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Synthetic generation of standard sky types series using Markov Transition Matrices
Renewable Energy 62 (2014) 731e736
Contents lists available at ScienceDirect
Renewable Energy journal homepage: www.elsevier.com/locate/renene
Technical note
Synthetic generation of standard sky types series using Markov Transition Matrices J.L. Torres a, *, M. de Blas a, L.M. Torres b, A. García a, A. de Francisco c a
Department of Projects and Rural Engineering, Los Olivos, Campus Arrosadia, Public University of Navarre, 31006 Pamplona, Spain Department of Automatic and Computers, Los Pinos, Campus Arrosadia, Public University of Navarre, 31006 Pamplona, Spain c Department of Forestry Engineering, Polytechnic University of Madrid, Ciudad Universitaria, 28040 Madrid, Spain b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 26 March 2013 Accepted 15 August 2013 Available online 23 September 2013
The paper describes a methodology for creating synthetic time series of the fifteen standard sky types considered by the Commission International d’Eclairage (CIE). Since the measurements of sky luminance and accordingly CIE sky types are limited to a small number of locations, it is important to develop models to generate the time series of these parameters. The method is applied to two different locations [Pamplona (Spain) and Arcavacata di Rende (Italy)] where experimental luminance data are available. Firstly, the standard sky types corresponding to the observed luminance values are determined every 10 min and from these, Markov’s Transition Matrices (MTM) are obtained corresponding to the four seasons. Secondly, it is statistically verified that the process follows a Markov’s chain of first order and that it is stationary. Thirdly, obtained MTMs are used as a basis for generating the synthetic series of types of sky. Finally, experimental and synthetic time series are compared for the two locations, exhibiting good fitting results. As a conclusion, it is verified that first order MTM method can be used to generate time series of occurrence of CIE standard sky types, for the two locations. To clarify the general applicability, it should be applied to different locations having different climates and in addition with longer data sets. Ó 2013 Elsevier Ltd. All rights reserved.
Keywords: Sky types Synthetic time series Natural lighting
1. Introduction The exact knowledge of outdoor natural lighting is very important for bioclimatic architecture as well as for calculating lighting installations in buildings, mainly indoor. For this, it is necessary to know the existing luminance at different points of the sky dome. Although luminance distribution can be measured by means of a number of ground instruments [1] or even estimated from satellite images [2], up to now such measurements are limited to a few places and to relative short periods of time. On the one hand, important research has been done [3e6] for modeling the luminance distribution and different expressions have been proposed which have mathematical equations and coefficients depending on the sky type. The disadvantage is that they consider different sky type classifications [7].
* Corresponding author. Tel.: þ34 948 169175; fax: þ34 948 169148. E-mail addresses: [email protected] (J.L. Torres), [email protected] (M. de Blas), [email protected] (L.M. Torres), [email protected] (A. García), [email protected] (A. de Francisco). 0960-1481/$ e see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.renene.2013.08.022
CIE standard general sky [8] uses a similar formulation to that of Perez et al. [5] and considers up to 36 different sky types, from which there are 15 standards regarded as usual, which are showed in Table 1. In order to apply this standard for determining the luminance distribution it is necessary to know previously the sky type under consideration, this task being complex even for the 15 more common sky types. Three procedures have been described Wittkopf and Soon [9], Kittler et al. [10] and Tregenza [11,12]. The last one consists of comparing, in 145 positions of the sky hemisphere, the theoretical relative luminance of each sky type and the one observed. The application of this procedure to the successive sky conditions allows calculating the statistical distribution of the General Sky Types that best fits to the local sky luminance patterns and therefore knowing the local daylight climate. Related published work is concentrated in a reduced number of locations, such as Sheffield and Garston (Central and South England) [11,13,14], Bratislava and Athens [15,16], Hong Kong [17], Bangkok [18], Singapore [9,11] and Fukuoka [11]. In addition, some work is focused on a defined season, i.e. summer [14,15] or winter [13,16,19,20]. On the other hand, procedures for synthetic generation of time series of variables of interest have been investigated with
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J.L. Torres et al. / Renewable Energy 62 (2014) 731e736
Nomenclature and list of symbols amsl CC MBE RMSE
a g c2
above mean sea level correlation coefficient mean bias error root mean square error statistic of Markov’s chain dependence statistic of stationary of Markov’s chain Pearson’s chi-squared test
the aim of simulating the behavior of installations that integrate renewable energies. The application of Markov Transition Matrices (MTM) is one of the techniques used for this purpose. Aguiar and Collares-Pereira [21] proposed to use them for generating sequences of daily radiation values and Musselli et al. [22] demonstrated that the sequences of typical days (classified into three categories, i.e. clear, overcast and cloudy) fit to the said model and from here on they suggested to use MTM to generate synthetic series of typical days. Ehnberg and Bollen [23] used these matrices to simulate the cloud clover. Widen et al. [24] employed MTM to model the occupation of dwellings and from there to assess the domestic lighting demand in a residential area. This paper describes a methodology to generate synthetic series of the 15 standard sky types considered by CIE [8] in 10 min intervals. The method was tested in two locations of the few in which observed luminance angular distribution data are available: Pamplona, Spain (42.83 N, 1.64 W, 465 m amsl) and Arcavacata di Rende, Italy (39.38 N, 16.23 E, 365 m amsl). These synthetic series can be used as input data in the design and calculation of lighting systems utilizing computer codes, like Radiance [25] or Dialux [26] for analyzing the behavior of them over time and in the proper design of certain architectural elements, where natural lighting plays an important role. 2. Data Data are measured in the station sited on the roof of a building belonging to the Public University of Navarre, in Pamplona (42.83 N, 1.6 W, 435 m amsl). Sky scanner model SSK EKO MS 321 LR registered luminance values in the 145 positions recommended by
CIE every 10 min from 2007 to 2009. A preliminary comparison was carried out to ensure that the three year period used in this research was representative of the local weather conditions in Pamplona. The same model of instrument was used in Arcavacata di Rende (39.38 N, 16.23 E, 365 m amsl) for measuring luminance in 2008. 3. Methodology and calculation Firstly, it is calculated the standard sky type (Table 1) corresponding to the observed luminance values in each record of the time series, using the method of Tregenza [11,12] with slight variations [19]. Fig. 1 shows the sky type time series for winter in Pamplona and Fig. 2 shows the frequencies of occurrence of the different sky types grouped by seasons in Pamplona and Arcavacata di Rende. Significant differences between the two locations can be appreciated. Secondly, the experimental MTM and the corresponding marginal probabilities for each of the seasons are obtained in Pamplona and Arcavacata di Rende from seasonal occurrence series and it is statistically verified that the process has the properties of a Markov chain of first order. Thirdly, the different sequences of sky types for each season are divided in three intervals to evaluate if the process is stationary or not. The probability of transition matrices (MTM) is calculated and compared for each subinterval. As a result, the matrices corresponding to the different subintervals are approximately equal, which proves that the process is stationary. Finally, obtained MTM in the two locations are used as a basis for generating the synthetic time series of types of sky and synthetic and experimental time series are compared. 4. Results Table 2 included the statistical values for a and g for the experimental MTM obtained in each season and location. Since the value of c2 at the 5% level with 196 degrees of freedom is much smaller than any of the a statistic, it is found that successive transitions between sky types exhibit the first order dependence of the Markov chain. On the other hand, the value of c2 at the 5% level with 420 degrees of freedom is considerably greater than any of the g statistic, so the stationarity is proven.
15 14
Table 1 Number and characteristics of the 15 standard sky types considered in this work.
13
Type Characteristics
12
1
11
7 8 9 10 11 12 13 14 15
10
Sky types
2 3 4 5 6
Overcast CIE Standard Overcast Sky, steep gradation and with azimuthal uniformity. Overcast, with steep gradation and slight brightening towards the sun Overcast, moderately graded, with azimuthal uniformity. Overcast, moderately graded, with slight brightening towards the sun Overcast, foggy or cloudy, with overall uniformity. Partly cloudy, with uniform gradation and slight brightening towards the sun Partly cloudy, with a bright circumsolar effect and uniform gradation. Partly cloudy, rather uniform, with a clear solar corona. Partly cloudy, with obscured sun Partly cloudy, with brighter circumsolar region White-blue sky with distinct solar corona CIE Standard Clear Sky, low illuminance turbidity CIE Standard Clear Sky, polluted atmosphere. Cloudless turbid sky with broad solar corona. White-blue turbid sky with broad solar corona.
9 8 7 6 5 4 3 2 1
Time winter Fig. 1. Series of sky types in Pamplona in winter time.
J.L. Torres et al. / Renewable Energy 62 (2014) 731e736
733
Fig. 2. Frequency of occurrence of standard sky types grouped by seasons in the two locations considered.
Fig. 3 illustrates the contour lines corresponding to the transition probabilities between the different types of sky and the seasons of the year in Pamplona. It can be seen that the highest transition probabilities are between skies of features close together. In fact, the transition probabilities from the skies 1 to 5 (covered) to the skies 11 to 15 (clear) are generally very low and even zero or vice versa. This demonstrates the relative stability of the sky conditions between consecutive intervals of 10 min in the localities considered. In addition, it is also seen that the transition between different covered sky types is smoother than between clear skies (the contour lines on the bottom left of the graphs are less grouped and exhibit lower values than on the top right). In regard to the degree of fit between the experimental time series and the synthetically generated, Fig. 4 includes the marginal probabilities of occurrence of the types of skies of these time series for Pamplona. Finally, Table 3 includes RMSE and correlation coefficient (CC) values corresponding to the comparison, in each location and season, between the MTMs obtained, on the one hand, from the time series of observed sky types and, on the other hand, from the simulated time series generated using the MTM obtained from experimental data in each location. The CC corresponding to summer time in Arcavacata di Rende is indeterminate because the
Table 2 Values of statistics a and g to test the properties of Markov chains. Season
Pamplona
Arcavacata di Rende
a Statistic
g Statistic
a Statistic
g Statistic
Spring Summer Autumn Winter
19505.1 12683.6 7680.8 8543.6
69.17 354.86 314.55 83.39
5112.86 2074.35 5305.23 4781.3
143.13 166.64 143.12 175.18
probability of occurrence of the sky type 15 according to the available experimental data is null. The low RMSE values ([0.017; 0.026] for Pamplona and [0.021; 0.048] for Arcavacata di Rende) and the high CC values ([0.977; 0.988] for Pamplona and [0.951; 0.987] for Arcavacata di Rende) reveal that the proposed method, based on 10min transitions of the standard sky types allows easy simulation of the sequence of occurrence of standard sky types, as it is expected. Notwithstanding, when experimental data of Arcavacata di Rende are compared to the ones coming from simulated MTM of Pamplona, RMSE values increase four times and CC values become lower. 5. Conclusions The type of standard sky type was determined for each of the luminance distributions recorded by the sky scanner in each of the locations considered and occurrence of these types of sky were carried out. A simple methodology for generating synthetic series of standard sky types was established based on the use of MTM. As expected if the process responds to the statistical model used, the synthetic series of occurrence of sky types fit very well to the actual series observed at both locations. Consequently, the methodology can be used by designers of lighting installations in buildings to simulate the behavior with respect to time. However, it would be interesting to test the goodness of the procedure in other locations, when the experimental data needed to perform the analysis are available. In addition, it would be of interest to try to obtain general MTMs classified as a function of an easy to obtain indicator that allows selecting and applying a defined MTM suitable for a particular location without having to obtain the experimental MTM in it.
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J.L. Torres et al. / Renewable Energy 62 (2014) 731e736
Fig. 3. Transition probabilities between sky types in the different seasons corresponding to the MTM for Pamplona.
J.L. Torres et al. / Renewable Energy 62 (2014) 731e736
735
Fig. 4. Marginal probabilities of the simulated and experimental time series in Pamplona.
Table 3 Statistics for assessing the agreement between the experimental and simulated transition probability matrices. Arcavacata di Rende experimental vs. Arcavacata di Rende simulated
Arcavacata di Rende experimental vs. Pamplona simulated
Season
Pamplona experimental vs. Pamplona simulated RMSE
CC
RMSE
CC
RMSE
CC
Spring Summer Autumn Winter
0.017 0.018 0.020 0.026
0.988 0.987 0.985 0.977
0.048 0.044 0.025 0.021
0.951 Indeterminate 0.983 0.987
0.083 0.121 0.082 0.10
0.812 0.615 0.818 0.73
Acknowledgments The authors would like to thank Prof. Valerio Marinelli from the University of Calabria for providing them with luminance data of Arcavacata di Rende. References [1] Badescu V. Modeling solar radiation at the earth surface. Heidelberg: Springer; 2008. [2] Janjai S, Masiria I, Nunezb M, Laksanaboonsonga J. Modeling sky luminance using satellite data to classify sky conditions. Build Environ 2008;43(12): 2059e73. [3] Perraudeau M. Luminance models. In: National Lighting Conference and daylighting Colloquium, Cambridge 1988. p. 291e2. [4] Perez R, Ineichen P, Seals R, Michalsky J, Stewart R. Modeling daylight availability and irradiance components from direct and global irradiance. Solar Energy 1990;44:271e89. [5] Perez R, Seals R, Michalsky J. All weather model for sky luminance distribution. Preliminary configuration and validation. Solar Energy 1993;50: 235e45.
[6] Igawa N, Koga Y, Matsuzawa T, Nakamura H. Models of sky radiance distribution and sky luminance distribution. Solar Energy 2004;77:137e57. [7] Gracia AM, Torres JL, De Blas M, Illanes R. Comparación de las distintas clasificaciones de estados de cielo empleados en diferentes modelos de distribución angular de radiancia o luminancia. In: Actas del XIV Congreso Ibérico y IX Congreso Iberoamericano de Energía Solar, Vigo 2008. p. 11074e112. [8] CIE S 011/E spatial distribution of daylight e CIE standard general sky; 2004. [9] Wittkopf SK, Soon LK. Analysing sky luminance scans and predicting frequent sky patterns in Singapore. Light Res Technol 2007;39:31e51. [10] Kittler R, Perez R, Darula S. A new generation of sky standards. In: 8th European Lighting Conference, Amsterdam 1997. p. 359e73. [11] Tregenza PR. Standard skies for maritime climates. Light Res Technol 1999;31: 97e106. [12] Tregenza PR. Analysing sky luminance scans to obtain frequency distributions of CIE Standard General Skies. Light Res Technol 2004;36(4):271e81. [13] Markou MT, Kambezidis HD, Katsoulis BD, Muneer T, Bartzokas A. Sky type classification in South England during the winter period. Build Res J 2004;52(1):19e30. [14] Markou MT, Kambezidis HD, .Bartzokas A, Katsoulis BD, Muneer T. Sky type classification in Central England during winter. Energy 2005;30: 1667e74. [15] Bartzokas A, Darula S, Kambezidis HD, Kittler R. Sky luminance distribution in central Europe and the Mediterranean area during the winter period. J Atmos Solar-Terr Phys 2003;65:113e9. [16] Bartzokas A, Kambezidis HD, Darula S, Kittler R. Sky luminance distribution in central Europe and the Mediterranean area during the summer period. In: Proceedings of the 8th International Conference on Environmental Science and Technology, Greece 2003. p. 79e85. [17] Li DHW, Lau CCS, Lam JC. A study of 15 sky luminance patterns against Hong Kong data. Archit Sci Rev 2003;46:61e8. [18] Chaiwiwatworakul P, Chirarattananon S. Distribution of sky luminance in tropical climate. In: Proceedings of the Joint International Conference on Sustainable Energy and Environment, Hua Hin 2004. p. 530e7. [19] Torres JL, de Blas M, García A, Gracia A, de Francisco A. Sky luminance distribution in Pamplona (Spain) during the summer period. J Atmos Solar-Terr Phys 2010;72:382e8. [20] Torres JL, de Blas M, García A, Gracia A, de Francisco A. Sky luminance distribution in North of Iberian Peninsula during winter. J Atmos Solar-Terr Phys 2010;72:1147e54. [21] Aguiar R, Collares-Pereira M. A simple procedure for generating sequences of daily radiation values using a library of Markov transition matrices. Solar Energy 1988;40(3):269e79.
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[22] Muselli M, Poggi P, Notton G, Louche A. First order Markov chain model for generating synthetic “typical days” series of global irradiation in order to design photovoltaic stand-alone systems. Energy Convers Manag 2001;42: 675e87. [23] Ehnberg J, Bollen M. Simulation of global solar radiation based on cloud observations. Solar Energy 2005;78:157e62.
[24] Widen J, Nilsson A, Wäckelgârd E. A combined Markov-chain and bottom-up approach to modelling of domestic lighting demand. Energy Build 2009;41(10):1001e12. [25] Ward G. Radiance reference manual. USA: lighting systems research group. Lawrence Berkeley Laboratory; 1988. [26] Dialux 4.0. Germany Dial GmbH; 2005.
Contents lists available at ScienceDirect
Renewable Energy journal homepage: www.elsevier.com/locate/renene
Technical note
Synthetic generation of standard sky types series using Markov Transition Matrices J.L. Torres a, *, M. de Blas a, L.M. Torres b, A. García a, A. de Francisco c a
Department of Projects and Rural Engineering, Los Olivos, Campus Arrosadia, Public University of Navarre, 31006 Pamplona, Spain Department of Automatic and Computers, Los Pinos, Campus Arrosadia, Public University of Navarre, 31006 Pamplona, Spain c Department of Forestry Engineering, Polytechnic University of Madrid, Ciudad Universitaria, 28040 Madrid, Spain b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 26 March 2013 Accepted 15 August 2013 Available online 23 September 2013
The paper describes a methodology for creating synthetic time series of the fifteen standard sky types considered by the Commission International d’Eclairage (CIE). Since the measurements of sky luminance and accordingly CIE sky types are limited to a small number of locations, it is important to develop models to generate the time series of these parameters. The method is applied to two different locations [Pamplona (Spain) and Arcavacata di Rende (Italy)] where experimental luminance data are available. Firstly, the standard sky types corresponding to the observed luminance values are determined every 10 min and from these, Markov’s Transition Matrices (MTM) are obtained corresponding to the four seasons. Secondly, it is statistically verified that the process follows a Markov’s chain of first order and that it is stationary. Thirdly, obtained MTMs are used as a basis for generating the synthetic series of types of sky. Finally, experimental and synthetic time series are compared for the two locations, exhibiting good fitting results. As a conclusion, it is verified that first order MTM method can be used to generate time series of occurrence of CIE standard sky types, for the two locations. To clarify the general applicability, it should be applied to different locations having different climates and in addition with longer data sets. Ó 2013 Elsevier Ltd. All rights reserved.
Keywords: Sky types Synthetic time series Natural lighting
1. Introduction The exact knowledge of outdoor natural lighting is very important for bioclimatic architecture as well as for calculating lighting installations in buildings, mainly indoor. For this, it is necessary to know the existing luminance at different points of the sky dome. Although luminance distribution can be measured by means of a number of ground instruments [1] or even estimated from satellite images [2], up to now such measurements are limited to a few places and to relative short periods of time. On the one hand, important research has been done [3e6] for modeling the luminance distribution and different expressions have been proposed which have mathematical equations and coefficients depending on the sky type. The disadvantage is that they consider different sky type classifications [7].
* Corresponding author. Tel.: þ34 948 169175; fax: þ34 948 169148. E-mail addresses: [email protected] (J.L. Torres), [email protected] (M. de Blas), [email protected] (L.M. Torres), [email protected] (A. García), [email protected] (A. de Francisco). 0960-1481/$ e see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.renene.2013.08.022
CIE standard general sky [8] uses a similar formulation to that of Perez et al. [5] and considers up to 36 different sky types, from which there are 15 standards regarded as usual, which are showed in Table 1. In order to apply this standard for determining the luminance distribution it is necessary to know previously the sky type under consideration, this task being complex even for the 15 more common sky types. Three procedures have been described Wittkopf and Soon [9], Kittler et al. [10] and Tregenza [11,12]. The last one consists of comparing, in 145 positions of the sky hemisphere, the theoretical relative luminance of each sky type and the one observed. The application of this procedure to the successive sky conditions allows calculating the statistical distribution of the General Sky Types that best fits to the local sky luminance patterns and therefore knowing the local daylight climate. Related published work is concentrated in a reduced number of locations, such as Sheffield and Garston (Central and South England) [11,13,14], Bratislava and Athens [15,16], Hong Kong [17], Bangkok [18], Singapore [9,11] and Fukuoka [11]. In addition, some work is focused on a defined season, i.e. summer [14,15] or winter [13,16,19,20]. On the other hand, procedures for synthetic generation of time series of variables of interest have been investigated with
732
J.L. Torres et al. / Renewable Energy 62 (2014) 731e736
Nomenclature and list of symbols amsl CC MBE RMSE
a g c2
above mean sea level correlation coefficient mean bias error root mean square error statistic of Markov’s chain dependence statistic of stationary of Markov’s chain Pearson’s chi-squared test
the aim of simulating the behavior of installations that integrate renewable energies. The application of Markov Transition Matrices (MTM) is one of the techniques used for this purpose. Aguiar and Collares-Pereira [21] proposed to use them for generating sequences of daily radiation values and Musselli et al. [22] demonstrated that the sequences of typical days (classified into three categories, i.e. clear, overcast and cloudy) fit to the said model and from here on they suggested to use MTM to generate synthetic series of typical days. Ehnberg and Bollen [23] used these matrices to simulate the cloud clover. Widen et al. [24] employed MTM to model the occupation of dwellings and from there to assess the domestic lighting demand in a residential area. This paper describes a methodology to generate synthetic series of the 15 standard sky types considered by CIE [8] in 10 min intervals. The method was tested in two locations of the few in which observed luminance angular distribution data are available: Pamplona, Spain (42.83 N, 1.64 W, 465 m amsl) and Arcavacata di Rende, Italy (39.38 N, 16.23 E, 365 m amsl). These synthetic series can be used as input data in the design and calculation of lighting systems utilizing computer codes, like Radiance [25] or Dialux [26] for analyzing the behavior of them over time and in the proper design of certain architectural elements, where natural lighting plays an important role. 2. Data Data are measured in the station sited on the roof of a building belonging to the Public University of Navarre, in Pamplona (42.83 N, 1.6 W, 435 m amsl). Sky scanner model SSK EKO MS 321 LR registered luminance values in the 145 positions recommended by
CIE every 10 min from 2007 to 2009. A preliminary comparison was carried out to ensure that the three year period used in this research was representative of the local weather conditions in Pamplona. The same model of instrument was used in Arcavacata di Rende (39.38 N, 16.23 E, 365 m amsl) for measuring luminance in 2008. 3. Methodology and calculation Firstly, it is calculated the standard sky type (Table 1) corresponding to the observed luminance values in each record of the time series, using the method of Tregenza [11,12] with slight variations [19]. Fig. 1 shows the sky type time series for winter in Pamplona and Fig. 2 shows the frequencies of occurrence of the different sky types grouped by seasons in Pamplona and Arcavacata di Rende. Significant differences between the two locations can be appreciated. Secondly, the experimental MTM and the corresponding marginal probabilities for each of the seasons are obtained in Pamplona and Arcavacata di Rende from seasonal occurrence series and it is statistically verified that the process has the properties of a Markov chain of first order. Thirdly, the different sequences of sky types for each season are divided in three intervals to evaluate if the process is stationary or not. The probability of transition matrices (MTM) is calculated and compared for each subinterval. As a result, the matrices corresponding to the different subintervals are approximately equal, which proves that the process is stationary. Finally, obtained MTM in the two locations are used as a basis for generating the synthetic time series of types of sky and synthetic and experimental time series are compared. 4. Results Table 2 included the statistical values for a and g for the experimental MTM obtained in each season and location. Since the value of c2 at the 5% level with 196 degrees of freedom is much smaller than any of the a statistic, it is found that successive transitions between sky types exhibit the first order dependence of the Markov chain. On the other hand, the value of c2 at the 5% level with 420 degrees of freedom is considerably greater than any of the g statistic, so the stationarity is proven.
15 14
Table 1 Number and characteristics of the 15 standard sky types considered in this work.
13
Type Characteristics
12
1
11
7 8 9 10 11 12 13 14 15
10
Sky types
2 3 4 5 6
Overcast CIE Standard Overcast Sky, steep gradation and with azimuthal uniformity. Overcast, with steep gradation and slight brightening towards the sun Overcast, moderately graded, with azimuthal uniformity. Overcast, moderately graded, with slight brightening towards the sun Overcast, foggy or cloudy, with overall uniformity. Partly cloudy, with uniform gradation and slight brightening towards the sun Partly cloudy, with a bright circumsolar effect and uniform gradation. Partly cloudy, rather uniform, with a clear solar corona. Partly cloudy, with obscured sun Partly cloudy, with brighter circumsolar region White-blue sky with distinct solar corona CIE Standard Clear Sky, low illuminance turbidity CIE Standard Clear Sky, polluted atmosphere. Cloudless turbid sky with broad solar corona. White-blue turbid sky with broad solar corona.
9 8 7 6 5 4 3 2 1
Time winter Fig. 1. Series of sky types in Pamplona in winter time.
J.L. Torres et al. / Renewable Energy 62 (2014) 731e736
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Fig. 2. Frequency of occurrence of standard sky types grouped by seasons in the two locations considered.
Fig. 3 illustrates the contour lines corresponding to the transition probabilities between the different types of sky and the seasons of the year in Pamplona. It can be seen that the highest transition probabilities are between skies of features close together. In fact, the transition probabilities from the skies 1 to 5 (covered) to the skies 11 to 15 (clear) are generally very low and even zero or vice versa. This demonstrates the relative stability of the sky conditions between consecutive intervals of 10 min in the localities considered. In addition, it is also seen that the transition between different covered sky types is smoother than between clear skies (the contour lines on the bottom left of the graphs are less grouped and exhibit lower values than on the top right). In regard to the degree of fit between the experimental time series and the synthetically generated, Fig. 4 includes the marginal probabilities of occurrence of the types of skies of these time series for Pamplona. Finally, Table 3 includes RMSE and correlation coefficient (CC) values corresponding to the comparison, in each location and season, between the MTMs obtained, on the one hand, from the time series of observed sky types and, on the other hand, from the simulated time series generated using the MTM obtained from experimental data in each location. The CC corresponding to summer time in Arcavacata di Rende is indeterminate because the
Table 2 Values of statistics a and g to test the properties of Markov chains. Season
Pamplona
Arcavacata di Rende
a Statistic
g Statistic
a Statistic
g Statistic
Spring Summer Autumn Winter
19505.1 12683.6 7680.8 8543.6
69.17 354.86 314.55 83.39
5112.86 2074.35 5305.23 4781.3
143.13 166.64 143.12 175.18
probability of occurrence of the sky type 15 according to the available experimental data is null. The low RMSE values ([0.017; 0.026] for Pamplona and [0.021; 0.048] for Arcavacata di Rende) and the high CC values ([0.977; 0.988] for Pamplona and [0.951; 0.987] for Arcavacata di Rende) reveal that the proposed method, based on 10min transitions of the standard sky types allows easy simulation of the sequence of occurrence of standard sky types, as it is expected. Notwithstanding, when experimental data of Arcavacata di Rende are compared to the ones coming from simulated MTM of Pamplona, RMSE values increase four times and CC values become lower. 5. Conclusions The type of standard sky type was determined for each of the luminance distributions recorded by the sky scanner in each of the locations considered and occurrence of these types of sky were carried out. A simple methodology for generating synthetic series of standard sky types was established based on the use of MTM. As expected if the process responds to the statistical model used, the synthetic series of occurrence of sky types fit very well to the actual series observed at both locations. Consequently, the methodology can be used by designers of lighting installations in buildings to simulate the behavior with respect to time. However, it would be interesting to test the goodness of the procedure in other locations, when the experimental data needed to perform the analysis are available. In addition, it would be of interest to try to obtain general MTMs classified as a function of an easy to obtain indicator that allows selecting and applying a defined MTM suitable for a particular location without having to obtain the experimental MTM in it.
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Fig. 3. Transition probabilities between sky types in the different seasons corresponding to the MTM for Pamplona.
J.L. Torres et al. / Renewable Energy 62 (2014) 731e736
735
Fig. 4. Marginal probabilities of the simulated and experimental time series in Pamplona.
Table 3 Statistics for assessing the agreement between the experimental and simulated transition probability matrices. Arcavacata di Rende experimental vs. Arcavacata di Rende simulated
Arcavacata di Rende experimental vs. Pamplona simulated
Season
Pamplona experimental vs. Pamplona simulated RMSE
CC
RMSE
CC
RMSE
CC
Spring Summer Autumn Winter
0.017 0.018 0.020 0.026
0.988 0.987 0.985 0.977
0.048 0.044 0.025 0.021
0.951 Indeterminate 0.983 0.987
0.083 0.121 0.082 0.10
0.812 0.615 0.818 0.73
Acknowledgments The authors would like to thank Prof. Valerio Marinelli from the University of Calabria for providing them with luminance data of Arcavacata di Rende. References [1] Badescu V. Modeling solar radiation at the earth surface. Heidelberg: Springer; 2008. [2] Janjai S, Masiria I, Nunezb M, Laksanaboonsonga J. Modeling sky luminance using satellite data to classify sky conditions. Build Environ 2008;43(12): 2059e73. [3] Perraudeau M. Luminance models. In: National Lighting Conference and daylighting Colloquium, Cambridge 1988. p. 291e2. [4] Perez R, Ineichen P, Seals R, Michalsky J, Stewart R. Modeling daylight availability and irradiance components from direct and global irradiance. Solar Energy 1990;44:271e89. [5] Perez R, Seals R, Michalsky J. All weather model for sky luminance distribution. Preliminary configuration and validation. Solar Energy 1993;50: 235e45.
[6] Igawa N, Koga Y, Matsuzawa T, Nakamura H. Models of sky radiance distribution and sky luminance distribution. Solar Energy 2004;77:137e57. [7] Gracia AM, Torres JL, De Blas M, Illanes R. Comparación de las distintas clasificaciones de estados de cielo empleados en diferentes modelos de distribución angular de radiancia o luminancia. In: Actas del XIV Congreso Ibérico y IX Congreso Iberoamericano de Energía Solar, Vigo 2008. p. 11074e112. [8] CIE S 011/E spatial distribution of daylight e CIE standard general sky; 2004. [9] Wittkopf SK, Soon LK. Analysing sky luminance scans and predicting frequent sky patterns in Singapore. Light Res Technol 2007;39:31e51. [10] Kittler R, Perez R, Darula S. A new generation of sky standards. In: 8th European Lighting Conference, Amsterdam 1997. p. 359e73. [11] Tregenza PR. Standard skies for maritime climates. Light Res Technol 1999;31: 97e106. [12] Tregenza PR. Analysing sky luminance scans to obtain frequency distributions of CIE Standard General Skies. Light Res Technol 2004;36(4):271e81. [13] Markou MT, Kambezidis HD, Katsoulis BD, Muneer T, Bartzokas A. Sky type classification in South England during the winter period. Build Res J 2004;52(1):19e30. [14] Markou MT, Kambezidis HD, .Bartzokas A, Katsoulis BD, Muneer T. Sky type classification in Central England during winter. Energy 2005;30: 1667e74. [15] Bartzokas A, Darula S, Kambezidis HD, Kittler R. Sky luminance distribution in central Europe and the Mediterranean area during the winter period. J Atmos Solar-Terr Phys 2003;65:113e9. [16] Bartzokas A, Kambezidis HD, Darula S, Kittler R. Sky luminance distribution in central Europe and the Mediterranean area during the summer period. In: Proceedings of the 8th International Conference on Environmental Science and Technology, Greece 2003. p. 79e85. [17] Li DHW, Lau CCS, Lam JC. A study of 15 sky luminance patterns against Hong Kong data. Archit Sci Rev 2003;46:61e8. [18] Chaiwiwatworakul P, Chirarattananon S. Distribution of sky luminance in tropical climate. In: Proceedings of the Joint International Conference on Sustainable Energy and Environment, Hua Hin 2004. p. 530e7. [19] Torres JL, de Blas M, García A, Gracia A, de Francisco A. Sky luminance distribution in Pamplona (Spain) during the summer period. J Atmos Solar-Terr Phys 2010;72:382e8. [20] Torres JL, de Blas M, García A, Gracia A, de Francisco A. Sky luminance distribution in North of Iberian Peninsula during winter. J Atmos Solar-Terr Phys 2010;72:1147e54. [21] Aguiar R, Collares-Pereira M. A simple procedure for generating sequences of daily radiation values using a library of Markov transition matrices. Solar Energy 1988;40(3):269e79.
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J.L. Torres et al. / Renewable Energy 62 (2014) 731e736
[22] Muselli M, Poggi P, Notton G, Louche A. First order Markov chain model for generating synthetic “typical days” series of global irradiation in order to design photovoltaic stand-alone systems. Energy Convers Manag 2001;42: 675e87. [23] Ehnberg J, Bollen M. Simulation of global solar radiation based on cloud observations. Solar Energy 2005;78:157e62.
[24] Widen J, Nilsson A, Wäckelgârd E. A combined Markov-chain and bottom-up approach to modelling of domestic lighting demand. Energy Build 2009;41(10):1001e12. [25] Ward G. Radiance reference manual. USA: lighting systems research group. Lawrence Berkeley Laboratory; 1988. [26] Dialux 4.0. Germany Dial GmbH; 2005.