**4. Approximate dependence of the four daylight situations on relative sunshine duration**

In the paper by Kittler & Darula (2002) a P-D-G diagram was published to show Bratislava 5-minute data covering the whole July 1996. From 5315 cases were 3113 with sunshine while 2202 measured cases were without sunshine according to the WMO (1983) classification. The monthly relative sunshine duration after 1-minute recordings was in July 1996 on the average *s* 0.52 with daily changes within the range 0.022 – 0.946 which indicates the possibility of half-day situations in all four categories. Due to the averaging distortion it would seem that the prevailing sunny 5-minute intervals 3113/5315 indicate the sunshine duration roughly 0.586.

The review of daily measured illuminance courses representing July 1996 by 62 half- days can be classified into:


It is evident that neither the number of sunshine or sunless cases within a month in a P-D-G diagram nor *L D vz v* / and *G E v v* / time-averaged ratios are capable to differentiate the halfday situations when data are summarised during a day, a week or month in these mixed groups. Therefore the first step to identify, select or classify the half-day situations is to check the overall courses of *Gv* and *Dv* illuminance trends and levels and their relative efficiencies compared to the momentary extraterrestrial availability levels expressed in *G E v v* / and *D E v v* / ratios. Of course the stable or discontinuous sunshine duration follows the changes in *G E v v* / and the momentary presence of *P E v v* / ratios indicating the penetration of available extraterrestrial sunshine intensity. These half-day courses roughly characterise also the range of prevailing sky luminance patterns that can be expected and principally belong to the particular half-day situation. While situation 1 and 3 and sometimes even 2 are approximately homogeneous with evenly distributed turbidities and cloudiness cover over the whole sky vault, the situation 4 is characteristic for its unstable dynamic illuminance changes caused by complex layers of different cloud types and

Parameterisation of the Four Half-Day Daylight Situations 167

Morning Afternoon Average *SD* Average *SD* Average *SD* Clear Clear 175 0,917 0,034 0,967 0,031 0,849 0,034 Clear Cloudy 87 0,796 0,091 0,944 0,054 0,586 0,131 Clear Overcast 4 0,506 0,027 0,868 0,032 0,003 0,004 Clear Dynamic 70 0,818 0,094 0,941 0,037 0,655 0,165 Cloudy Clear 9 0,649 0,148 0,512 0,185 0,839 0,038 Cloudy Cloudy 128 0,261 0,224 0,265 0,220 0,254 0,208 Cloudy Overcast 201 0,049 0,074 0,082 0,080 0,000 0,001 Cloudy Dynamic 38 0,200 0,122 0,109 0,097 0,324 0,153 Overcast Clear 0 - - - - - - Overcast Cloudy 53 0,056 0,055 0,003 0,003 0,134 0,108 Overcast Overcast 311 0,002 0,002 0,003 0,003 0,000 0,000 Overcast Dynamic 34 0,096 0,065 0,004 0,003 0,219 0,128 Dynamic Clear 90 0,753 0,154 0,690 0,220 0,841 0,035 Dynamic Cloudy 160 0,383 0,208 0,467 0,227 0,267 0,210 Dynamic Overcast 72 0,149 0,140 0,257 0,201 0,001 0,002 Dynamic Dynamic 750 0,504 0,219 0,536 0,252 0,462 0,205

*s sm sa* 

Situation sequence Number

Sum of cases 2182

 *situation* =

where

Table 1. Statistical parameters of typical courses in Bratislava, 1994 -1999

*situation 1, situation 2* or *situation 4* by introducing an additional *U* parameter. Thus

 

1 0.75 <7, 2 <0.75 <7,

*if s and U if s and U*

, (21)

, (20)

4 0.01 7,

1 *i i i*

*x x n* 

*if s and U*

3 <0.01,

*if s*

*U* = 1 1

<sup>1</sup> ln

and *<sup>i</sup> x* and *<sup>i</sup>* <sup>1</sup> *x* are consecutive illuminance values in the half-day course.

Anyhow it can be assumed that in simulation programs of a daylight reference year the halfday sequences or changes will allow to model in series of about sixty cases during a specific month either the fluent and gradual or sudden changes in weather or sky types corresponding to the probability of occurrence with its proportionality to monthly averages of relative sunshine duration. At least the mentioned four half-day daylight situations have to be foreseen for modelling the complex sun-sky coexistence with cloudiness patterns in any daylight climate, although typical cases were selected only from measurements collected in Athens and Bratislava. A research report (Darula et al., 2004) contains the detail analysis with proposals of several parameters to identify the four relevant situations from measured half-day illuminance courses and the daily average relative sunshine duration. It is evident that the stable and homogeneous *situation 1* and *situation 3* can be defined by the *s* instead of sm and sa. However, the dynamically changing illuminance courses had to be identified and classified or selected to

of cases

distribution as well as patch movements. Thus under situation 4 can happen locally many accidental variations between quite low turbidity pockets with white-blue sky background through which direct sunshine temporarily can reach the ground while in other intervals the cloud patches cover and shade the sun beam penetration considerably.

Under homogeneous atmospheric conditions the *L D vz v* / ratio is quite a safe indicator of the sky luminance pattern, but during the dynamic half-day the zenith luminance as well as the sun position are influenced by passing clouds or cloud patches in several following sequence intervals.

However, for general practice and local characterisation of daylight conditions year-round longterm data are needed and should be locally available. Daylight data are also measured at the CIE IDMP stations or can be taken from the satellite database. In this respect besides global irradiation recorded in short-term variability or hourly averages at ground meteorological stations or recalculated from satellite measurements, only relative sunshine duration in daily or monthly averages have a very long tradition and are evaluated in many stations world-wide.

When inspecting monthly graphs of daily illuminance courses it becomes obvious that especially during winter and summer seasons typical weather patterns last for several days with changes either during night-time or noon. Even during perfectly clear days the symmetry around noon seems to be broken by higher turbidity in afternoon hours caused by water vapour evaporated due to rising air temperature and sunshine. Furthermore, in equatorial climate have to be expected changes in cloud cover at around noon, i.e. frequent mostly clear mornings and hours before noon but rather cloudy afternoons. During the Slovak-Greek cooperation simultaneously collected data at the CIE IDMP stations in Bratislava and Athens could serve to compare four half-day situations occurring in the temperate climate of Central Europe to those in the Mediterranean region (Darula et al., 2004). Available data was gathered during relatively long period 1994-1999.

The whole set of measured data was used to analyse the relation between sunshine duration and daily courses of illuminance. Relative sunshine duration with standard deviation *SD* for four typical situations were investigated in number with respect to their sequence of occurrence and results are documented in Table 1. Symbol *s* is relative sunshine duration calculated for the whole day while *sm* is for the morning period when local clock time was less than 12 o´clock and *sa* for the afternoon relative sunshine duration when local clock time was from 12 hours to sunset.

Except for the rapid change from overcast to clear all possible changes from morning to afternoon situations were found during the long-term of six years, i.e. 2182 days or 4364 half-days. The average relative sunshine duration corresponds perfectly with the change from the morning situation to the afternoon one respecting the tendency of the following situation change.

Although the half-day characteristics and their sequences in one or few days can form a typical year simulation, within this span any time subdivision can be utilised, i.e. Bratislava 1-minute data or Bratislava and Athens 5-minute average data can serve for analysis and comparison studies of several descriptor interrelations. However, to reach an absolute symmetry in halfdays due to perfect noon time all measured momentary or average values are to be recalculated from local clock time in which these were recorded to true solar time. Of course, it has to be realised that because the daytime span between sunrise and sunset is changing during the year as well as with the local latitude the relative time of a half-day element is not constant.

distribution as well as patch movements. Thus under situation 4 can happen locally many accidental variations between quite low turbidity pockets with white-blue sky background through which direct sunshine temporarily can reach the ground while in other intervals the

Under homogeneous atmospheric conditions the *L D vz v* / ratio is quite a safe indicator of the sky luminance pattern, but during the dynamic half-day the zenith luminance as well as the sun position are influenced by passing clouds or cloud patches in several following

However, for general practice and local characterisation of daylight conditions year-round longterm data are needed and should be locally available. Daylight data are also measured at the CIE IDMP stations or can be taken from the satellite database. In this respect besides global irradiation recorded in short-term variability or hourly averages at ground meteorological stations or recalculated from satellite measurements, only relative sunshine duration in daily or monthly averages have a very long tradition and are evaluated in many stations world-wide. When inspecting monthly graphs of daily illuminance courses it becomes obvious that especially during winter and summer seasons typical weather patterns last for several days with changes either during night-time or noon. Even during perfectly clear days the symmetry around noon seems to be broken by higher turbidity in afternoon hours caused by water vapour evaporated due to rising air temperature and sunshine. Furthermore, in equatorial climate have to be expected changes in cloud cover at around noon, i.e. frequent mostly clear mornings and hours before noon but rather cloudy afternoons. During the Slovak-Greek cooperation simultaneously collected data at the CIE IDMP stations in Bratislava and Athens could serve to compare four half-day situations occurring in the temperate climate of Central Europe to those in the Mediterranean region (Darula et al.,

cloud patches cover and shade the sun beam penetration considerably.

2004). Available data was gathered during relatively long period 1994-1999.

The whole set of measured data was used to analyse the relation between sunshine duration and daily courses of illuminance. Relative sunshine duration with standard deviation *SD* for four typical situations were investigated in number with respect to their sequence of occurrence and results are documented in Table 1. Symbol *s* is relative sunshine duration calculated for the whole day while *sm* is for the morning period when local clock time was less than 12 o´clock and *sa* for the afternoon relative sunshine duration when local clock

Except for the rapid change from overcast to clear all possible changes from morning to afternoon situations were found during the long-term of six years, i.e. 2182 days or 4364 half-days. The average relative sunshine duration corresponds perfectly with the change from the morning situation to the afternoon one respecting the tendency of the following

Although the half-day characteristics and their sequences in one or few days can form a typical year simulation, within this span any time subdivision can be utilised, i.e. Bratislava 1-minute data or Bratislava and Athens 5-minute average data can serve for analysis and comparison studies of several descriptor interrelations. However, to reach an absolute symmetry in halfdays due to perfect noon time all measured momentary or average values are to be recalculated from local clock time in which these were recorded to true solar time. Of course, it has to be realised that because the daytime span between sunrise and sunset is changing during the year

as well as with the local latitude the relative time of a half-day element is not constant.

sequence intervals.

time was from 12 hours to sunset.

situation change.


Table 1. Statistical parameters of typical courses in Bratislava, 1994 -1999

Anyhow it can be assumed that in simulation programs of a daylight reference year the halfday sequences or changes will allow to model in series of about sixty cases during a specific month either the fluent and gradual or sudden changes in weather or sky types corresponding to the probability of occurrence with its proportionality to monthly averages of relative sunshine duration. At least the mentioned four half-day daylight situations have to be foreseen for modelling the complex sun-sky coexistence with cloudiness patterns in any daylight climate, although typical cases were selected only from measurements collected in Athens and Bratislava. A research report (Darula et al., 2004) contains the detail analysis with proposals of several parameters to identify the four relevant situations from measured half-day illuminance courses and the daily average relative sunshine duration. It is evident that the stable and homogeneous *situation 1* and *situation 3* can be defined by the *s* instead of sm and sa. However, the dynamically changing illuminance courses had to be identified and classified or selected to *situation 1, situation 2* or *situation 4* by introducing an additional *U* parameter. Thus

$$
\begin{array}{c}
\text{saturation} = \begin{cases}
1 & \text{if } s \ge 0.75 \text{ and } \text{Ut-7}, \tau \\
2 & \text{if } s \le 0.75 \text{ and } \text{Ut-7}, \tau \\
3 & \text{if } s \le 0.01, \tau \\
4 & \text{if } s \ge 0.01 \text{ and } \text{Ut-7}, \tau
\end{array}
\end{array}
\tag{20}
$$

where

$$\mathcal{U}\_{\mathcal{U}} = \ln \left( \frac{1}{n-1} \sum\_{i=1}^{n} |\mathbf{x}\_i - \mathbf{x}\_{i+1}| \right) \tag{21}$$

and *<sup>i</sup> x* and *<sup>i</sup>* <sup>1</sup> *x* are consecutive illuminance values in the half-day course.

Parameterisation of the Four Half-Day Daylight Situations 169

Athens 1994 data for clear mornings

 Overcast Dynamic Morning Bratislava data 1994 in half-days:

Cloudy Clear

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Dynamic

 Clear Athens 1994 data for clear afternoons

Afternoon Bratislava data 1994 in half-days:

 Overcast Cloudy

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Gv / Ev = 0.182 + 1.038s - 1.385s2

The probability of occurrence of each of the four daylight situations in each month can be

if *s*  0.93 2 0 *Pm* %, (25)

Half-day sunshine duration

Half-day sunshine duration

Gv / Ev = 0.182 + 1.038s - 1.385s<sup>2</sup>

+ 0.883s3

+ 0.883s3

2 3 *Pm*1 100 0.55 0.95 1.65 *ss s* %, (22)

if *s* = 0 - 0.5, then *Pm s s* 2 100 1.5 1.85 %, (23)

if *s* = 0.5 - 0.93, then *Pm*2 66.86 0.93 *s* %, (24)

2.47 *Pm s* 3 100 1 %, (26)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Fig. 35. Morning data for four half-day situations

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Fig. 36. Afternoon data for four half-day situations

approximately estimated - for morning half-days by:

except if *s*  0.93, then 1 *Pm* = 100

Half-day average Gv / Ev

Half-day average Gv / Ev

Using these classification parameters all four daylight types are interrelated by a fluent course of half-day average *G E v v* / and *D E v v* / dependent on the half-day sunshine duration as documented in Fig. 33 and 34 containing all 1994 morning and afternoon data recorded in Bratislava and in Athens first. In the second step a more detail separation of half-day situations dependent on half-day relative sunshine duration was made for morning 1994 data (Fig. 35) and afternoon (Fig. 36) in relation to *G E v v* / parameter repeated for also *D E v v* / and *P E v v* / with the best fit simulation of their dependence on the half-day relative sunshine duration. However, as most frequently are available only monthly relative sunshine durations in meteorological station reports the probability of occurrence of the morning and afternoon halfday situations was sought first for 1994 data (example in Fig. 37 for *situation 1*) and checked for 1991-2001 data. Thus best fit probability for the monthly redistribution simulation of morning and afternoon *situations 1* to *4* were predetermined solely dependent on the monthly relative sunshine duration using curves in Fig. 38 for morning half-days or in Fig. 39 for afternoons (Darula et al., 2004 and Darula & Kittler, 2005b).

Fig. 33. Morning and afternoon *G E v v* / data after Bratislava and Athens measurements during 1994

Fig. 34. Morning and afternoon *D E v v* / data after Bratislava and Athens measurements during 1994

Using these classification parameters all four daylight types are interrelated by a fluent course of half-day average *G E v v* / and *D E v v* / dependent on the half-day sunshine duration as documented in Fig. 33 and 34 containing all 1994 morning and afternoon data recorded in Bratislava and in Athens first. In the second step a more detail separation of half-day situations dependent on half-day relative sunshine duration was made for morning 1994 data (Fig. 35) and afternoon (Fig. 36) in relation to *G E v v* / parameter repeated for also *D E v v* / and *P E v v* / with the best fit simulation of their dependence on the half-day relative sunshine duration. However, as most frequently are available only monthly relative sunshine durations in meteorological station reports the probability of occurrence of the morning and afternoon halfday situations was sought first for 1994 data (example in Fig. 37 for *situation 1*) and checked for 1991-2001 data. Thus best fit probability for the monthly redistribution simulation of morning and afternoon *situations 1* to *4* were predetermined solely dependent on the monthly relative sunshine duration using curves in Fig. 38 for morning half-days or in Fig. 39 for afternoons

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Half-day sunshine duration

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Half-day sunshine duration

Fig. 34. Morning and afternoon *D E v v* / data after Bratislava and Athens measurements

Morning and afternoon 1994 data

+ 0,883 s<sup>3</sup>

+ 0.126 s<sup>3</sup>

Gv / Ev = 0,182+1,038 s - 1,385 s2

Fig. 33. Morning and afternoon *G E v v* / data after Bratislava and Athens measurements

Dv / Ev = 0.182+0.693 s - 0.759 s<sup>2</sup>

Morning and afternoon 1994 data

Best fit with R = 0,955:

Best fit with R = 0.72:

(Darula et al., 2004 and Darula & Kittler, 2005b).

Half-day average Gv / Ev

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Half-day average Dv / Ev

during 1994

during 1994

Fig. 35. Morning data for four half-day situations

Fig. 36. Afternoon data for four half-day situations

The probability of occurrence of each of the four daylight situations in each month can be approximately estimated


$$Pm1 = 100\left\{0.55s - 0.95s^2 + 1.65s^3\right\} \text{ [\%]}.\tag{22}$$

except if *s*  0.93, then 1 *Pm* = 100

$$\text{if } s = 0 \text{ - 0.5, then } Pm2 = 100s(1.5 - 1.85s) \text{ [\%]}.\tag{23}$$

$$\text{if } s = 0.5 \text{ -} 0.93 \text{, then } Pm2 = 66.86 \text{(} 0.93 \text{ $-s$ ) } \text{[}\%\text{]}\_{\text{}} \tag{24}$$

$$\text{if } s > 0.9\\$ \text{ } Pm2 = 0 \text{ } \{\%\} \text{ } \tag{25}$$

2.47 *Pm s* 3 100 1 %, (26)

Parameterisation of the Four Half-Day Daylight Situations 171

Pm1

Pm4

Pm2

Pm3

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Monthly average relative sunshine duration, s

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Monthly average relative sunshine duration, s

These probabilities of the occurrence of typical four daylight situations were derived from measurements in two different climate zones, i.e. in Bratislava as well as in Athens. So, it can be assumed that the dependence on monthly sunshine durations during morning and afternoon half-days could be valid not only in Central Europe and European Mediterranean

dynamic

clear

half-days

overcast

cloudy

dynamic

clear

half-days

Probability of occurrence in %

regions but also world-wide.

Fig. 38. Occurrence probability of half-day situations during mornings

Pa1

Pa4

Pa2

Pa3

Fig. 39. Occurrence probability of half-day situations during afternoons

overcast

cloudy

Probability of occurrence in %

$$Pm4 = 100 - \left(Pm1 + Pm2 + Pm3\right) \left\{ \, ^{\otimes}\_{\bullet} \right\} \tag{27}$$


$$\text{Pa1} = 100 \left( 0.62s - 0.77s^2 + 1.26s^3 \right) \text{ [\%]}\_{\prime} \tag{28}$$

except if *s*  0.97, then 1 *Pa* = 100

$$\text{if } s = 0 \text{ - 0.5, then } Pa2 = 100s(1.2 - 1.6s) \text{ [\%]}\text{.}\tag{29}$$

$$\text{if } s = 0.5 \text{ -} 0.93 \text{, then } Pa2 = 46.51 \text{(} 0.93 \text{ $-s$ ) [\%]} \text{.} \tag{30}$$

$$\text{if } s > 0.9\\\text{\textbullet } Pa2 = 0 \quad \text{\textbullet } \text{\textbullet } \text{\textbullet } \tag{31}$$

$$Pa\text{3} = 100(1 - s)^{2.7} \text{ [\%]} \tag{32}$$

$$Pa4 = 100 - \left(Pa1 + Pa2 + Pa3\right) \text{ [\%]}.\tag{33}$$

Fig. 37. Relation of clear situation to monthly relative sunshine duration in Bratislava and Athens

if *s*  0.93 2 0 *Pa* %, (31)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Pm1 = 0.55s - 0.95s2

Monthly relative sunshine duration s

Fig. 37. Relation of clear situation to monthly relative sunshine duration in Bratislava and

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Athens

Probability of occurrence in %

afternoon

afternoon

Athens 5-mi data, 1994 morning

during half-days Bratislava 5-mi data, 1994 morning

Clear sky conditions - Type 1


except if *s*  0.97, then 1 *Pa* = 100

*Pm Pm Pm Pm* 4 100 1 2 3 %, (27)

2 3 *Pa*1 100 0.62 0.77 1.26 *ss s* %, (28)

if *s* = 0 - 0.5, then *Pa s s* 2 100 1.2 1.6 %, (29)

if *s* = 0.5 - 0.93, then *Pa*2 46.51 0.93 *s* %, (30)

2.7 *Pa s* 3 100 1 %, (32)

*Pa Pa Pa Pa* 4 100 1 2 3 %. (33)

+ 1.65s3

Pa1 = 0.62s - 0.77s2

+ 1.26s3

Fig. 38. Occurrence probability of half-day situations during mornings

Fig. 39. Occurrence probability of half-day situations during afternoons

These probabilities of the occurrence of typical four daylight situations were derived from measurements in two different climate zones, i.e. in Bratislava as well as in Athens. So, it can be assumed that the dependence on monthly sunshine durations during morning and afternoon half-days could be valid not only in Central Europe and European Mediterranean regions but also world-wide.

Parameterisation of the Four Half-Day Daylight Situations 173

In these figures besides the probability percentage notation 1 4 *Pm Pm* and 1 4 *Pa Pa* a similar notation for the number of half-days is used 1 4 *Nm Nm* and 1 4 *Na Na* while the overall number of morning half-days in a particular month is *Nm* for mornings and *Na* for afternoons in Fig. 40 and 41. These document and confirm the redistribution model that approximates the participation of the main three situations on sunlight presence and monthly sunshine duration within the particular half-day assuming that the overcast half-

This redistribution of half-day situations during mornings and afternoons was calculated for Bratislava and Athens data and as examples are shown in Table 2 and 3 only those for morning half-days. Although the verification of these redistributions for other localities is rather complicated it is evident that the ranges of mornings *sm* and those measured during afternoons *sa* can be in every month specific too. While during overcast situations the range of *s* 0.05 is relatively small with *G E v v* / within the spread 0.05 - 0.35 (Fig. 35 and 36), the *s* ranges in dynamic situations are quite large i.e. 0.3 – 0.76 while Gv/Ev spread is

Thus eq. (34) and (35) characterise the redistribution of *sm* and *sa* due to four half-day situations simulating Central European and Mediterranean daylight conditions. In other climate regions (like maritime and equatorial) or during rainy (April or May) or during

Therefore in the application of this redistribution it is recommended to test whether the *sm* and *sa* for appropriate situations are within their usual ranges. Approximately this is done by checking 4 *sm* and 4 *sa* ranges after eq. (34) and (35). During dynamic half-days both *sm*4 and 4 *sa* should be in the range 0.3 to 0.75 to be related to the rise of *G E v v* / from 0.35

For an example of such a check can be taken the ten-year (1995-2004) average of relative sunshine duration in Prague, which is for May 0.502 *s* . In the book by Darula et al., (2009) after percentage probabilities the number of four half-day situations was determined (on

*Nm*1 8 , 2 9 *Nm* , 3 6 *Nm* and 4 8 *Nm* with the full number of morning half-days in

 0.92 1 0.25 2 31(0.502) 0.92 8 0.25 9 <sup>4</sup> 0.744 4 8

*sm Nm Nm Nm Nm* 0.92 1 0.25 2 0.61 4 / , (34)

*sa Na Na Na Na* 0.9 1 0.25 2 0.5 4 / . (35)

*sm sm Nm sm Nm sm Nm Nm* 1 1 2 2 4 4/ , (36)

*sa sa Na sa Na sa Na Na* 1 1 2 2 4 4/ . (37)

day is absolutely without any sunshine, thus

approximately within 0.32 – 0.61.

monsoon months more general relations might be valid as

*smNm Nm Nm*

*Nm*

and

and

to 0.6 respectively.

May 31 *Nm* ; So, after eq. (34)

*sm*

p. 64, Tab. 5.4.1) as follows:

### **5. Approximate redistribution of the four daylight situations in the yearly simulation of their occurrence**

In accordance with the probability study of the four daylight situations in Bratislava morning and afternoon data during 1994-2001 the check was done using Athens data gathered in a five year period 1992-1996 (Darula et al., 2004). Because the calculated probability had to be substituted by a concrete number of days within a particular month, i.e. in integer numbers, these had to correspond with sum of half-days in that actual month. The redistribution into half-days had to dependent also on the overall monthly sunshine duration, so the redistribution model correlating the probability percentage and number of half-day situations had to be found. The best fit final solution is documented for the morning redistribution model with results shown in Fig. 40 as well as for afternoon in Fig. 41 with monthly relative sunshine duration data measured during mornings *sm* and measured during afternoons *sa.* 

Fig. 40. Redistribution model after Bratislava and Athens morning data

Fig. 41. Similar redistribution for afternoon half-days

In these figures besides the probability percentage notation 1 4 *Pm Pm* and 1 4 *Pa Pa* a similar notation for the number of half-days is used 1 4 *Nm Nm* and 1 4 *Na Na* while the overall number of morning half-days in a particular month is *Nm* for mornings and *Na* for afternoons in Fig. 40 and 41. These document and confirm the redistribution model that approximates the participation of the main three situations on sunlight presence and monthly sunshine duration within the particular half-day assuming that the overcast halfday is absolutely without any sunshine, thus

$$\text{Lsm} = \left(0.92\,\text{N}m1 + 0.25\,\text{N}m2 + 0.61\,\text{N}m4\right) / \,\text{N}m \,\text{ }\tag{34}$$

and

172 Sustainable Growth and Applications in Renewable Energy Sources

In accordance with the probability study of the four daylight situations in Bratislava morning and afternoon data during 1994-2001 the check was done using Athens data gathered in a five year period 1992-1996 (Darula et al., 2004). Because the calculated probability had to be substituted by a concrete number of days within a particular month, i.e. in integer numbers, these had to correspond with sum of half-days in that actual month. The redistribution into half-days had to dependent also on the overall monthly sunshine duration, so the redistribution model correlating the probability percentage and number of half-day situations had to be found. The best fit final solution is documented for the morning redistribution model with results shown in Fig. 40 as well as for afternoon in Fig. 41 with monthly relative sunshine duration data measured during mornings *sm* and

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Athens morning averages 1992-1996

Bratislava morning averages 1994-2001

Five-year average relative sunshine duration

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Bratislava afternoon averages 1994-2001 Athens afternoon averages 1992-1996

Five-year average relative sunshine duration

**5. Approximate redistribution of the four daylight situations in the yearly** 

Redistribution model:

Fig. 40. Redistribution model after Bratislava and Athens morning data

Redistribution model:

 sa = 0,92 Pa1 + 0,05 Pa2 + 0,61 Pa4 sa = (0,9 Na1 + 0,25 Na2 + 0,5 Na4)/Na

sm=0,92 Pm1+0,21 Pm2+0,56 Pm4 sm=(0,92 Nm1+0,25 Nm2+0,61 Nm4)/Nm

**simulation of their occurrence** 

measured during afternoons *sa.* 

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Fig. 41. Similar redistribution for afternoon half-days

Modelled sunshine duration

Modelled sunshine duration

$$sa = \left(0.9Na1 + 0.25Na2 + 0.5Na4\right) / Na\cdot \tag{39}$$

This redistribution of half-day situations during mornings and afternoons was calculated for Bratislava and Athens data and as examples are shown in Table 2 and 3 only those for morning half-days. Although the verification of these redistributions for other localities is rather complicated it is evident that the ranges of mornings *sm* and those measured during afternoons *sa* can be in every month specific too. While during overcast situations the range of *s* 0.05 is relatively small with *G E v v* / within the spread 0.05 - 0.35 (Fig. 35 and 36), the *s* ranges in dynamic situations are quite large i.e. 0.3 – 0.76 while Gv/Ev spread is approximately within 0.32 – 0.61.

Thus eq. (34) and (35) characterise the redistribution of *sm* and *sa* due to four half-day situations simulating Central European and Mediterranean daylight conditions. In other climate regions (like maritime and equatorial) or during rainy (April or May) or during monsoon months more general relations might be valid as

$$\text{sm} = \left( \text{sm}1\text{N}m1 + \text{sm}2\text{N}m2 + \text{sm}4\text{N}m4 \right) / \text{Nm} \,\tag{36}$$

and

$$\mathbf{a}s\mathbf{a} = \left(\mathbf{s}a\mathbf{1}\mathbf{N}a\mathbf{1} + \mathbf{s}a\mathbf{2}\mathbf{N}a\mathbf{2} + \mathbf{s}a\mathbf{4}\mathbf{N}a\mathbf{4}\right) / \text{ Na}\cdot\text{ }\tag{37}$$

Therefore in the application of this redistribution it is recommended to test whether the *sm* and *sa* for appropriate situations are within their usual ranges. Approximately this is done by checking 4 *sm* and 4 *sa* ranges after eq. (34) and (35). During dynamic half-days both *sm*4 and 4 *sa* should be in the range 0.3 to 0.75 to be related to the rise of *G E v v* / from 0.35 to 0.6 respectively.

For an example of such a check can be taken the ten-year (1995-2004) average of relative sunshine duration in Prague, which is for May 0.502 *s* . In the book by Darula et al., (2009) after percentage probabilities the number of four half-day situations was determined (on p. 64, Tab. 5.4.1) as follows:

*Nm*1 8 , 2 9 *Nm* , 3 6 *Nm* and 4 8 *Nm* with the full number of morning half-days in May 31 *Nm* ;

So, after eq. (34)

$$\text{sm4} = \frac{\text{smNm} - \left(0.92 \text{Nm} \text{1} + 0.25 \text{Nm} \text{2}\right)}{\text{Nm} \text{4}} = \frac{\text{31}(0.502) - \left\lfloor \left(0.92\right) \text{8} + \left(0.25\right) \text{9} \right\rfloor}{\text{8}} = 0.744$$

Parameterisation of the Four Half-Day Daylight Situations 175

which means that *sm*4 = 0.744 falls to the upper range 0.75, but *sm*2 = 0.25 is suspect due to probably more sunshine intervals in May. Under such conditions probably *sm*4 = 0.61 as in

> 31(0.502) 0.92 8 0.61 8 <sup>2</sup> 0.369 9

Similarly a November check can be done using 0.195 *s* for Prague with 1 3 *Nm* , *Nm*2 7 , 3 18 *Nm* and 4 2 *Nm* with the full number of morning half-days in November

> 0.92 1 0.25 2 30 (0.195) 0.92 3 0.25 7 <sup>4</sup> 0.67 4 2

In accordance with the already approximated monthly averaged values *G E v v* / in Fig. 33, 35 and 36 as well as *D E v v* / in Fig. 34 can be simulated also roughly half-day illuminance

sin *<sup>v</sup> v s*

sin *<sup>v</sup> v s*

It is evident that the course distribution of illuminances is caused by the sine of the solar angle with either the momentary sine value for the moment or for the chosen time interval.

*TST* can be used. For a short time period a straight-line interpolation can be applied when

A further possible step to specify the site and situation dependent illuminance stimulated a study that would show the relation of the four situations on typical sky patterns or ISO (2004)/CIE (2003) standard general skies if possible. Originally these standards were derived with the specification of indicatrix and gradation function in Kittler (1995) and

After a detail number of 5-minute measured cases in Bratislava specifying every year within the five year 1994 - 1998 span all four daylight situations were analysed with the following results:


2 3 / 0.182 1.038 1.385 0.883 *G E v v ss s* -, (39)

2 3 / 0.182 0.693 0.759 0.126 *D E v v ss s* -. (41)

*<sup>s</sup>*after eq. (2) for any hour number *H* during daytime in

lx, (38)

lx, (40)

*v <sup>G</sup> G LSC <sup>E</sup>*

*v <sup>D</sup> D LSC <sup>E</sup>*

finally recommended for standardisation in Kittler et al., (1997).

*HHH* 1 2 / 2 or a value after eq. (16) is more precise.

which suites the dynamic range and is quite close to the assumed sm4 = 0.61.

.

eq. (34) and in May *sm*2 is higher:

*Nm* 30 where after eq. (34) is

This sine of the solar altitude

1. under *situation 1* were present

of sky type 12,

*sm*

courses as

where

and

where

*sm*

*smNm Nm Nm*

*Nm*


Table 2. Redistribution of half-day situations according to Bratislava morning 8 – year data related to monthly average relative sunshine duration


Table 3. Redistribution of half-day situations according to Athens morning 5 – year data related to monthly average relative sunshine duration

which means that *sm*4 = 0.744 falls to the upper range 0.75, but *sm*2 = 0.25 is suspect due to probably more sunshine intervals in May. Under such conditions probably *sm*4 = 0.61 as in eq. (34) and in May *sm*2 is higher:

$$sm2 = \frac{31(0.502) - \left[ (0.92)\,\,8 + (0.61)\,8 \right]}{9} = 0.369\,\,\dots$$

Similarly a November check can be done using 0.195 *s* for Prague with 1 3 *Nm* , *Nm*2 7 , 3 18 *Nm* and 4 2 *Nm* with the full number of morning half-days in November *Nm* 30 where after eq. (34) is

$$\text{sm4} = \frac{\text{smNm} - \left(0.92\text{Nm}1 + 0.25\text{Nm}2\right)}{\text{Nm}4} = \frac{\text{30}\left(0.195\right) - \left[\left(0.92\right)3 + \left(0.25\right)7\right]}{2} = 0.67$$

which suites the dynamic range and is quite close to the assumed sm4 = 0.61. In accordance with the already approximated monthly averaged values *G E v v* / in Fig. 33, 35 and 36 as well as *D E v v* / in Fig. 34 can be simulated also roughly half-day illuminance courses as

$$G\_v = \frac{G\_v}{E\_v} \left( LSC \in \sin \chi\_s \right) \tag{38}$$

where

174 Sustainable Growth and Applications in Renewable Energy Sources

Month s Pm1 Nm1 Pm2 Nm2 Pm3 Nm3 Pm4 Nm4 Nm sm

Table 2. Redistribution of half-day situations according to Bratislava morning 8 – year data

Month s Pm1 Nm1 Pm2 Nm2 Pm3 Nm3 Pm4 Nm4 Nm sm

Table 3. Redistribution of half-day situations according to Athens morning 5 – year data

1 0,451 20,62 6 30,02 9 22,74 7 26,62 9 31 0,436 2 0,480 22,76 6 29,38 8 19,89 6 27,98 8 28 0,451 3 0,516 25,75 8 27,68 9 16,66 5 29,91 9 31 0,496 4 0,643 39,95 12 19,19 6 7,85 2 33,00 10 30 0,631 5 0,666 43,23 13 17,65 5 6,66 2 32,45 11 31 0,653 6 0,797 67,02 20 8,89 3 1,95 1 22,14 6 30 0,766 7 0,844 77,95 24 5,75 2 1,02 0 15,29 5 31 0,832 8 0,854 80,45 25 5,08 2 0,86 0 13,60 4 31 0,841 9 0,783 64,03 19 9,83 3 2,30 1 23,85 7 30 0,757 10 0,605 35,04 11 21,73 7 10,08 3 33,15 10 31 0,589 11 0,458 21,11 6 29,89 9 22,03 7 26,96 8 30 0,430 12 0,400 17,36 5 30,40 9 28,32 9 23,92 8 31 0,386

related to monthly average relative sunshine duration

related to monthly average relative sunshine duration

1 0,204 8,67 3 22,90 7 56,92 18 11,51 3 31 0,207 2 0,404 17,59 5 30,41 9 27,85 8 24,15 6 28 0,382 3 0,367 15,55 5 30,13 9 32,32 10 22,00 7 31 0,365 4 0,466 21,70 7 29,73 9 21,23 6 27,34 8 30 0,460 5 0,541 28,08 9 26,01 8 14,61 5 31,30 9 31 0,517 6 0,522 26,29 8 27,28 8 16,15 5 30,28 9 30 0,504 7 0,525 26,57 8 27,08 8 15,90 5 30,45 10 31 0,508 8 0,609 35,53 11 21,46 7 9,83 3 33,18 10 31 0,589 9 0,426 18,95 6 30,33 9 25,38 8 25,35 7 30 0,408 10 0,420 18,57 6 30,37 9 26,04 8 25,03 8 31 0,416 11 0,244 10,16 3 25,59 8 50,11 15 14,14 4 30 0,244 12 0,192 8,23 3 21,98 7 59,06 18 10,73 3 31 0,207

$$\{G\_v \mid E\_v = 0.182 + 1.038s - 1.385s^2 + 0.883s^3 \text{ [\text{\textdegree l}]} \}\tag{39}$$

and

$$D\_v = \frac{D\_v}{E\_v} \text{(LSC} \in \sin \gamma\_s\text{)} \text{ [lux]} \tag{40}$$

where

$$D\_v \;/\; E\_v = 0.182 + 0.693s - 0.759s^2 + 0.126s^3 \; \text{[r]} \cdot \text{[r]} \cdot \tag{41}$$

It is evident that the course distribution of illuminances is caused by the sine of the solar angle with either the momentary sine value for the moment or for the chosen time interval. This sine of the solar altitude *<sup>s</sup>*after eq. (2) for any hour number *H* during daytime in *TST* can be used. For a short time period a straight-line interpolation can be applied when *HHH* 1 2 / 2 or a value after eq. (16) is more precise.

A further possible step to specify the site and situation dependent illuminance stimulated a study that would show the relation of the four situations on typical sky patterns or ISO (2004)/CIE (2003) standard general skies if possible. Originally these standards were derived with the specification of indicatrix and gradation function in Kittler (1995) and finally recommended for standardisation in Kittler et al., (1997).

After a detail number of 5-minute measured cases in Bratislava specifying every year within the five year 1994 - 1998 span all four daylight situations were analysed with the following results:


Parameterisation of the Four Half-Day Daylight Situations 177

Situation sequence Winter Summer Spring and autumn

Clear Clear 44 10,35 55 12,20 52 11,38 Cloudy Cloudy 82 19,29 56 12,42 81 17,72 Overcast Overcast 55 12,94 1 0,22 11 2,41 Dynamic Dynamic 29 6,82 56 12,42 62 13,57 Clear Cloudy 13 3,06 20 4,44 13 2,85 Clear Overcast 2 0,47 0 0,00 0 0,00 Clear Dynamic 30 7,06 109 24,17 65 14,22 Cloudy Clear 15 3,53 6 1,33 12 2,63 Cloudy Overcast 18 4,24 2 0,44 4 0,88 Cloudy Dynamic 55 12,94 75 16,63 70 15,32 Overcast Clear 0 0,00 0 0,00 0 0,00 Overcast Cloudy 28 6,59 0 0,00 3 0,66 Overcast Dynamic 5 1,18 0 0,00 3 0,66 Dynamic Clear 23 5,41 31 6,87 36 7,88 Dynamic Cloudy 25 5,88 40 8,87 45 9,85 Dynamic Overcast 1 0,24 0 0,00 0 0,00 Sum of cases 425 100 451 100 457 100

of cases % Number

of cases % Number

of cases %

of cases %

of cases % Number

Table 4. Occurrence of daylight situations with typical sequences in one whole day

of cases % Number

Table 5. Repetition of four half-day situations in conscutive two days

Situation sequence Winter Summer Spring and autumn

Clear Clear 13 13,27 18 21,95 20 19,05 Cloudy Cloudy 39 39,80 14 17,07 26 24,76 Overcast Overcast 24 24,49 0 0,00 3 2,86 Dynamic Dynamic 3 3,06 14 17,07 18 17,14 Clear Cloudy 0 0,00 2 2,44 0 0,00 Clear Overcast 0 0,00 0 0,00 0 0,00 Clear Dynamic 3 3,06 13 15,85 14 13,33 Cloudy Clear 3 3,06 0 0,00 0 0,00 Cloudy Overcast 1 1,02 0 0,00 0 0,00 Cloudy Dynamic 7 7,14 10 12,20 15 14,29 Overcast Clear 0 0,00 0 0,00 0 0,00 Overcast Cloudy 2 2,04 0 0,00 1 0,95 Overcast Dynamic 0 0,00 0 0,00 0 0,00 Dynamic Clear 1 1,02 3 3,66 2 1,91 Dynamic Cloudy 2 2,04 8 9,76 6 5,71 Dynamic Overcast 0 0,00 0 0,00 0 0,00 Sum of cases 98 100 82 100 105 100

Morning Afternoon Number

Morning Afternoon Number


This sky type prevalence (Darula & Kittler, 2008a) was in coherence considerably also with the seasonal frequency of dominant sky types found in the seasonal distribution (Kitttler et al., 2001) with prevailing overcast skies in type 2 and 3 and clear sky types 12 and 11 in Bratislava, while in Athens the highest frequency of clear polluted sky type 13 was documented, while uniform cloudy skies 5 and 6 were the most often occurring in dull seasons. Of course, the seasonal changes in occurrence frequency of clear and overcast skies is linked with relative sunshine duration and therefore with the number of half-days in any locality. However, it is interesting that in any daylight climate there exists a number of (Lambert) overcast sky type 5 with uniform luminance sky patterns, e.g. in Bratislava five year long-term these represented 12.6 % whithin cloudy *situation 2* during morning halfdays and over 14 % during afternoons whithin overcast *situation 3* these were represented by morning 8.08 % and afternoon 7.74 % presence.

More and further measurements in different locations are expected to demonstrate the sitespecific and short-term variability of illuminance levels (as recently was shown for irradiance by Perez et al., 2011). Due to dynamic situations it is important to evaluate shortterm (momentary 1 or 5-minute regular measurements) because estimations of using hourly insolation data from satellite-based sources can be problematic and less accurate when subhourly variability is uncertain and especially if irradiance data are recalculated via luminous efficacy into illuminances (Darula & Kittler, 2008b). Therefore long-term regular measurements in absolute illuminance values are so important to have site-specific fundamental data with the possibility to derive also half-day situations. When modelling year-round situation frequencies it is also important to randomly distribute also some sequential ocurrence of specific situations (Darula & Kittler 2002) which can occur several half-days or even days after each other as is documented in Table 4 and 5. Of course, one situation can last during the whole day, i.e. the morning situation is the same in the afternoon, but quite frequent are also changes from clear to dynamic or cloudy to dynamic and vice versa especially in summer as shown in Table 4. In winter are typical lasting same situations except dynamic in two adjacent days, while in summer all consecutive days with the same situation are quite often except overcast.








This sky type prevalence (Darula & Kittler, 2008a) was in coherence considerably also with the seasonal frequency of dominant sky types found in the seasonal distribution (Kitttler et al., 2001) with prevailing overcast skies in type 2 and 3 and clear sky types 12 and 11 in Bratislava, while in Athens the highest frequency of clear polluted sky type 13 was documented, while uniform cloudy skies 5 and 6 were the most often occurring in dull seasons. Of course, the seasonal changes in occurrence frequency of clear and overcast skies is linked with relative sunshine duration and therefore with the number of half-days in any locality. However, it is interesting that in any daylight climate there exists a number of (Lambert) overcast sky type 5 with uniform luminance sky patterns, e.g. in Bratislava five year long-term these represented 12.6 % whithin cloudy *situation 2* during morning halfdays and over 14 % during afternoons whithin overcast *situation 3* these were represented

More and further measurements in different locations are expected to demonstrate the sitespecific and short-term variability of illuminance levels (as recently was shown for irradiance by Perez et al., 2011). Due to dynamic situations it is important to evaluate shortterm (momentary 1 or 5-minute regular measurements) because estimations of using hourly insolation data from satellite-based sources can be problematic and less accurate when subhourly variability is uncertain and especially if irradiance data are recalculated via luminous efficacy into illuminances (Darula & Kittler, 2008b). Therefore long-term regular measurements in absolute illuminance values are so important to have site-specific fundamental data with the possibility to derive also half-day situations. When modelling year-round situation frequencies it is also important to randomly distribute also some sequential ocurrence of specific situations (Darula & Kittler 2002) which can occur several half-days or even days after each other as is documented in Table 4 and 5. Of course, one situation can last during the whole day, i.e. the morning situation is the same in the afternoon, but quite frequent are also changes from clear to dynamic or cloudy to dynamic and vice versa especially in summer as shown in Table 4. In winter are typical lasting same situations except dynamic in two adjacent days, while in summer all consecutive days with

4. under *sitation 4* were very changeable sky patterns, but the most present were

type 12,

sky type 3,

type 3,

type 2,

sky type 12,

of sky type 12.

2. under *situation 2* were occurring

3. under *situation 3* were present

almost 27 % of sky type 2,

by morning 8.08 % and afternoon 7.74 % presence.

the same situation are quite often except overcast.


Table 4. Occurrence of daylight situations with typical sequences in one whole day


Table 5. Repetition of four half-day situations in conscutive two days

Parameterisation of the Four Half-Day Daylight Situations 179

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