.70 0.71–0.80 0.81–0.90 0.91–0.99 1.00
Number of days 6 12 27 33 34 25 36 28 28 27 14 2
Average
PC day 0.00 0.058 0.143 0.243 0.335 0.433 0.533 0.647 0.727 0.834 0.933 1.00
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a
Average air temperature (°C ) 25.7 26.2 26.0 26.3 26.6 25.9 24.7 23.8 23.4 21.6 20.8 18.9
of point cloudiness for the given class. Classification of days according to their cloudiness is justified by the persistence of cloud cover amount, especially during the days with high cloud cover (Davies and McKay, 1982; Young et al., 1995). This persistence is the main reason for the good performance of some solar radiation computing models when 3 or even 6-hourly cloud cover data are used (Davies et al., 1988; Young et al., 1995). Next, we applied the cloudless sky models for all the ‘‘instantaneous’’ data belonging to all the days from classes 1 to 5. Figures 3(a) and 3(b) show the results for the best models, which proved again to be ABCG and H. In July the ABCG model gives better results and a very small systematic error. Its accuracy decreases by increasing the cloudiness class but the statistical indicators are quite acceptable even for class 5. The same is true for the H model. In January, the H model is more accurate, with a small systematic error. The ABCG model systematically underestimates the solar irradiance. There is a decrease in accuracy for the higher cloudiness classes but the statistical indicators remain reasonably small. More details about the scattering of the results can be found in Fig. 4 which refers to the ABCG model applied in July at Bucharest. The number of larger errors is similar for the neighbor classes (compare classes 1 and 2 in Fig. 4(a) or classes 4 and 5 in Fig. 4(b)). However, the classes of higher cloudiness have a larger number of large errors (compare classes 1 and 2 on the one hand ( Fig. 4(a)), and classes 4 and 5 on the other hand (Fig. 4(b))). It is interesting to note in Fig. 4(b) that the larger errors occur at lower global irradiance, i.e. near sunrise and sunset. In addition to other possible
b
Fig. 3. Statistical indicators for the very simple models ABCG and H applied to days belonging to different cloudiness classes at Jassy. (a) July; (b) January.
a
b Fig. 4. Scatter plot of the results obtained by using the very simple model ABCG for days belonging to different cloudiness classes during July at Bucharest. (a) Classes 1 and 2; (b) classes 4 and 5.
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causes (see the end of Section 3.1), this could be a result of the increase in the cloud amount at the beginning and at the end of the day. 4. SIMPLE AND VERY SIMPLE CLOUDY SKY MODELS
Extensive testing performed by the IEA Task IX ranked the cloud layer models as the best simple solar radiation models (Davies et al., 1988). Cloud layer recordings are kept by few stations in the world. However, the models based on total cloud cover amount, which are ranked the second best by the IEA studies, are attractive as this meteorological parameter is estimated regularly by many stations throughout the world. As an example, the number of actinometric stations in Romania (with a surface area of 237,500 km2) is less than ten while long-term measurements of relative sunshine and total cloud amount are carried out in over one hundred and fifty stations. On the other hand, cloud layer data is not routinely recorded in any station. As a first step, we selected three simple cloudy sky models. The first is the BCLS model (Barbaro et al., 1979), which we extensively tested under Romanian climate and latitudes (Badescu, 1987). Our previous studies referred to the monthly average values of daily and hourly direct, diffuse and global radiation. Here we analyze the accuracy of the BCLS model to compute ‘‘instantaneous’’ global irradiance, which is a feature the original model lacked. Note that the BCLS model was also tested under the IEA Task IX and the conclusion was that it performs ‘‘surprisingly poorly’’ (Davies et al., 1988, p. 85). The reason could be an error in the extinction factor for the direct solar irradiance in the absence of scattering (Badescu, 1987). Note that BCLS is a sunshine-based model. We estimated the relative sunshine S as the complement of point cloudiness (S= 1−PC ). This is a usual approximation. Details on the modified version of the model (BCLSM ) are given in Appendix C. A second model is the modified version of the Kasten model ( KASM ), also tested under the IEA Task IX (Davies et al., 1988). The third model we selected was that proposed by Munro (M ) (Munro, 1991). This model was not tested by the IEA team. All these simple models ( KASM, BCLSM and M ) use several meteorological quantities
such as point cloudiness and atmospheric air pressure, temperature, and relative humidity as input data. In addition, we tested two very simple models (ABCGK and HK below) which require point cloudiness only as input data. These are ad hoc combinations of existing models. The five simple and very simple models are described in Appendix C. Before the model evaluation we tested three different formulae to compute the standard air mass (see Appendix A). All of them give practically the same results. In computations we used the formula from Gueymard (1993). From the vast literature on the subject we selected three different formulae to evaluate the precipitable water depth u (Appendix A). They w were proposed by Hann (1901), Barbaro et al. (1979) (slightly modified in Badescu (1987)) and Leckner (1978). Table 4 shows the accuracy of the simple model KASM when using these formulae. The model was applied for those readings of global irradiance associated with clear sky conditions (PC=0). The water depth formulae induce a small difference between the solar radiation estimations. The Hann’s formula give slightly better results. Thus, one confirms the conclusions of Davies et al. (1975) that generally the solar radiation models are not very sensitive to substantial errors in precipitable water. This is rather useful for our computations. Indeed, because of the lack of available ‘‘instantaneous’’ measurements we used the multi-year monthly averages as inputs for air relative humidity. They are 0.72 and 0.90 for July and January at Bucharest and 0.73 and 0.88 for the same months at Jassy. In the final computations we used Leckner’s formula because recent measurements of vertical water vapor density variation seem to confirm it (Psigloglou et al., 1994). The daily and hourly variation of point cloudiness is important, even if it is not comparable in magnitude to the variation of global (or beam) solar irradiance. Figure 5 shows a smoothed representation of point cloudiness variation in July 1960 at Bucharest. We performed three types of model evaluations, described in Sections 4.1–4.3. The three evaluations differed in the way the input data (point cloudiness and air temperature) were averaged. The purpose of the three evaluations was to outline the most appropriate way for solar radiation models to utilize cloud fraction information. The first model evaluation,
Clear and cloudy sky models to evaluate global solar irradiance
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Table 4. Statistical indicators of accuracy for the simple model KASM when applied at Jassy and Bucharest (values in parentheses) for clear sky situations (PC=0) Precipitable water formula Hann (1901) Barbaro et al. (1979) Leckner (1978)
MAE
July MBE
RMSE
MAE
January MBE
RMSE
0.08 (0.05) 0.10 (0.07) 0.08 (0.05)
0.03 (0.00) 0.08 (0.05) 0.06 (0.02)
0.10 (0.09) 0.12 (0.10) 0.11 (0.09)
0.15 (0.13) 0.15 (0.14) 0.15 (0.13)
0.01 (0.07) 0.04 (0.09) 0.03 (0.08)
0.17 (0.16) 0.18 (0.18) 0.18 (0.17)
Three different formulae for the precipitable water depth were used (see Appendix A). RMSE is the root mean square error, MBE is the mean bias error and MAE is the mean absolute error.
Fig. 5. Smoothed variation of point cloudiness in July 1960 at Bucharest derived from estimations performed at 6.00, 9.00, 12.00, 15.00 and 18.00 local standard time.
described in Section 4.1, used instantaneous input data, averaged for each of the 12 cloudiness classes from Section 3.2. These data represented the mean daily variation of cloud fraction and air temperature for each cloudiness class. The second evaluation, described in Section 4.2, used instantaneous input data, with no averaging applied. It also assessed how the estimated global irradiance was affected by monthly averaging of the input data. The third evaluation, described in Section 4.3, used instantaneous input data, but stratified the evaluation by the 12 cloudiness classes from Section 3.2. It also assessed the estimation of daily mean global irradiance from daily mean input data. 4.1. Average global irradiance per cloudiness classes First, by averaging the data from all the days belonging to a given cloudiness class, a mean daily variation of global irradiance, point cloud-
iness and ambient temperature were computed. The five cloudy sky models were applied to these 12 mean days attached to the 12 classes of cloudiness. The MBE and RMSE of the results obtained for Bucharest are shown in Fig. 6. The accuracy is better in July, when the dependence on the point cloudiness is small (except for the last two classes). The simple model KASM and the very simple model ABCGK generally perform the best. The BCLSM model is the worst. This model is characterized by a relatively large negative systematic error due mainly to the assumption S= 1−PC. Indeed, in the case of an ideal cloud cover observer we proved, using techniques of geometrical probabilities, that this approximation would be good, apart from some corrections depending on hour and cloud shape (Badescu, 1992). In fact, the estimation of total cloud cover amount by real observers is subject to perspective errors and the relationship
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a
b
c
et al., 1995 and references therein). Therefore, the approximation (S=1−PC ) tends to underestimate the sunshine periods, diminishing the contribution of the direct solar radiation. In January, the simple model KASM and the very simple model HK performed the best. Generally, all the models have negative systematic errors. The accuracy depends obviously on the class of cloudiness, but not in a linear manner. Practically, the classes 1 to 6 on the one hand, and 7 to 12 on the other one hand, are characterized by a relatively similar accuracy. As a next step of verification we tested the models for the same 12 mean days under two new assumptions: (A) the input for air temperature was the average value (arithmetic mean) for each mean day; and (B) the inputs for both air temperature and point cloudiness were the average values for each mean day. Thus, the daily variation of the computed global irradiance in case (B) is determined (for a given cloudiness class) by the daily variation of the astronomical parameters. a
d
b
Fig. 6. Statistical indicators for the simple cloudy sky models BCLSM, KASM and M and the very simple cloudy sky models ABCGK and HK applied to the 12 mean days attached to the 12 cloudiness classes at Bucharest. (a) July – MBE; (b) July – RMSE; (c) January – MBE; (d ) January – RMSE.
between S and the estimated PC is non-linear. Consequently, S could exceed 1−PC by as much as 0.2–0.3 (see Badescu, 1991; Gueymard
Fig. 7. Statistical indicators for the simple model KASM when applied in Bucharest for the 12 mean days attached to the 12 cloudiness classes. Computations were made for two cases: (i) the general case (i.e. the case of Fig. 6) and (ii) the inputs for both air temperature and point cloudiness were the average values for each mean day. The latter case is denoted PC. (a) July; (b) January.
Clear and cloudy sky models to evaluate global solar irradiance
Use of approximation (A) did not decrease the accuracy of the models. Assumption (B) has, as expected, more serious consequences. Figures 7 and 8 show the results obtained in July at Bucharest by using the simple model KASM and the very simple model ABCGK, respectively. In July both the systematic and non-systematic errors of the KASM model increase for the cloudiness classes 1 to 9 (Fig. 7(a)). For higher cloudiness classes MBE and RMSE decrease (or do not change). With a few exceptions, in January both MBE and RMSE decrease (Fig. 7(b)). Surprisingly, assumption (B) improves the performance of the very simple model ABCGK: both MBE and RMSE decrease in both July and January (Figs 8(a) and (b)). 4.2. ‘‘Instantaneous’’ global irradiance The above results are useful for the evaluation of solar collector performance during days with different point cloudiness (see e.g. Zamfir et al., 1994). However, the mean daily variation (of global irradiance and/or other meteorological and actinometric parameters), even classified according to the mean daily value of point cloudiness, may not be useful for some simula-
a
b
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tion procedures, where more realistic data should be used as inputs. Therefore, we tested the five simple and very simple models for the evaluation of ‘‘instantaneous’’ (not averaged) global irradiance. Three different cases were analyzed: (A) the input data were primary readings of point cloudiness and ambient temperature (monthly average values for air humidity and atmospheric pressure were used); (B) the same as (A) except the input data for air temperature were monthly average values; and (C ) all the input data (point cloudiness, air pressure, temperature and humidity) were monthly average values. Table 5 shows the statistical indicators of the simple model KASM and the very simple model ABCGK, ranked the first and the second best, respectively. In case (A) the systematic error is reasonable low, except for model ABCGK in January at Jassy. The RMSE varies roughly between 35 and 45%, with the KASM model being slightly more accurate. These values are in agreement with those obtained with a total cloud amount model by Young et al. (1995) in summer (RMSE between 40 and 50%) and with some of the results of Davies et al. (1988). Figure 9 shows the scattering of the observed and computed global irradiance for July at Bucharest for the two models. In both cases there is an overestimation for the range of low irradiance and an underestimation when the observed irradiance increases. The global radiation shows a weak dependence on the ambient temperature. Indeed, the monthly averaged temperature can be used instead of the instantaneous values, without a significant decrease in accuracy (compare cases (A) and (B) in Table 4). This observation considerably simplifies the practical applications. As expected, assumption (C ) has important negative consequences on the accuracy (see especially the RMSE values during January in Table 4). Thus, this approximation is not recommended when instantaneous global irradiance is required. 4.3. ‘‘Instantaneous’’ global irradiance stratified per cloudiness classes
Fig. 8. The same as Fig. 7 for the case of the very simple model ABCGK.
Is the model accuracy dependent on the daily average cloudiness degree? We applied the models to all the days of each cloudiness class. Two cases were considered: (A) the input data were primary readings of point cloudiness and ambient temperature (monthly average values for air humidity and atmospheric pressure were
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V. Badescu Table 5. Statistical indicators of accuracy for the simple model KASM and the very simple model ABCGK, applied to evaluate instantaneous global irradiance July Bucharest KASM (simple model )
MBE RMSE
ABCGK (very simple model )
MBE RMSE
A B C A B C A B C A B C
−0.02 −0.02 0.20 0.34 0.34 0.48 −0.07 −0.07 0.13 0.35 0.35 0.44
Jassy −0.03 −0.02 0.12 0.36 0.36 0.50 0.03 0.04 0.04 0.39 0.38 0.47
January Bucharest Jassy −0.10 −0.10 −0.26 0.42 0.42 0.81 −0.22 −0.22 −0.35 0.46 0.46 0.85
−0.08 −0.08 −0.20 0.41 0.41 0.67 −0.20 −0.20 −0.30 0.46 0.46 0.73
RMSE is the root mean square error and MBE is the mean bias error. Case A: the input data are primary readings of point cloudiness and ambient temperature (monthly average values of air humidity and atmospheric pressure were used ); case B: the same as (A) except that monthly average values are used for air temperature; case C: all input data (point cloudiness, air pressure, temperature and humidity) are monthly average values.
a
a b
b Fig. 9. Scatter plot of the observed and computed global irradiance for July at Bucharest for (a) the simple model KASM and (b) the very simple model ABCGK.
Fig. 10. The dependence of the statistical indicators on the cloudiness class for two models applied in July at Bucharest: (a) the simple model KASM; (b) the very simple model ABCGK.
used) and (B) all the input data (point cloudiness, air pressure, temperature and humidity) were monthly average values. The dependence of the statistical indicators on the cloud amount is shown in Fig. 10 in the
case of the simple model KASM and the very simple model ABCGK. The classes of cloudiness can be roughly divided into two groups: classes 1 to 6 on the one hand, and classes 7 to 12 on the other one hand. Both models show a
Clear and cloudy sky models to evaluate global solar irradiance
comparable accuracy for the first group of classes. Approximation (B) slightly decreases the accuracy. When classes 7 to 12 are considered, both the systematic and non-systematic errors have generally higher values. Procedure (B) clearly leads to less precise results compared to procedure (A), especially for classes 9, 10 and 11. It is interesting to compare these results with the accuracy of daily solar radiation estimates obtained by Degaetano et al. (1995) in the U.S.A. Their Table 3 shows that the largest MAE (between 28.3 and 52.6%) correspond to the lowest level of daily radiation (which is obviously obtained during days with a higher cloud cover amount). Thus, our results and the results of Degaetano et al. are in good concordance, taking into account that by daily averaging the errors decrease.
5. CONCLUSIONS
Five very simple models of clear sky solar global irradiance were tested under the climate and latitudes of Romania ( Eastern Europe). These models do not require meteorological data as inputs. All of them perform acceptably well. The best model (Adnot et al., 1979) originates from ( Western) Europe. The clear sky models can be used successfully for days with small cloud cover amount (point cloudiness less than 0.4). We identified two very simple cloudy sky models (which use only total cloud amount data) with sufficient accuracy. The root mean square error of the computed instantaneous global irradiance is in the range 35–45%. This is in agreement with the performance reported in the literature and/or evaluated in this paper for the more complicated models. The accuracy is higher in summer and lower in winter. The very simple models have two advantages for users involved in practical solar energy applications. First, they allow a quick evaluation of instantaneous global irradiance, simply by eye estimation of the total cloud cover amount. This was already stated in Young et al. (1995) who selected cloud layer and total cloud cover model approaches because this information is easily obtainable by scientific fields camps without extensive instrumentation. The second advantage is related to the persistence of the cloud cover amount. This allows to predict the global solar irradiance for the next few hours (and, perhaps, for the next day).
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The individual days were stratified into twelve classes of mean cloudiness. The very simple and simple solar radiation models performed better for the lower cloudiness classes. Two levels of accuracy were roughly outlined: higher accuracy for the first six classes (with small cloud amounts) and a lower accuracy for the other six classes. Several simplifications can be made without a significant loss in accuracy. The most important one is to replace the instantaneous values by the monthly averaged air temperature. Acknowledgements—I thank Dr I. Ciocoiu (Romanian Meteorological and Hydrological Institute) for providing access to the data base. I received strong material and moral support from Dr Elena Zamfir ( University of Bucharest). I thank Dr D. C. McKay (Environment Service of Canada) for keeping me in touch with the activities of Task IX of the IEA and Drs R. Festa and C. F. Ratto ( University of Genova) for supplying the 1993 IEA Task IX Report. Also, I thank the referees and Dr A. Zemel (Ben-Gurion University of the Negev) for valuable comments and suggestions which improved the final form of the manuscript.
REFERENCES Adnot J., Bourges B., Campana D. and Gicquel R. (1979). Utilisation des courbes de frequence cumulees pour le calcul des installation solaires. In Analise Statistique des Processus Meteorologiques Appliquee a l’Energie Solaire, Lestienne R. ( Ed), pp. 9–40. CNRS, Paris. Badescu V. (1987) Can the model proposed by Barbaro et al. be used to compute global solar radiation on the Romanian territory?. Solar Energy 38, 4, 247–254. Badescu V. (1991) Studies concerning the empirical relationship of cloud shade to point cloudiness (Romania). Theor. Appl. Climatol 44, 187–200. Badescu V. (1992) Over and under estimation of cloud amount: theory and Romanian observations. Int. J. Sol. Energy 11, 201–209. Barbaro S., Coppolino S., Leone C. and Sinagra E. (1979) An atmospheric model for computing direct and diffuse solar radiation. Solar Energy 22, 225–228. Bener P. (1984). Survey and comments on various methods to compute the components of solar irradiance on horizontal and inclined surfaces. In Handbook of Methods of Estimating Solar Radiation, pp. 47–77. Swedish Council for Building Research, Stockholm, Sweden. Berger X. (1979). Etude du Climat en Region Nicoise en vue d’Applications a l’Habitat Solaire. CNRS, Paris. Ciocoiu I., Elekes I. and Glodeanu F. (1974). Corelatia dintre radiatia globala si durata de stralucire a soarelui. In Culegerea Lucrarilor RMHI pe Anul. 1972, pp. 265–275. RMHI, Bucharest. Creteanu V. (1984). Cadastrul energiei solare destinat nevoilor energetice, St. Cerc. Meteorol., pp. 33–41. RMHI, Bucharest. Daneshyar M. (1978) Solar radiation statistics for Iran. Solar Energy 21, 345–349. Davies J. A., Schertzer W. and Nunez M. (1975) Estimating global solar radiation. Boundary-Layer Meteorol 9, 33–52. Davies J. A. and McKay D. C. (1982) Estimating solar irradiance and components. Solar Energy 29, 55–64.
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Davies J. A., McKay D. C., Luciani G. and Abdel-Wahab M. (1988). Validation of models for estimating solar radiation on horizontal surfaces, Vol. 1, IEA Task IX, Final Report. Atmospheric Environment Service of Canada, Downsview, Ontario, Canada. Degaetano A. T., Eggleston K. L. and Knapp W. W. (1995) A comparison of daily solar radiation estimates for the Northeastern United States using the Northeast Regional Climate Center and National Renewable Energy Laboratory models. Solar Energy 55, 3, 185–194. Festa R. and Ratto C. F. (1993). Solar radiation statistical properties. Technical Report for IEA Task IX, University of Genova. Gueymard C. (1993) Critical analysis and performance assessment of clear sky solar irradiance models using theoretical and measured data. Solar Energy 51, 2, 121–138. Gueymard C., Jindra P. and Estrada-Cajigal V. (1995). A ˚ ngstro¨m critical look at recent interpretations of the A approach and its future in global solar radiation. 54(5), 357–363. Hann J. (1901). Lehrbuch der Meteorologie, p. 225. Leipzig, Tauchnitz; quoted in May et al. (1984). Hourwitz B. (1945) Insolation in relation to cloudiness and cloud density. J. Met 2, 154–156. Hourwitz B. (1946) Insolation in relation to cloud type. J. Met 3, 123–124. Kasten H. (1966). A new table and approximation formula for the relative optical air mass. Archiv fur Meteorol. Geophys. und Bioklim., B, 206–223. Kasten F. (1983) Parametrisierung der globalstrahlung durch bedeckungsgrad und trubungsfactor. Ann. Met 20, 49–50. Kasten F. and Czeplak G. (1980) Solar and terrestrial radiation dependent on the amount and type of clouds. Solar Energy 24, 177–189. Leckner B. (1978) The spectral distribution of solar radiation at the Earth’s surface – elements of a model. Solar Energy 20, 143–150. Lumb F. E. (1964) The influence of cloud on hourly amounts of total solar radiation at the sea level. Q. J. R. Met. Soc 90, 43–56. May B. R., Collingbourne R. H. and McKay D. C. (1984). Catalogue of estimating methods. In Handbook of Methods of Estimating Solar Radiation, pp. 4–32. Swedish Council for Building Research, Stockholm, Sweden. Munro D. S. (1991) A surface energy exchange model of glacier melt and net mass balance. Int. J. Clim 11, 689–700. Neacsa O. and Susan V. (1984). Unele caracteristici ale duratei stralucirii soarelui deasupra teritoriului Romaniei, St. Cerc. Meteorol., pp. 99–113. RIMH, Bucharest. Paltridge G. W. and Proctor D. (1976) Monthly mean solar radiation statistics for Australia. Solar Energy 18, 235–243. Psigloglou B. E., Santamouris M. and Asimakopoulos D. N. (1994) On the atmospheric water vapor transmission function for solar radiation models. Solar Energy 53, 5, 445–453. RMHI (1974). Basis of Meteorological Data. Romanian Meteorological and Hydrological Institute, Bucharest. Suehrcke H. (1994) The effect of time errors on the accuracy of solar radiation measurements. Solar Energy 53, 4, 353–357. Suehrcke H. and McCormick P. G. (1988) The diffuse fraction of instantaneous radiation. Solar Energy 40, 423–430. Young K. L., Woo M. K. and Munro D. S. (1995) Simple approaches to modelling solar radiation in the Arctic. Solar Energy 54, 1, 33–40. Zamfir E., Oancea C. and Badescu V. (1994) Cloud cover influence on long-term performances of flat plate solar collectors. Renewable Energy 4, 3, 339–347.
APPENDIX A This Appendix gives some astronomical and general use formulae. We mainly follow the relationships from the IEA studies (Davies et al., 1988). The value adopted for the solar constant is I =1376 W m−2. For a given Julian day SC n, the intensity of the extraterrestrial solar radiation I 0 ( W m−2) is computed with (h=2p(n−1)/365): I 0=I (1.00011+0.034221 cos h+0.00128 sin h SC −0.000719 cos 2h+0.000077 sin 2h) (A1) The extraterrestrial solar irradiance on a horizontal surface G0 ( W m−2) is given by G0=I 0 cos z
(A2)
where the cosine of the zenith angle z was computed with cos z=sin w sin d+cos w cos d cos v
(A3)
Here w, d and v are the geographical latitude, the solar declination and the solar hour angle, respectively. d (radians) was evaluated with: d=0.006918−0.399912 cos h+0.070257 sin h −0.006759 cos 2h+0.000907 sin 2h −0.002697 cos 3h+0.00148 sin 3h
(A4)
The solar hour angle (degrees) is given by v=15|12−LAT|
(A5)
where LAT is the local apparent (true solar) time, determined from the local standard time LST, the equation of time ET (minutes), the geographical longitude of the site LS (degrees) and the standard meridian LSM (degrees) for the time zone: LAT=LST+ET/60+(LSM−LS)/15
(A6)
For Romania LSM=30°E. ET (minutes) is computed with: ET=0.000075+0.001868 cos h−0.032077 sin h −0.14615 cos 2h−0.04084 sin 2h
(A7)
The standard air mass m was computed with three st different formulae (z in degrees): m ( Kasten)=1/[cos z+0.15/(93.885−z)1.253] (A8) st m (Gueymard )=1/[cos z+a z(a +90−z)a3 ], st 1 2 a =0.00176759, a =4.37515, a =−1.21563 (A9) 1 2 3 m (Badescu)=[−cos z+(cos2 z+f 2−1)1/2]/( f−1), st f=1+11/6371.2 (A10) taken from Kasten (1966), Gueymard (1993) and Badescu (1987), respectively. The air mass m was obtained from the standard air mass m after a correction which took into st account the standard and actual atmospheric pressure, p(h) and p (h), respectively, depending on the altitude h of the st site. A procedure originating in the international atmospheric model CIRA 1961 was used (Badescu, 1987): m=[ p(h)/p (h)]m (A11) st st The standard pressure p (h) is computed by st p (h)=p (0)[T(h)/T (0)]5.2561 (A12) st st st where the standard pressure at sea level is p (0)=1017.085h Pa while the standard ambient temperst ature at sea level can be evaluated from the measured value T(h) ( K ) at altitude h (meters) by T (0)=T(h)+0.0065h st
(A13)
Clear and cloudy sky models to evaluate global solar irradiance Three different formulae were used to evaluate the thickness of precipitable water layer u (mm): w u (Hann)=0.025p u (A14) w sat u (Leckner)=4.93u/T exp(26.23−5416/T ) (A15) w u (Barbaro et al.)=2 · 106ur (T )/r (T ) (A16) w vap water where p (Pa) is the saturation pressure of water vapor, u sat (underunitary) and T ( K ) are air relative humidity and temperature at ground level, while r (kg m−3) and vap r (kg m−3) are the densities of vapor and liquid water. water The above formulae are due to Hann (1901), Leckner (1978) and Barbaro et al. (1979), respectively (the last one being slightly modified in Badescu (1987)).
APPENDIX B The very simple cloudless sky models are briefly described below. We denote the clear sky beam, diffuse and global irradiance, respectively by I , D and G (all of them in 0 0 0 W m−2) on a horizontal surface while z (radians) is the zenith angle.
The Daneshyar–Paltridge–Proctor model (DPP) (Daneshyar, 1978; Paltridge and Proctor, 1976) I =950.2{1−exp [−0.075(p/2−z)]} 0 D =14.29+21.04(p/2−z) 0 G =I +D 0 0 0
(B1) (B2) (B3)
(B4)
(B5)
The Berger–Duffie model (BD) (Berger, 1979) G =1350 · 0.70 · cos z 0
(B6)
The Adnot–Bourges–Campana–Gicquel model (ABCG) (Adnot et al., 1979) G =951.39 cos1.15 z 0
I =G0[0.938 exp(−0.00154X )]+[2.97X2.1 w 1 1 −773.24 · 10−5X3 +85058.73(1+X ) 1 1 /(1+10X2 )] · 10−3 (C3) 1 where X =mu . 1 w We evaluated short-term values of beam and diffuse irradiance for cloudy skies (I and D ( W m−2), respectively) by using formulae similar to those used in Barbaro et al. (1979) to evaluate daily totals. If one denotes by S the relative sunshine then: I=SI (C4) 0 D*=SD +t(1−S) (I +D ) (C5) 0 0 0 where the * denotes that an albedo correction is necessary in the case of the diffuse component. The cloud transmissivity factor t is well described for latitudes L (degrees) between 40 and 50°N by t=0.61−0.015L+0.0002L2
The Hourwitz model (H) (Hourwitz, 1945, Hourwitz, 1946) G =1098 cos z exp(−0.057/cos z) 0
Here d is the dust content (=200 particles cm−3), u is the w depth of the precipitable water layer (mm), k is an empirical coefficient given by k=0.5 cos1/3 z and I is the direct irradiw ance transmitted in the absence of scattering, given in W m−2 by
(C6)
The global irradiance is given by
The Kasten–Czeplak model (KC) (Kasten and Czeplak, 1980) G =910 cos z−30 0
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(B7)
APPENDIX C The simple and very simple cloudy sky models are briefly described below.
Simple models The modified BCLS model (BCLSM) (Barbaro et al., 1979; Badescu, 1987) The direct beam and diffuse irradiance for cloudless skies on a horizontal surface are given, in W m−2, by I and D below 0 0 I =G0[a +b u −a (d−400)] · exp {−[a +b u 0 1 1 w 3 2 2 w +b (d−400)]m}, a =−0.13491, a =0.13708, 3 1 2 a =3.68 · 10−5, b =−4.28 · 10−3, b =2.61 · 10−3, 3 1 2 b =1.131 · 10−4 (C1) 3 D =k(I −I ) (C2) 0 w 0
G=
(I+D*) 1−a[0.2+0.5(1−S)]
(C7)
where a (=0.2) is the ground albedo. The relative sunshine S was evaluated by using S=1−PC where PC is the point cloudiness (underunitary). The modified Kasten model (KASM) (Kasten, 1983; Davies et al., 1988) The global irradiance G for cloudy sky on a horizontal surface is given, in W m−2, by G=G (1−0.72PC3.2) (C8) 0 where the global irradiance for cloudless sky G was eval0 uated according to the MAC model (Davies and McKay, 1982, Davies et al., 1988): G =I +D +D (C9) 0 0 r a where I , D and D are the direct component and the 0 r a diffuse components due to Rayleigh and aerosol scattering, respectively, given in W m−2 by I =G0(T T −a )T (C10) 0 0 r w a D =G0T (1−T )/2 (C11) r 0 r D =G0(T T −a ) (1−T )a g (C12) a 0 r w a a Here G0 is extraterrestrial irradiance on a horizontal surface, T , T and T are the transmissivities for absorption 0 r a by ozone, for Rayleigh scattering and for extinction by aerosols, respectively, a is the absorptivity by water vapor, w a is the spectrally-averaged single-scattering albedo for a aerosols, while g is the ratio of forward to total scattering by aerosols. A depth of ozone layer u =3.5 mm was assumed. Then, 0 the transmissivity T is computed by: 0 x =u m (C13) 0 0 0.1082x 0.00658x 0 + 0 a = 0 1+13.86x0.805 1+(10.36x )3 0 0 0.00218 + (C14) 1+0.0042x +0.00000323x2 0 0 T =1−a (C15) 0 0 The relationship proposed for T in Davies et al. (1988, r p. 18) for the MAC model is in need of correction.
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Therefore, we used the equation proposed for the Jossefson model (Davies et al., 1988, p. 21): T =0.9768−0.0874m+0.010607552m2 r −8.46205 · 10−4m3+3.57246 · 10−5m4 −6.0176 · 10−7m5
(C16)
The absorptivity by water vapor is given by: x =mu (C17) 2 w 0.29x 2 (C18) a = w (1+14.15x )0.635+0.5925x 2 2 with u in mm. w The transmissivity T is given by a T =km (C19) a a where the unit air mass aerosol transmissivity k was chosen a to be 0.90 as an average between the values 0.91, 0.94, 0.87 and 0.90 from the four European localities analyzed in Davies et al. (1988) (i.e. De Bilt, Hamburg, Kew and Zurich). We assumed a =0.75 (Davies et al., 1988, p. 36) a and g=0.93−0.21 ln m (C20) The Munro model (M) (Munro, 1991) The direct irradiance under cloudy skies I on a horizontal surface is given by I=I (1−PC ) (C21) 0 where I is the direct irradiance under clear sky. We used 0 eqn (C10) to evaluate I . The diffuse irradiance under cloudy 0 sky D is given by D=D +D (C22) 1 2 where D and D correspond to the diffuse radiation from 1 2 the cloudless portion of the sky plus that passing through
the cloud layer and, respectively, the diffuse radiation due to multiple reflections between the cloud base and the ground. They are computed with D =D (1−PC )+PCI (1−a −a ) (C23) 1 0 0 n n (I+D )PCa a 1 b D = (C24) 2 1−a a b Here D is the cloudless sky diffuse irradiance, a is the 0 n absorptance of the cloud, and a , a and a are the reflectance n b of the cloud top, the cloud base and the albedo of the ground, respectively. D was evaluated as the sum of D 0 r and D , given by eqns (C11) and (C12), respectively. Other a values used are a =0.18, a =0.6 and a=0.2. a was comn b n puted with a =1+{exp(−0.5656mx)[ x(1−1.48m)+3.54m−1 n −2.62]−3.54m−1−2.62}(x+5.2)−1 (C25) where x is the ratio of cloud thickness to the mean free path of light through the cloud. The value x=1.5 was adopted. The global irradiance on cloudy sky G on a horizontal surface is given by G=I+D
(C26)
Very simple models The Adnot et al.–Kasten model (ABCGK) This model evaluates the global irradiance for cloudy sky G by using eqn (C8) from the KASM model and the cloudless sky irradiance G computed with the ABCG model (eqn (B7)). 0 The Hourwitz–Kasten model (HK) This model evaluates the global irradiance for cloudy sky G by using eqn (C8) from the KASM model and the cloudless sky irradiance G computed with the H model (eqn (B5)). 0
Solar Energy Vol. 61, No. 4, pp. 251–264, 1997 © 1997 Elsevier Science Ltd All rights reserved. Printed in Great Britain 0038-092X/97 $17.00+0.00
VERIFICATION OF SOME VERY SIMPLE CLEAR AND CLOUDY SKY MODELS TO EVALUATE GLOBAL SOLAR IRRADIANCE VIOREL BADESCU† Candida Oancea Institute of Solar Energy, Faculty of Mechanical Engineering, Polytechnic University of Bucharest, Spl Independentei 313, Bucharest 79590, Romania ( E-mail: [email protected]) Received 14 June 1996; revised version accepted 25 May 1997 Communicated by AMOS ZEMEL Abstract—Several very simple clear sky and cloudy sky global irradiance models were tested under the climate and latitudes of Romania (Eastern Europe). The very simple clear sky models do not require meteorological data while the very simple cloudy sky models need only data on the total cloud amount. Three slightly more complex cloudy sky models which use additional meteorological inputs were also tested. The mean absolute error of the very simple clear sky models varies between 7 and 14% in July and between 12 and 19% in January. The best model originates from Western Europe. The performance of the very simple cloudy sky models is comparable to that of the more complicated ones. This is the main conclusion of the paper. Generally, their root mean square error varies between 35 and 45% in close agreement with results from the literature. Twelve classes of daily average cloudiness were defined. The model accuracy is better for the first six classes (with smaller cloud amount) and worse for the other six classes. © 1997 Elsevier Science Ltd.
1. INTRODUCTION
With an increasing number of solar energy applications the need for solar radiation data becomes more and more important. The kind of data required depends on application and user. For example, average monthly or daily data are required for climatological studies or to conduct feasibility studies for solar energy systems. Data for hourly (or shorter) periods are needed to simulate the performance of solar devices or during collector testing and other activities. In most countries, however, the spatial density of the actinometric station is inadequate (Davies et al., 1988). This situation prompted the development of calculation procedures to provide radiation estimates for places where measurements are not carried out and for places where there are gaps in the measurement records. There are numerous models for solar radiation computation, ranging from very complicated computer codes to empirical relations (for reviews see e.g. the reports of the Task IX of the International Energy Agency (IEA): May et al., 1984; Bener, 1984; Davies et al., 1988; Festa and Ratto, 1993). Choosing among these models usually takes into account two features: (1) the availability of meteorological and other kind of data required as the model input and † ISES member. 251
(2) the model accuracy. For most practical purposes and users the first criterion renders the sophisticated programs based on the solution of the radiative transfer equation unusable. As a consequence, the other models (which we call here ‘‘simple models’’) were widely tested. This paper deals with very simple models for computing global solar irradiance. Here a ‘‘very simple model’’ is defined as follows: (i) very simple clear sky models do not require any meteorological parameter as input and (ii) very simple cloudy sky models require as input a single meteorological parameter associated to the cloudiness degree (e.g. the cloud amount or the relative sunshine). The very simple models are important because the majority of the people involved in practical solar energy applications have access (for various reasons) to this kind of models only. The paper has two main purposes. First, one intends to prove that very simple models with sufficient accuracy can be identified for a given location. These models have to obey the current users exigencies (i.e. to evaluate ‘‘instantaneous’’ global irradiance). Second, one studies the dependence of their accuracy on different procedures of averaging the input data. Indeed, many users have access to averaged meteorological data only. The models we test in this work belong to two categories. First, there are the very simple
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models, which are our main interest. Second, there are some simple models. Their testing is important from two points of view. First, because the literature is surprisingly silent concerning testing of solar radiation models in Eastern Europe. Second, these models can be used as a reference level for the first category of procedures. Actinometric and meteorological data from Romania are used here. However, both the subject and the results are of general significance. Indeed, Romania serves here as an arbitrarily chosen environment for model application, and it is expected that very simple (but relatively accurate) models can be identified for any other location. In Section 2 we give details about the meteorological and actinometric data bases used in this work. Also, we present the statistical indicators of accuracy used to evaluate the models performance. Section 3 deals with the evaluation of five very simple cloudless sky models. First, they are used to compute global irradiance for clear sky. Then, their accuracy to predict the global irradiance for sky with a small cloud amount is tested. Five simple and two very simple cloudy sky models are verified in Section 4. Three types of model evaluations are performed. They differ in the way the input data (cloud amount and atmospheric air temperature) are averaged. Section 5 contains the conclusion of this work. 2. DATA AND INDICATORS OF ACCURACY
Meteorological and actinometric data measured in the following two Romanian localities were used in this work: Bucharest ( latitude 44.5°N, longitude 26.2°E, altitude 131 m asl ) and Jassy (47.2°N, 27.6°E, 130 m asl ). The climate of both localities is temperate-continental. The climatic index of continentality (Ivanov) is 131.9% at Bucharest and 129.9% at Jassy (Badescu, 1991). Data measured on about 1200 days during January and July were used in the analysis. In Bucharest we used data collected in the years 1960–1969 while in Jassy the data we used were collected during 1964–1973 ( RMHI, 1974). The data consist of global solar radiation, total amount of cloud cover and ambient temperature. They were measured at 6.00, 9.00, 12.00, 15.00 and 18.00 local standard time (LST ) in July and at 9.00, 12.00 and 15.00 LST in January. Mean monthly values of atmospheric
air pressure and relative humidity were also used. The readings of global solar irradiance were performed on Robitzsch actinographs whose maximum relative error was evaluated at 5% (Neacsa and Susan, 1984; Ciocoiu et al., 1974; Creteanu, 1984). The total cloud amount was estimated by eye by trained weather observers (in tenths of the celestial vault). Here we denote by PC (point cloudiness) the fractional total cloud amount. The daily average of point cloudiness ( PC ) was simply computed as the day arithmetic mean of the three or five ‘‘instantaneous’’ estimations. We used several statistical indicators for testing the models. They include the usual root mean square and mean bias errors (RMSE and MBE, respectively) and the mean absolute error (MAE ). To obtain dimensionless statistical indicators we expressed MBE, RMSE and MAE as fractions of the mean solar global irradiance during the respective time interval. This was the current practice during the major testing activities performed under the IEA Task IX (Davies et al., 1988). Thus, MBE, MAE and RMSE are given by: n ∑ (G −G ) i,comp i,meas MBE= i=1 , n ∑ G i,meas i=1 n ∑ |G −G | i,comp i,meas MAE= i=1 , n ∑ G i,meas i=1 1 n 1/2 ∑ (G −G )2 i,comp i,meas n i=1 RMSE= 1 n ∑ G (1) i,meas n i=1 where G and G are the i-th measured i,meas i,comp and computed values of global solar irradiance, respectively, while n is the number of measurements taken into account.
C
D
3. VERY SIMPLE CLOUDLESS SKY MODELS
Numerous very simple cloudless sky solar radiation models have been previously proposed. Seven models were selected in a first stage. After a short testing procedure we decided not to use the models of Lumb (1964) and
Clear and cloudy sky models to evaluate global solar irradiance
Suehrcke and McCormick (1988), which perform poorly under the temperate-continental climatic conditions of Romania. Finally, five very simple models were tested. They are: (1) Daneshyar–Paltridge–Proctor model (DPP) (Daneshyar, 1978; Paltridge and Proctor, 1976); (2) Kasten–Czeplak model ( KC ) ( Kasten and Czeplak, 1980); (3) Hourwitz model (H ) (Hourwitz, 1945; Hourwitz, 1946); (4) Berger–Duffie model (BD) (Berger, 1979); and (5) Adnot–Bourges–Campana–Gicquel model (ABCG) (Adnot et al., 1979). A short description of these models is given in Appendix B. We performed two types of model evaluations, described in Sections 3.1 and 3.2. These evaluations refer to skies with different degree of cloudiness. The purpose of the two evaluations was to outline the models performance and to study the possibility of using the models beyond their ‘‘design’’ limit (i.e. for cloudy skies), respectively. In Section 3.1 the models are used to compute global irradiance on clear skies. Their accuracy in predicting the global irradiance for skies with (a small ) cloud amount is tested in Section 3.2. 3.1. Very simple clear sky models applied for clear skies As a first step, we selected from the data set those readings of global solar irradiance associated with clear sky situations (PC=0). The statistics of these readings are summarized in Table 1. They reflect the dependence of the measured global irradiance on both month and latitude. No official report is available concerning the dependence of the measurement accuracy on the level of solar irradiance. We applied the five very simple models to all clear sky readings (for general computing relationships see Appendix A). The models show a comparable accuracy ( Table 2). A ranking procedure similar to that of Davies et al. (1988) Table 1. Statistics of measured global solar irradiance associated with clear sky situations (PC=0). Data from Bucharest and Jassy during January and July were used Bucharest Month Number of measurements Minimum ( W m−2) Maximum ( W m−2) Average ( W m−2) Standard deviation ( W m−2)
Jassy
July 258
January 147
July 175
January 94
36.6 975.7 485.0 292.7
27.9 692.0 253.3 131.0
39.1 961.7 441.3 277.3
27.18 719.95 213.7 123.2
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was used. The models were ranked for each performance statistic (MBE, MAE and RMSE ). For each statistic a weight of 5 was assigned to the model with the best performance, 4 to the one with the second best, and so on. Then, the ranking counts were pooled for both months and both localities and the final hierarchy was established. The ABCG and H models perform the best. The accuracy is higher in the summer and lower in the winter, as other authors already observed (from the recent literature see e.g. Degaetano et al., 1995). There is a slight dependence on locality. The result obtained at Jassy are worse than those obtained at Bucharest, where the standard deviation of the observed data is also higher (see Table 1). It is interesting to compare our results with those obtained in Canada by Gueymard (1993), who tested eleven simple clear sky models which use meteorological data as inputs to compute hourly average global irradiance (among others). Generally, for the eleven models he obtained a MBE between 4.6 and 7.8% while the RMSE varied between 4.4 and 10.5%. However, for Montreal, which is closer in latitude to Romania, the MBE and RMSE varied between 0.3 and 13.6% and between 6.6 and 16.6%, respectively. Thus, the very simple models we test here are (relatively) not far away from the more complicated models. Figure 1 shows the correlation between the measured data and the global irradiance computed by using the ABCG model. There is a systematic underestimation of the measured data. The separate ‘‘clouds’’ of filled squares can be related to the finite (and small ) number of readings per day. The first left cloud corresponds to the minimum values measured in July and January. It also contains the values below the mean from January. This first cloud is relatively free of systematic errors. The second cloud comprises the values below the mean from July, while the other two clouds correspond to the maximum values from January and July, respectively. All these three clouds have systematic underestimation, especially in July. A more complete description of the ABCG model accuracy is given in Fig. 2, where an ad hoc ‘‘average clear month’’ July was built by using those readings of global irradiance collected in Bucharest during the months of July of 1960–1969 associated with a null point cloudiness. As we see, the accuracy of the model is generally better in the middle of the day. This
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V. Badescu Table 2. Statistical indicators of accuracy for five very simple clear sky models
Model DPP KC H BD ABCG
July Bucharest MAE MBE RMSE MAE 0.14 0.11 0.09 0.09 0.07
−0.09 −0.06 0.08 0.03 −0.03
0.17 0.15 0.12 0.12 0.11
0.14 0.11 0.14 0.12 0.09
Jassy MBE
RMSE
−0.06 −0.02 0.13 0.08 0.01
0.17 0.14 0.16 0.15 0.11
January Bucharest MAE MBE RMSE MAE 0.17 0.14 0.12 0.14 0.14
−0.09 −0.06 0.05 0.09 −0.07
0.22 0.18 0.16 0.19 0.18
0.19 0.16 0.12 0.15 0.16
Jassy MBE
RMSE
−0.09 −0.10 0.00 0.08 −0.11
0.24 0.20 0.15 0.19 0.21
The models were used to compute global solar irradiance for clear skies (PC=0) at Bucharest and Jassy during January and July. RMSE is the root mean square error, MBE is the mean bias error and MAE is the mean absolute error. The models are: DPP: Daneshyar–Paltridge–Proctor (Daneshyar, 1978; Paltridge and Proctor, 1976); KC: Kasten–Czeplak ( Kasten and Czeplak, 1980); H: Hourwitz (Hourwitz, 1945, Hourwitz, 1946); BD: Berger–Duffie (Berger, 1979); ABCG: Adnot–Bourges–Campana–Gicquel (Adnot et al., 1979).
model intrinsic limitations but also of the decrease in the measurement accuracy, which usually occurs at a lower level of global irradiance. It is interesting to note, however, that recent studies showed that a possible cause of such a systematic error can be simply a few minutes error in the reading times (Suehrcke, 1994).
Fig. 1. Scatter plot of observed and computed values of global irradiance on clear sky. The very simple model ABCG was used. It was applied to all clear sky situations (PC=0) from Bucharest and Jassy during all the months of January and July considered in this study.
is, of course, a good recommendation for the potential users of the model engaged in solar energy applications. Indeed, the model has a better accuracy during that period of the day when usually the solar devices provide the most part of their useful energy. The larger differences between the observed and computed values near sunrise and (especially) near sunset seem to be systematic. This could be a consequence of the
3.2. Very simple clear sky models applied for cloudy skies It is known that the procedures to evaluate solar energy for clear skies are generally more precise and easy to use than those dealing with solar radiation for cloudy skies. Thus, a natural question is: what is the accuracy of the very simple clear sky models during days characterized by a low degree of cloudiness? As a first step we classified the 1200 days with observed data in twelve classes, according to the values of their daily average point cloudiness PC . As an example, Table 3 gives day details on the cloudiness classes from July at Bucharest. ‘‘The average’’ PC in this table day is the mean of all the average daily values
Fig. 2. Relative errors for global irradiance values computed by using the very simple model ABCG. An average ‘‘clear month’’ July was built by using only those readings of global irradiance collected in Bucharest during the months of July in 1960–1969 associated to a null point cloudiness. The relative error is defined as: (computed value/measured value)−1.
Clear and cloudy sky models to evaluate global solar irradiance Table 3. Features of the cloudiness classes in July at Bucharest (1960–69)
Class of cloudiness
Range of
PC day
1 2 3 4 5 6 7 8 9 10 11 12
0.00 0.01–0.10 0.11–0.20 0.21–0.30 0.31–0.40 0.41–0.50 0.51–0.60 0.61–0.70 0.71–0.80 0.81–0.90 0.91–0.99 1.00
Number of days 6 12 27 33 34 25 36 28 28 27 14 2
Average
PC day 0.00 0.058 0.143 0.243 0.335 0.433 0.533 0.647 0.727 0.834 0.933 1.00
255
a
Average air temperature (°C ) 25.7 26.2 26.0 26.3 26.6 25.9 24.7 23.8 23.4 21.6 20.8 18.9
of point cloudiness for the given class. Classification of days according to their cloudiness is justified by the persistence of cloud cover amount, especially during the days with high cloud cover (Davies and McKay, 1982; Young et al., 1995). This persistence is the main reason for the good performance of some solar radiation computing models when 3 or even 6-hourly cloud cover data are used (Davies et al., 1988; Young et al., 1995). Next, we applied the cloudless sky models for all the ‘‘instantaneous’’ data belonging to all the days from classes 1 to 5. Figures 3(a) and 3(b) show the results for the best models, which proved again to be ABCG and H. In July the ABCG model gives better results and a very small systematic error. Its accuracy decreases by increasing the cloudiness class but the statistical indicators are quite acceptable even for class 5. The same is true for the H model. In January, the H model is more accurate, with a small systematic error. The ABCG model systematically underestimates the solar irradiance. There is a decrease in accuracy for the higher cloudiness classes but the statistical indicators remain reasonably small. More details about the scattering of the results can be found in Fig. 4 which refers to the ABCG model applied in July at Bucharest. The number of larger errors is similar for the neighbor classes (compare classes 1 and 2 in Fig. 4(a) or classes 4 and 5 in Fig. 4(b)). However, the classes of higher cloudiness have a larger number of large errors (compare classes 1 and 2 on the one hand ( Fig. 4(a)), and classes 4 and 5 on the other hand (Fig. 4(b))). It is interesting to note in Fig. 4(b) that the larger errors occur at lower global irradiance, i.e. near sunrise and sunset. In addition to other possible
b
Fig. 3. Statistical indicators for the very simple models ABCG and H applied to days belonging to different cloudiness classes at Jassy. (a) July; (b) January.
a
b Fig. 4. Scatter plot of the results obtained by using the very simple model ABCG for days belonging to different cloudiness classes during July at Bucharest. (a) Classes 1 and 2; (b) classes 4 and 5.
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V. Badescu
causes (see the end of Section 3.1), this could be a result of the increase in the cloud amount at the beginning and at the end of the day. 4. SIMPLE AND VERY SIMPLE CLOUDY SKY MODELS
Extensive testing performed by the IEA Task IX ranked the cloud layer models as the best simple solar radiation models (Davies et al., 1988). Cloud layer recordings are kept by few stations in the world. However, the models based on total cloud cover amount, which are ranked the second best by the IEA studies, are attractive as this meteorological parameter is estimated regularly by many stations throughout the world. As an example, the number of actinometric stations in Romania (with a surface area of 237,500 km2) is less than ten while long-term measurements of relative sunshine and total cloud amount are carried out in over one hundred and fifty stations. On the other hand, cloud layer data is not routinely recorded in any station. As a first step, we selected three simple cloudy sky models. The first is the BCLS model (Barbaro et al., 1979), which we extensively tested under Romanian climate and latitudes (Badescu, 1987). Our previous studies referred to the monthly average values of daily and hourly direct, diffuse and global radiation. Here we analyze the accuracy of the BCLS model to compute ‘‘instantaneous’’ global irradiance, which is a feature the original model lacked. Note that the BCLS model was also tested under the IEA Task IX and the conclusion was that it performs ‘‘surprisingly poorly’’ (Davies et al., 1988, p. 85). The reason could be an error in the extinction factor for the direct solar irradiance in the absence of scattering (Badescu, 1987). Note that BCLS is a sunshine-based model. We estimated the relative sunshine S as the complement of point cloudiness (S= 1−PC ). This is a usual approximation. Details on the modified version of the model (BCLSM ) are given in Appendix C. A second model is the modified version of the Kasten model ( KASM ), also tested under the IEA Task IX (Davies et al., 1988). The third model we selected was that proposed by Munro (M ) (Munro, 1991). This model was not tested by the IEA team. All these simple models ( KASM, BCLSM and M ) use several meteorological quantities
such as point cloudiness and atmospheric air pressure, temperature, and relative humidity as input data. In addition, we tested two very simple models (ABCGK and HK below) which require point cloudiness only as input data. These are ad hoc combinations of existing models. The five simple and very simple models are described in Appendix C. Before the model evaluation we tested three different formulae to compute the standard air mass (see Appendix A). All of them give practically the same results. In computations we used the formula from Gueymard (1993). From the vast literature on the subject we selected three different formulae to evaluate the precipitable water depth u (Appendix A). They w were proposed by Hann (1901), Barbaro et al. (1979) (slightly modified in Badescu (1987)) and Leckner (1978). Table 4 shows the accuracy of the simple model KASM when using these formulae. The model was applied for those readings of global irradiance associated with clear sky conditions (PC=0). The water depth formulae induce a small difference between the solar radiation estimations. The Hann’s formula give slightly better results. Thus, one confirms the conclusions of Davies et al. (1975) that generally the solar radiation models are not very sensitive to substantial errors in precipitable water. This is rather useful for our computations. Indeed, because of the lack of available ‘‘instantaneous’’ measurements we used the multi-year monthly averages as inputs for air relative humidity. They are 0.72 and 0.90 for July and January at Bucharest and 0.73 and 0.88 for the same months at Jassy. In the final computations we used Leckner’s formula because recent measurements of vertical water vapor density variation seem to confirm it (Psigloglou et al., 1994). The daily and hourly variation of point cloudiness is important, even if it is not comparable in magnitude to the variation of global (or beam) solar irradiance. Figure 5 shows a smoothed representation of point cloudiness variation in July 1960 at Bucharest. We performed three types of model evaluations, described in Sections 4.1–4.3. The three evaluations differed in the way the input data (point cloudiness and air temperature) were averaged. The purpose of the three evaluations was to outline the most appropriate way for solar radiation models to utilize cloud fraction information. The first model evaluation,
Clear and cloudy sky models to evaluate global solar irradiance
257
Table 4. Statistical indicators of accuracy for the simple model KASM when applied at Jassy and Bucharest (values in parentheses) for clear sky situations (PC=0) Precipitable water formula Hann (1901) Barbaro et al. (1979) Leckner (1978)
MAE
July MBE
RMSE
MAE
January MBE
RMSE
0.08 (0.05) 0.10 (0.07) 0.08 (0.05)
0.03 (0.00) 0.08 (0.05) 0.06 (0.02)
0.10 (0.09) 0.12 (0.10) 0.11 (0.09)
0.15 (0.13) 0.15 (0.14) 0.15 (0.13)
0.01 (0.07) 0.04 (0.09) 0.03 (0.08)
0.17 (0.16) 0.18 (0.18) 0.18 (0.17)
Three different formulae for the precipitable water depth were used (see Appendix A). RMSE is the root mean square error, MBE is the mean bias error and MAE is the mean absolute error.
Fig. 5. Smoothed variation of point cloudiness in July 1960 at Bucharest derived from estimations performed at 6.00, 9.00, 12.00, 15.00 and 18.00 local standard time.
described in Section 4.1, used instantaneous input data, averaged for each of the 12 cloudiness classes from Section 3.2. These data represented the mean daily variation of cloud fraction and air temperature for each cloudiness class. The second evaluation, described in Section 4.2, used instantaneous input data, with no averaging applied. It also assessed how the estimated global irradiance was affected by monthly averaging of the input data. The third evaluation, described in Section 4.3, used instantaneous input data, but stratified the evaluation by the 12 cloudiness classes from Section 3.2. It also assessed the estimation of daily mean global irradiance from daily mean input data. 4.1. Average global irradiance per cloudiness classes First, by averaging the data from all the days belonging to a given cloudiness class, a mean daily variation of global irradiance, point cloud-
iness and ambient temperature were computed. The five cloudy sky models were applied to these 12 mean days attached to the 12 classes of cloudiness. The MBE and RMSE of the results obtained for Bucharest are shown in Fig. 6. The accuracy is better in July, when the dependence on the point cloudiness is small (except for the last two classes). The simple model KASM and the very simple model ABCGK generally perform the best. The BCLSM model is the worst. This model is characterized by a relatively large negative systematic error due mainly to the assumption S= 1−PC. Indeed, in the case of an ideal cloud cover observer we proved, using techniques of geometrical probabilities, that this approximation would be good, apart from some corrections depending on hour and cloud shape (Badescu, 1992). In fact, the estimation of total cloud cover amount by real observers is subject to perspective errors and the relationship
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V. Badescu
a
b
c
et al., 1995 and references therein). Therefore, the approximation (S=1−PC ) tends to underestimate the sunshine periods, diminishing the contribution of the direct solar radiation. In January, the simple model KASM and the very simple model HK performed the best. Generally, all the models have negative systematic errors. The accuracy depends obviously on the class of cloudiness, but not in a linear manner. Practically, the classes 1 to 6 on the one hand, and 7 to 12 on the other one hand, are characterized by a relatively similar accuracy. As a next step of verification we tested the models for the same 12 mean days under two new assumptions: (A) the input for air temperature was the average value (arithmetic mean) for each mean day; and (B) the inputs for both air temperature and point cloudiness were the average values for each mean day. Thus, the daily variation of the computed global irradiance in case (B) is determined (for a given cloudiness class) by the daily variation of the astronomical parameters. a
d
b
Fig. 6. Statistical indicators for the simple cloudy sky models BCLSM, KASM and M and the very simple cloudy sky models ABCGK and HK applied to the 12 mean days attached to the 12 cloudiness classes at Bucharest. (a) July – MBE; (b) July – RMSE; (c) January – MBE; (d ) January – RMSE.
between S and the estimated PC is non-linear. Consequently, S could exceed 1−PC by as much as 0.2–0.3 (see Badescu, 1991; Gueymard
Fig. 7. Statistical indicators for the simple model KASM when applied in Bucharest for the 12 mean days attached to the 12 cloudiness classes. Computations were made for two cases: (i) the general case (i.e. the case of Fig. 6) and (ii) the inputs for both air temperature and point cloudiness were the average values for each mean day. The latter case is denoted PC. (a) July; (b) January.
Clear and cloudy sky models to evaluate global solar irradiance
Use of approximation (A) did not decrease the accuracy of the models. Assumption (B) has, as expected, more serious consequences. Figures 7 and 8 show the results obtained in July at Bucharest by using the simple model KASM and the very simple model ABCGK, respectively. In July both the systematic and non-systematic errors of the KASM model increase for the cloudiness classes 1 to 9 (Fig. 7(a)). For higher cloudiness classes MBE and RMSE decrease (or do not change). With a few exceptions, in January both MBE and RMSE decrease (Fig. 7(b)). Surprisingly, assumption (B) improves the performance of the very simple model ABCGK: both MBE and RMSE decrease in both July and January (Figs 8(a) and (b)). 4.2. ‘‘Instantaneous’’ global irradiance The above results are useful for the evaluation of solar collector performance during days with different point cloudiness (see e.g. Zamfir et al., 1994). However, the mean daily variation (of global irradiance and/or other meteorological and actinometric parameters), even classified according to the mean daily value of point cloudiness, may not be useful for some simula-
a
b
259
tion procedures, where more realistic data should be used as inputs. Therefore, we tested the five simple and very simple models for the evaluation of ‘‘instantaneous’’ (not averaged) global irradiance. Three different cases were analyzed: (A) the input data were primary readings of point cloudiness and ambient temperature (monthly average values for air humidity and atmospheric pressure were used); (B) the same as (A) except the input data for air temperature were monthly average values; and (C ) all the input data (point cloudiness, air pressure, temperature and humidity) were monthly average values. Table 5 shows the statistical indicators of the simple model KASM and the very simple model ABCGK, ranked the first and the second best, respectively. In case (A) the systematic error is reasonable low, except for model ABCGK in January at Jassy. The RMSE varies roughly between 35 and 45%, with the KASM model being slightly more accurate. These values are in agreement with those obtained with a total cloud amount model by Young et al. (1995) in summer (RMSE between 40 and 50%) and with some of the results of Davies et al. (1988). Figure 9 shows the scattering of the observed and computed global irradiance for July at Bucharest for the two models. In both cases there is an overestimation for the range of low irradiance and an underestimation when the observed irradiance increases. The global radiation shows a weak dependence on the ambient temperature. Indeed, the monthly averaged temperature can be used instead of the instantaneous values, without a significant decrease in accuracy (compare cases (A) and (B) in Table 4). This observation considerably simplifies the practical applications. As expected, assumption (C ) has important negative consequences on the accuracy (see especially the RMSE values during January in Table 4). Thus, this approximation is not recommended when instantaneous global irradiance is required. 4.3. ‘‘Instantaneous’’ global irradiance stratified per cloudiness classes
Fig. 8. The same as Fig. 7 for the case of the very simple model ABCGK.
Is the model accuracy dependent on the daily average cloudiness degree? We applied the models to all the days of each cloudiness class. Two cases were considered: (A) the input data were primary readings of point cloudiness and ambient temperature (monthly average values for air humidity and atmospheric pressure were
260
V. Badescu Table 5. Statistical indicators of accuracy for the simple model KASM and the very simple model ABCGK, applied to evaluate instantaneous global irradiance July Bucharest KASM (simple model )
MBE RMSE
ABCGK (very simple model )
MBE RMSE
A B C A B C A B C A B C
−0.02 −0.02 0.20 0.34 0.34 0.48 −0.07 −0.07 0.13 0.35 0.35 0.44
Jassy −0.03 −0.02 0.12 0.36 0.36 0.50 0.03 0.04 0.04 0.39 0.38 0.47
January Bucharest Jassy −0.10 −0.10 −0.26 0.42 0.42 0.81 −0.22 −0.22 −0.35 0.46 0.46 0.85
−0.08 −0.08 −0.20 0.41 0.41 0.67 −0.20 −0.20 −0.30 0.46 0.46 0.73
RMSE is the root mean square error and MBE is the mean bias error. Case A: the input data are primary readings of point cloudiness and ambient temperature (monthly average values of air humidity and atmospheric pressure were used ); case B: the same as (A) except that monthly average values are used for air temperature; case C: all input data (point cloudiness, air pressure, temperature and humidity) are monthly average values.
a
a b
b Fig. 9. Scatter plot of the observed and computed global irradiance for July at Bucharest for (a) the simple model KASM and (b) the very simple model ABCGK.
Fig. 10. The dependence of the statistical indicators on the cloudiness class for two models applied in July at Bucharest: (a) the simple model KASM; (b) the very simple model ABCGK.
used) and (B) all the input data (point cloudiness, air pressure, temperature and humidity) were monthly average values. The dependence of the statistical indicators on the cloud amount is shown in Fig. 10 in the
case of the simple model KASM and the very simple model ABCGK. The classes of cloudiness can be roughly divided into two groups: classes 1 to 6 on the one hand, and classes 7 to 12 on the other one hand. Both models show a
Clear and cloudy sky models to evaluate global solar irradiance
comparable accuracy for the first group of classes. Approximation (B) slightly decreases the accuracy. When classes 7 to 12 are considered, both the systematic and non-systematic errors have generally higher values. Procedure (B) clearly leads to less precise results compared to procedure (A), especially for classes 9, 10 and 11. It is interesting to compare these results with the accuracy of daily solar radiation estimates obtained by Degaetano et al. (1995) in the U.S.A. Their Table 3 shows that the largest MAE (between 28.3 and 52.6%) correspond to the lowest level of daily radiation (which is obviously obtained during days with a higher cloud cover amount). Thus, our results and the results of Degaetano et al. are in good concordance, taking into account that by daily averaging the errors decrease.
5. CONCLUSIONS
Five very simple models of clear sky solar global irradiance were tested under the climate and latitudes of Romania ( Eastern Europe). These models do not require meteorological data as inputs. All of them perform acceptably well. The best model (Adnot et al., 1979) originates from ( Western) Europe. The clear sky models can be used successfully for days with small cloud cover amount (point cloudiness less than 0.4). We identified two very simple cloudy sky models (which use only total cloud amount data) with sufficient accuracy. The root mean square error of the computed instantaneous global irradiance is in the range 35–45%. This is in agreement with the performance reported in the literature and/or evaluated in this paper for the more complicated models. The accuracy is higher in summer and lower in winter. The very simple models have two advantages for users involved in practical solar energy applications. First, they allow a quick evaluation of instantaneous global irradiance, simply by eye estimation of the total cloud cover amount. This was already stated in Young et al. (1995) who selected cloud layer and total cloud cover model approaches because this information is easily obtainable by scientific fields camps without extensive instrumentation. The second advantage is related to the persistence of the cloud cover amount. This allows to predict the global solar irradiance for the next few hours (and, perhaps, for the next day).
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The individual days were stratified into twelve classes of mean cloudiness. The very simple and simple solar radiation models performed better for the lower cloudiness classes. Two levels of accuracy were roughly outlined: higher accuracy for the first six classes (with small cloud amounts) and a lower accuracy for the other six classes. Several simplifications can be made without a significant loss in accuracy. The most important one is to replace the instantaneous values by the monthly averaged air temperature. Acknowledgements—I thank Dr I. Ciocoiu (Romanian Meteorological and Hydrological Institute) for providing access to the data base. I received strong material and moral support from Dr Elena Zamfir ( University of Bucharest). I thank Dr D. C. McKay (Environment Service of Canada) for keeping me in touch with the activities of Task IX of the IEA and Drs R. Festa and C. F. Ratto ( University of Genova) for supplying the 1993 IEA Task IX Report. Also, I thank the referees and Dr A. Zemel (Ben-Gurion University of the Negev) for valuable comments and suggestions which improved the final form of the manuscript.
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Davies J. A., McKay D. C., Luciani G. and Abdel-Wahab M. (1988). Validation of models for estimating solar radiation on horizontal surfaces, Vol. 1, IEA Task IX, Final Report. Atmospheric Environment Service of Canada, Downsview, Ontario, Canada. Degaetano A. T., Eggleston K. L. and Knapp W. W. (1995) A comparison of daily solar radiation estimates for the Northeastern United States using the Northeast Regional Climate Center and National Renewable Energy Laboratory models. Solar Energy 55, 3, 185–194. Festa R. and Ratto C. F. (1993). Solar radiation statistical properties. Technical Report for IEA Task IX, University of Genova. Gueymard C. (1993) Critical analysis and performance assessment of clear sky solar irradiance models using theoretical and measured data. Solar Energy 51, 2, 121–138. Gueymard C., Jindra P. and Estrada-Cajigal V. (1995). A ˚ ngstro¨m critical look at recent interpretations of the A approach and its future in global solar radiation. 54(5), 357–363. Hann J. (1901). Lehrbuch der Meteorologie, p. 225. Leipzig, Tauchnitz; quoted in May et al. (1984). Hourwitz B. (1945) Insolation in relation to cloudiness and cloud density. J. Met 2, 154–156. Hourwitz B. (1946) Insolation in relation to cloud type. J. Met 3, 123–124. Kasten H. (1966). A new table and approximation formula for the relative optical air mass. Archiv fur Meteorol. Geophys. und Bioklim., B, 206–223. Kasten F. (1983) Parametrisierung der globalstrahlung durch bedeckungsgrad und trubungsfactor. Ann. Met 20, 49–50. Kasten F. and Czeplak G. (1980) Solar and terrestrial radiation dependent on the amount and type of clouds. Solar Energy 24, 177–189. Leckner B. (1978) The spectral distribution of solar radiation at the Earth’s surface – elements of a model. Solar Energy 20, 143–150. Lumb F. E. (1964) The influence of cloud on hourly amounts of total solar radiation at the sea level. Q. J. R. Met. Soc 90, 43–56. May B. R., Collingbourne R. H. and McKay D. C. (1984). Catalogue of estimating methods. In Handbook of Methods of Estimating Solar Radiation, pp. 4–32. Swedish Council for Building Research, Stockholm, Sweden. Munro D. S. (1991) A surface energy exchange model of glacier melt and net mass balance. Int. J. Clim 11, 689–700. Neacsa O. and Susan V. (1984). Unele caracteristici ale duratei stralucirii soarelui deasupra teritoriului Romaniei, St. Cerc. Meteorol., pp. 99–113. RIMH, Bucharest. Paltridge G. W. and Proctor D. (1976) Monthly mean solar radiation statistics for Australia. Solar Energy 18, 235–243. Psigloglou B. E., Santamouris M. and Asimakopoulos D. N. (1994) On the atmospheric water vapor transmission function for solar radiation models. Solar Energy 53, 5, 445–453. RMHI (1974). Basis of Meteorological Data. Romanian Meteorological and Hydrological Institute, Bucharest. Suehrcke H. (1994) The effect of time errors on the accuracy of solar radiation measurements. Solar Energy 53, 4, 353–357. Suehrcke H. and McCormick P. G. (1988) The diffuse fraction of instantaneous radiation. Solar Energy 40, 423–430. Young K. L., Woo M. K. and Munro D. S. (1995) Simple approaches to modelling solar radiation in the Arctic. Solar Energy 54, 1, 33–40. Zamfir E., Oancea C. and Badescu V. (1994) Cloud cover influence on long-term performances of flat plate solar collectors. Renewable Energy 4, 3, 339–347.
APPENDIX A This Appendix gives some astronomical and general use formulae. We mainly follow the relationships from the IEA studies (Davies et al., 1988). The value adopted for the solar constant is I =1376 W m−2. For a given Julian day SC n, the intensity of the extraterrestrial solar radiation I 0 ( W m−2) is computed with (h=2p(n−1)/365): I 0=I (1.00011+0.034221 cos h+0.00128 sin h SC −0.000719 cos 2h+0.000077 sin 2h) (A1) The extraterrestrial solar irradiance on a horizontal surface G0 ( W m−2) is given by G0=I 0 cos z
(A2)
where the cosine of the zenith angle z was computed with cos z=sin w sin d+cos w cos d cos v
(A3)
Here w, d and v are the geographical latitude, the solar declination and the solar hour angle, respectively. d (radians) was evaluated with: d=0.006918−0.399912 cos h+0.070257 sin h −0.006759 cos 2h+0.000907 sin 2h −0.002697 cos 3h+0.00148 sin 3h
(A4)
The solar hour angle (degrees) is given by v=15|12−LAT|
(A5)
where LAT is the local apparent (true solar) time, determined from the local standard time LST, the equation of time ET (minutes), the geographical longitude of the site LS (degrees) and the standard meridian LSM (degrees) for the time zone: LAT=LST+ET/60+(LSM−LS)/15
(A6)
For Romania LSM=30°E. ET (minutes) is computed with: ET=0.000075+0.001868 cos h−0.032077 sin h −0.14615 cos 2h−0.04084 sin 2h
(A7)
The standard air mass m was computed with three st different formulae (z in degrees): m ( Kasten)=1/[cos z+0.15/(93.885−z)1.253] (A8) st m (Gueymard )=1/[cos z+a z(a +90−z)a3 ], st 1 2 a =0.00176759, a =4.37515, a =−1.21563 (A9) 1 2 3 m (Badescu)=[−cos z+(cos2 z+f 2−1)1/2]/( f−1), st f=1+11/6371.2 (A10) taken from Kasten (1966), Gueymard (1993) and Badescu (1987), respectively. The air mass m was obtained from the standard air mass m after a correction which took into st account the standard and actual atmospheric pressure, p(h) and p (h), respectively, depending on the altitude h of the st site. A procedure originating in the international atmospheric model CIRA 1961 was used (Badescu, 1987): m=[ p(h)/p (h)]m (A11) st st The standard pressure p (h) is computed by st p (h)=p (0)[T(h)/T (0)]5.2561 (A12) st st st where the standard pressure at sea level is p (0)=1017.085h Pa while the standard ambient temperst ature at sea level can be evaluated from the measured value T(h) ( K ) at altitude h (meters) by T (0)=T(h)+0.0065h st
(A13)
Clear and cloudy sky models to evaluate global solar irradiance Three different formulae were used to evaluate the thickness of precipitable water layer u (mm): w u (Hann)=0.025p u (A14) w sat u (Leckner)=4.93u/T exp(26.23−5416/T ) (A15) w u (Barbaro et al.)=2 · 106ur (T )/r (T ) (A16) w vap water where p (Pa) is the saturation pressure of water vapor, u sat (underunitary) and T ( K ) are air relative humidity and temperature at ground level, while r (kg m−3) and vap r (kg m−3) are the densities of vapor and liquid water. water The above formulae are due to Hann (1901), Leckner (1978) and Barbaro et al. (1979), respectively (the last one being slightly modified in Badescu (1987)).
APPENDIX B The very simple cloudless sky models are briefly described below. We denote the clear sky beam, diffuse and global irradiance, respectively by I , D and G (all of them in 0 0 0 W m−2) on a horizontal surface while z (radians) is the zenith angle.
The Daneshyar–Paltridge–Proctor model (DPP) (Daneshyar, 1978; Paltridge and Proctor, 1976) I =950.2{1−exp [−0.075(p/2−z)]} 0 D =14.29+21.04(p/2−z) 0 G =I +D 0 0 0
(B1) (B2) (B3)
(B4)
(B5)
The Berger–Duffie model (BD) (Berger, 1979) G =1350 · 0.70 · cos z 0
(B6)
The Adnot–Bourges–Campana–Gicquel model (ABCG) (Adnot et al., 1979) G =951.39 cos1.15 z 0
I =G0[0.938 exp(−0.00154X )]+[2.97X2.1 w 1 1 −773.24 · 10−5X3 +85058.73(1+X ) 1 1 /(1+10X2 )] · 10−3 (C3) 1 where X =mu . 1 w We evaluated short-term values of beam and diffuse irradiance for cloudy skies (I and D ( W m−2), respectively) by using formulae similar to those used in Barbaro et al. (1979) to evaluate daily totals. If one denotes by S the relative sunshine then: I=SI (C4) 0 D*=SD +t(1−S) (I +D ) (C5) 0 0 0 where the * denotes that an albedo correction is necessary in the case of the diffuse component. The cloud transmissivity factor t is well described for latitudes L (degrees) between 40 and 50°N by t=0.61−0.015L+0.0002L2
The Hourwitz model (H) (Hourwitz, 1945, Hourwitz, 1946) G =1098 cos z exp(−0.057/cos z) 0
Here d is the dust content (=200 particles cm−3), u is the w depth of the precipitable water layer (mm), k is an empirical coefficient given by k=0.5 cos1/3 z and I is the direct irradiw ance transmitted in the absence of scattering, given in W m−2 by
(C6)
The global irradiance is given by
The Kasten–Czeplak model (KC) (Kasten and Czeplak, 1980) G =910 cos z−30 0
263
(B7)
APPENDIX C The simple and very simple cloudy sky models are briefly described below.
Simple models The modified BCLS model (BCLSM) (Barbaro et al., 1979; Badescu, 1987) The direct beam and diffuse irradiance for cloudless skies on a horizontal surface are given, in W m−2, by I and D below 0 0 I =G0[a +b u −a (d−400)] · exp {−[a +b u 0 1 1 w 3 2 2 w +b (d−400)]m}, a =−0.13491, a =0.13708, 3 1 2 a =3.68 · 10−5, b =−4.28 · 10−3, b =2.61 · 10−3, 3 1 2 b =1.131 · 10−4 (C1) 3 D =k(I −I ) (C2) 0 w 0
G=
(I+D*) 1−a[0.2+0.5(1−S)]
(C7)
where a (=0.2) is the ground albedo. The relative sunshine S was evaluated by using S=1−PC where PC is the point cloudiness (underunitary). The modified Kasten model (KASM) (Kasten, 1983; Davies et al., 1988) The global irradiance G for cloudy sky on a horizontal surface is given, in W m−2, by G=G (1−0.72PC3.2) (C8) 0 where the global irradiance for cloudless sky G was eval0 uated according to the MAC model (Davies and McKay, 1982, Davies et al., 1988): G =I +D +D (C9) 0 0 r a where I , D and D are the direct component and the 0 r a diffuse components due to Rayleigh and aerosol scattering, respectively, given in W m−2 by I =G0(T T −a )T (C10) 0 0 r w a D =G0T (1−T )/2 (C11) r 0 r D =G0(T T −a ) (1−T )a g (C12) a 0 r w a a Here G0 is extraterrestrial irradiance on a horizontal surface, T , T and T are the transmissivities for absorption 0 r a by ozone, for Rayleigh scattering and for extinction by aerosols, respectively, a is the absorptivity by water vapor, w a is the spectrally-averaged single-scattering albedo for a aerosols, while g is the ratio of forward to total scattering by aerosols. A depth of ozone layer u =3.5 mm was assumed. Then, 0 the transmissivity T is computed by: 0 x =u m (C13) 0 0 0.1082x 0.00658x 0 + 0 a = 0 1+13.86x0.805 1+(10.36x )3 0 0 0.00218 + (C14) 1+0.0042x +0.00000323x2 0 0 T =1−a (C15) 0 0 The relationship proposed for T in Davies et al. (1988, r p. 18) for the MAC model is in need of correction.
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V. Badescu
Therefore, we used the equation proposed for the Jossefson model (Davies et al., 1988, p. 21): T =0.9768−0.0874m+0.010607552m2 r −8.46205 · 10−4m3+3.57246 · 10−5m4 −6.0176 · 10−7m5
(C16)
The absorptivity by water vapor is given by: x =mu (C17) 2 w 0.29x 2 (C18) a = w (1+14.15x )0.635+0.5925x 2 2 with u in mm. w The transmissivity T is given by a T =km (C19) a a where the unit air mass aerosol transmissivity k was chosen a to be 0.90 as an average between the values 0.91, 0.94, 0.87 and 0.90 from the four European localities analyzed in Davies et al. (1988) (i.e. De Bilt, Hamburg, Kew and Zurich). We assumed a =0.75 (Davies et al., 1988, p. 36) a and g=0.93−0.21 ln m (C20) The Munro model (M) (Munro, 1991) The direct irradiance under cloudy skies I on a horizontal surface is given by I=I (1−PC ) (C21) 0 where I is the direct irradiance under clear sky. We used 0 eqn (C10) to evaluate I . The diffuse irradiance under cloudy 0 sky D is given by D=D +D (C22) 1 2 where D and D correspond to the diffuse radiation from 1 2 the cloudless portion of the sky plus that passing through
the cloud layer and, respectively, the diffuse radiation due to multiple reflections between the cloud base and the ground. They are computed with D =D (1−PC )+PCI (1−a −a ) (C23) 1 0 0 n n (I+D )PCa a 1 b D = (C24) 2 1−a a b Here D is the cloudless sky diffuse irradiance, a is the 0 n absorptance of the cloud, and a , a and a are the reflectance n b of the cloud top, the cloud base and the albedo of the ground, respectively. D was evaluated as the sum of D 0 r and D , given by eqns (C11) and (C12), respectively. Other a values used are a =0.18, a =0.6 and a=0.2. a was comn b n puted with a =1+{exp(−0.5656mx)[ x(1−1.48m)+3.54m−1 n −2.62]−3.54m−1−2.62}(x+5.2)−1 (C25) where x is the ratio of cloud thickness to the mean free path of light through the cloud. The value x=1.5 was adopted. The global irradiance on cloudy sky G on a horizontal surface is given by G=I+D
(C26)
Very simple models The Adnot et al.–Kasten model (ABCGK) This model evaluates the global irradiance for cloudy sky G by using eqn (C8) from the KASM model and the cloudless sky irradiance G computed with the ABCG model (eqn (B7)). 0 The Hourwitz–Kasten model (HK) This model evaluates the global irradiance for cloudy sky G by using eqn (C8) from the KASM model and the cloudless sky irradiance G computed with the H model (eqn (B5)). 0