ERB and NEFRIEND Proj.5 Conf. Interdisciplinary Approaches in Small Catchment Hydrology: Monitoring and
Research, Slovak NC IHP UNESCO/UH SAV, 2002, 127-135.
SOLAR ENERGY INCOME MODELLING IN MOUNTAINOUS
AREAS
Ivan Mészároš1, Pavol Miklánek1, Juraj Parajka1
Introduction
Mountainous areas are typical for their complex relief and great elevation changes.
Solar energy income is hardly to measure in mountainous environment. Especially the
practical possibilities of direct measurement of the energy income, as the most
important parameter for calculating different elements of water balance equation, like
evapotranspiration, snowmelt etc. on the slopes is very limited. For that reasons the
SOLEI-32 model which can calculate potential energy income to the slopes with
different orientation and taking into account the shadowing of the surrounding
topography was developed.
SOLEI32 model
First version of the SOLEI model was described by Miklánek and Mészároš (1993) and
was developed for DOS operating system. The second version of the model was
reprogrammed into the new operating system Windows 95/98/NT/2000 using of
Fortran PowerStation Version 4.0 programming language.
The input for the model is the matrix of square-grid elevation data (Digital Elevation
Model) and optionally vegetation, cloudiness and albedo grids in the same matrix shape.
Recent version of SOLEI-32 model uses the IDRISI (16 and 32 bit versions) raster
and/or SURFER grid format.
Energy income is represented by sum of direct, diffuse and reflected radiation. The
radiation components are calculated separately for the plots with general slope
(Miklánek and Mészároš, 1998).
Calculation of sunshine duration
During the first step of calculations the basic spatial geometrical characteristics for each
pixel are determined (Mészároš 1998). Then it is checked whether the sunshine can
theoretically reach the observed grid plot and whether there is no obstacle between the
sun and this surface. The plot can be located on the shaded side of the hill itself, or it
can be hidden by the slope, which is in front of the observed surface (shadowing effect
of terrain). This algorithm is applied for each grid plot.
This calculation is repeated in each time step for the actual sun position (angle and
declination), which depends on local time. Output from this subroutine is sunshine
duration during the optional time range, e.g. hours, one or several days.
1
Institute of Hydrology of SAS, Racianska 75, 830 08 Bratislava 38, Slovakia
Tel./fax.: (+4212) 44259 311, E-mails: meszaros@uh.savba.sk, miklanek@uh.savba.sk,
parajka@svslm.sk
Calculation of potential energy income
During the next step the potential energy income is calculated for each grid plot
(Miklánek 1993). The potential energy income is output from this subroutine. For the
plot with general slope  the equation (1) was used as it is described in Kittler and
Mikler (1986).
[W m-2]
Eb  p Eb  d Eb  r Eb
c
where
Eb
global radiation,
p
Eb
direct radiation,
c
(1)
d
Eb
Eb
r
diffuse radiation,
reflected radiation.
The component of the direct radiation pEb on grid cell with slope β is calculated
according (2), based on direct radiation incidence pEbK on surface normal to sun beam
p
Eb  pEbK cosi  ,
[Wm-2]
(2)
while
p
0.1T (Tm  1)
30
 E0 
sin( h0 )  0.106Tm
sin( h0 ) 
EbK
[Wm-2]
cos i  cos   sinh 0  sin   cosh 0  cos Ans  A0s
(3)
(4)
where
p
EbK
Tm
E0
h0
i
Ans
direct radiation incident to plane normal to sun beam direction,
Linke`s coefficient of atmospheric turbidity (mean monthly value),
solar constant on the upper boundary of the atmosphere (daily value),
Sun elevation (angle between horizontal plane and sun beam),
angle of the direct radiation incidence (spatial angle between sun beam and
the normal of the given plane),
azimuth of the normal of the plane,
A0s
Sun azimuth.
The Sun azimuth is calculated according (5):
As0 = arccos [ cos / cosh0 ( cos tg + sin cos(15°. H) ) ]
where 

H
(5)
declination,
latitude,
time.
The component of the diffuse radiation d E b incident to plane with slope β during
cloudless conditions is expressed by (6)
d
while
Eb  0.5d EbH  u2
[Wm-2]
(6)


1.84
u2  sin    0.94  ecos i 
 1.44   1  cos 
Tm


d


EbH  kb E0  p Ebk sin h0
(7)
[Wm-2]
(8)
kb  0.22  0.025  Tm 
(9)
where
d
EbH
kb
diffuse radiation incident to horizontal plane,
coefficient expressing the portion of the radiation diffused by the
atmosphere.
The reflected radiation r E b due to surrounding terrain is calculated for the plane with
slope β>0
r
where

Eb  0.5 1  cos   d EbH  pEbk  pEbk sin h0
α

[Wm-2]
(10)
albedo. (coefficient expressing the portion of the radiation reflected by
the surrounding surface).
On the plots, where direct sunshine duration (described in previous section) is equal to
zero, the potential energy income is assumed to be equal only to diffuse radiation.
It is possible to change different parameters needed for energy calculations:
The most important is vegetation file, which can characterise albedo constant as one
“value” for whole basin, or for each grid plot separately, in the same shape as the input
grid file. Output from this subroutine could be not only potential energy income to the
grid plots, but actual energy income, too.
Actual energy is calculated from the potential one, based on relation to measured
radiation at terrestrial control station, or based on cloudiness or relative sunshine
duration from terrestrial or remote sensing observations. Input data for cloudiness could
be represented from the most simple “one number” version for whole area and time
range, or in time series. This time series is based on data, where cloudiness coefficient
is related to date and time range. It means, if enough data are available, cloudiness
coefficients for every hour could be set up and used during simulations.
The solar radiation is being calculated for each grid point (or elementary plot) with
eligible time step for the selected time interval within the day. The calculated values are
being summarised for the selected time interval to obtain the quantity of solar energy
(irradiation) on the Earth surface. One of the possible outputs from the model are time
series for the selected grid points. There are no time range limitations for the
simulations.
The output of the subroutine is the gridded information on potential or actual energy
income integrated through the chosen time interval [W hrs m-2].
Case study example
The applicability and accuracy of developed SOLEI-32 model have been tested in a
case study in the mountainous Jalovecky creek basin, Western Tatra Mountains,
Slovakia (49.10 N, 19.40 E). The Jalovecky creek basin is one of the experimental
basins of the Institute of Hydrology SAS. Its area covers 22.2 km sq. and the elevations
range from 800 to 2178 m a.s.l.. This region is a typical high mountain area of the
central Europe.
Field measurements of incoming solar radiation were carried out from 1.1.1998 to
31.12.1998 at two stations – Červenec and Parichvost (Fig. 1). Measured daily and
hourly sums of incoming solar radiation were used to compare the simulation accuracy
of SOLEI-32 computation scheme. The calculations were done on the basis of a digital
elevation model with 10  10 m grid resolution. The land cover map (Phare, 1996) was
used for the determination of different albedo values. The estimation of mean daily
cloudiness of the basin was based on the algorithm that uses the mean daily air
temperature range (Kostka, 2000).
Červenec
Parichvost
[m a.s.l.]
2000
1500
1000
Fig. 1
Jalovecky creek basin and stations (Červenec and Parichvost), where direct
measurements of incoming solar radiation were carried out.
Comparison of measured and simulated daily sums of incoming solar radiation for
selected time period are presented in Fig. 2, 3, 4 and 5. Results showed that SOLEI-32
precisely simulates the solar energy income for both stations. Good agreement between
simulated and measured values for the Parichvost station, especially in winter months,
indicates, that topographic shading incorporated in SOLEI-32 model correctly simulates
the solar income in varying mountain terrain.
Červenec
Parichvost
JANUARY 1998
[Wh.m-2]
10000
10000
8000
8000
6000
6000
4000
4000
2000
2000
0
0
1
5
9
13
17
21
25
1
29
5
9
13
17
21
25
29
5
9
13
17
21
25
29
5
9
13
17
21
25
29
FEBRUARY 1998
[Wh.m-2]
10000
10000
8000
8000
6000
6000
4000
4000
2000
2000
0
0
1
5
9
13
17
21
25
1
29
MARCH 1998
[Wh.m-2]
10000
10000
8000
8000
6000
6000
4000
4000
2000
2000
0
0
1
5
9
13
17
21
25
29
1
measured
computed
Fig. 2 Comparison between measured and simulated daily sums of incoming solar
radiation [Whm-2] for stations Červenec and Parichvost.
Červenec
Parichvost
APRIL 1998
[Wh.m-2]
10000
10000
8000
8000
6000
6000
4000
4000
2000
2000
0
0
1
5
9
13
17
21
25
1
5
9
13
17
21
25
29
1
5
9
13
17
21
25
29
1
5
9
13
17
21
25
29
29
MAY 1998
[Wh.m-2]
10000
10000
8000
8000
6000
6000
4000
4000
2000
2000
0
0
1
5
9
13
17
21
25
29
JUNE 1998
[Wh.m-2]
10000
10000
8000
8000
6000
6000
4000
4000
2000
2000
0
0
1
5
9
13
17
21
25
29
measured
computed
Fig. 3
Comparison between measured and simulated daily sums of incoming solar radiation
[Whm-2] for stations Červenec and Parichvost.
Červenec
Parichvost
JULY 1998
[Wh.m-2]
10000
10000
8000
8000
6000
6000
4000
4000
2000
2000
0
0
1
5
9
13
17
21
25
1
29
5
9
13
17
21
25
29
5
9
13
17
21
25
29
5
9
13
17
21
25
29
AUGUST 1998
[Wh.m-2]
10000
10000
8000
8000
6000
6000
4000
4000
2000
2000
0
0
1
5
9
13
17
21
25
1
29
SEPTEMBER 1998
[Wh.m-2]
10000
10000
8000
8000
6000
6000
4000
4000
2000
2000
0
0
1
5
9
13
17
21
25
29
1
measured
computed
Fig. 4
Comparison between measured and simulated daily sums of incoming solar radiation
[Whm-2] for stations Červenec and Parichvost.
Červenec
Parichvost
OCTOBER 1998
[Wh.m-2]
10000
10000
8000
8000
6000
6000
4000
4000
2000
2000
0
0
1
5
9
13
17
21
25
1
29
5
9
13
17
21
25
29
5
9
13
17
21
25
29
5
9
13
17
21
25
29
NOVEMBER 1998
[Wh.m-2]
10000
10000
8000
8000
6000
6000
4000
4000
2000
2000
0
0
1
5
9
13
17
21
25
1
29
DECEMBER 1998
[Wh.m-2]
10000
10000
8000
8000
6000
6000
4000
4000
2000
2000
0
0
1
5
9
13
17
21
25
29
1
measured
computed
Fig. 5
Comparison between measured and simulated daily sums of incoming solar radiation
[Whm-2] for stations Červenec and Parichvost.
The applicability of the SOLEI-32 for the computation of hourly intensities of incoming
solar radiation was tested for cloudless days: 11.August, 22.July and 10.May 1998.
Simulations for the Červenec and Parichvost stations for selected days are presented in
Fig. 6. Comparison between measured and computed hourly intensities showed that
SOLEI-32 model is applicable also for the simulation of hourly intensities of incoming
solar radiation.
Červenec
Parichvost
11.8.1998
[W.m-2]
[W.m-2 ]
1000
1000
800
800
600
600
400
400
200
200
0
0
4
8
4
12 16 20
8
12 16 20
22.7.1998
[W.m-2 ]
1000
800
computed
measured
600
400
200
0
4
[W.m-2 ]
8
12 16 20
10.5.1998
[W.m-2 ]
1000
1000
800
800
600
600
400
400
200
200
0
0
4 8 12 16 20
4 8 12 16 20
Fig. 6
Comparison between measured and computed mean hourly intensities of incoming solar
radiation for stations Červenec and Parichvost.
Conclusions
In this paper, we have presented an effective and efficient algorithm for the computation
of the solar energy income in topographically varying terrain. Based on raster maps of
elevation, vegetation and possibly cloud cover, SOLEI-32 model simulates the potential
and actual solar energy income in the form of raster maps, as well as time series for
selected points. Comparisons between measured and computed daily sums of incoming
solar radiation and mean hourly intensities for selected cloudless days in Jalovecky
creek basin showed very good agreement between simulations and direct measurements.
Acknowledgement
The Slovak Grant Agency VEGA supported the research presented in the study through
grant 2/7149/20. The authors gratefully acknowledge the support from VEGA.
References
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Kostka, Z. (2000): Methods of cloudiness index estimation for the solar radiation energy
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