Mapping global, diffuse and beam solar radiation over Zimbabwe

Data bank

Mapping global, di�use and beam solarradiation over Zimbabwe

T. Hovea, J. GoÈ ttscheb,*aDepartment of Rural and Urban Planning, University of Zimbabwe, PO Box MP 167, Mt. Pleasant,

Harare, ZimbabwebDepartment of Mechanical Engineering, University of Zimbabwe, PO Box MP167, Mt. Pleasant,

Harare, Zimbabwe

Received 10 September 1998; accepted 2 October 1998

Abstract

A database for long-term monthly radiation over Zimbabwe is developed. Themeteorological raw data inputs are long-term monthly average records of pyranometer-measured hemispherical radiation, monthly average sunshine records, and satellite-measuredhemispherical records over a 2-year period. The sunshine records are incorporated into the

database by use of Angstrom-type correlations developed for Zimbabwe, and the short-term satellite data are `cultured to long-term ground-measurement basis by means of anempirically derived correlation' and a `time series factor'. Di�use radiation values are

generated from the resulting hemispherical radiation database by a locally developedcorrelation of the monthly average di�use fraction of hemispherical radiation with monthlyaverage clearness index. Normal beam radiation is computed from the hemispherical and

di�use radiation using two di�erent methods. The two methods are found to agreegenerally to within 7%. The results are presented as country-border-contained isolines ofradiation. The sensitivity of beam radiation to the accuracy of estimating di�use radiationis inspected. The sensitivity is quite high, about 1:1 for months and locations with very low

clearness. This translates to an almost equal sensitivity of insolation available to trackingtilted apertures, underscoring the importance of developing a local di�use fraction-clearnessindex correlation rather than relying on a correlation developed elsewhere. # 1999 Elsevier

Science Ltd. All rights reserved.

0960-1481/99/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved.

PII: S0960 -1481 (98)00782 -4

Renewable Energy 18 (1999) 535±556

www.elsevier.com/locate/renene

* Corresponding author. Tel.: +263-4-303 211; fax: +263-4-303 280.

E-mail address: [email protected] (J. GoÈ ttsche)

Nomenclature

a, b coe�cients in the empirical correlation (Angstrom) ofclearness index with relative sunshine duration

f shadow-band correction factor for di�use radiationassuming a two-component sky radiance model

fi shadow-band correction factor for di�use radiationassuming isotropic sky di�use radiance

H irradiation (or total insolation) [J/m2]Hd long-term monthly average di�use irradiation on a

horizontal surface (daily total) [MJ/m2]Hh long-term monthly average hemispherical irradiation on a

horizontal surface (daily total) [MJ/m2]Ho monthly average extraterrestrial irradiation on a horizontal

surface (daily total) [MJ/m2]Hground,1985/86 monthly average daily hemispherical radiation as would be

measured by ground-based pyranometers for 1985 and 1986(derived from correlation of measured ground and satellitedata for 1985 and 1986)

Hground,long-term predicted long-term monthly average daily hemisphericalradiation at locations with given 1985/86 satellite data(=TSF � Hground,1985/86)

Hsat,1985/86 monthly average daily hemispherical radiation as measuredby satellite for 1985 and 1986

I hourly insolation [J/m2]�Ibn long-term monthly average hourly beam irradiation at

normal incidence [MJ/m2]�Id long-term monthly average hourly di�use irradiation on a

horizontal surface [MJ/m2]�Ih long-term monthly average hourly hemispherical irradiation

on a horizontal surface [MJ/m2]Kh Hh/Ho=long-term monthly average clearness indexrd �Id= �H d=ratio of the monthly average hourly to the monthly

average daily di�use irradiation on a horizontal planerh �Ih= �H h=ratio of the monthly average hourly to the monthly

average daily hemispherical irradiation on a horizontal planeTSF a country-universal time series factor relating the ratio

between long-term and 1985/86 hemispherical irradiationlong-term averages

Subscriptsbn normal beamd di�useh hemispherical radiation

T. Hove, J. GoÈttsche / Renewable Energy 18 (1999) 535±556536

1. Introduction

In research problems attempting to verify the detailed performance of a

particular solar energy system, instantaneous ¯uxes of hemispherical (also called

global), di�use and/or beam solar radiation may be required. However, for the

purpose of engineering design and the basic assessment of the economics of solar

energy systems, only the monthly averages of the above quantities are required.

The objective of this paper is to present the distribution of the intensity of

monthly average hemispherical, di�use and normal beam solar radiation over

Zimbabwe.

In Zimbabwe hemispherical radiation is measured by pyranometers at 20

stations monitored by the Meteorological Department (Met. Dept)1 These stations

also record sunshine duration. Another nine stations record sunshine duration

only. The sunshine data can be converted into hemispherical radiation data using

Angstrom-type correlations, which were developed using sunshine and radiation

data for nearby stations [1]. In addition, Raschke et al. reported values of

satellite-measured hemispherical radiation for Africa, for the years 1985 and 1986,

at latitude±longitude grid points with a spatial resolution of 2.5 � 2.5 degrees [2].

These three data sources, pyranometer-measured radiation records, sunshine

duration records and satellite-measured radiation records, are blended, after some

adjustments as described in section 2, to build a consolidated hemispherical

radiation database.

In order to compute the insolation available to inclined planes, one needs, in

addition to hemispherical radiation, the corresponding di�use component. Di�use

radiation is recorded at only two locations in Zimbabwe. To determine the di�use

component of radiation when only the hemispherical radiation is known, the

correlation of the ratio, Hd/Hh, of monthly average di�use to hemispherical

radiation, with monthly average clearness index, Kh must be determined.

Several correlations between these quantities have been developed in the

literature using data for speci®c locations [3±6]. However, the general validity of

these correlations for applicability to locations situated very di�erently from the

locations for which they were derived is not known. Therefore, in this study a

region-speci®c correlation, based on the local data is developed, to be used in

generating di�use radiation values from hemispherical values for all locations in

Zimbabwe. An important step in the development of this correlation is the

appropriate correction of the measured di�use radiation for the e�ect that by

obscuring part of the sky dome from the pyranometer, the shade-ring cuts o� not

only the beam radiation but also a signi®cant proportion of the di�use radiation.

Without local measurements of true di�use radiation, the approach chosen here is

to select from the literature a correction model which returns the highest

1 The Meteorological Department is a government organisation responsible for recording and inter-

preting weather statistics.

T. Hove, J. GoÈttsche / Renewable Energy 18 (1999) 535±556 537

correction factors, in order to make conservative estimates of available beamradiation.

The hemispherical radiation database is built in section 2 using all the availabledata. The development of the correlation of Hd/Hh with Kh as outlined above isthe subject of section 3. In section 4, values of beam radiation at normalincidence, are calculated from the results of sections 2 and 3. The results of thestudy are presented in section 5.

2. Building the hemispherical radiation database

For the determination of the monthly average hemispherical radiation, Hh,three data sources were used.

1. Records of pyranometer-measurements of daily hemispherical radiation fromthe Department of Meteorological Services of Zimbabwe (elsewhere called theMet. Dept).

2. Records of monthly average sunshine hours per day from the Met. Dept.3. Hemispherical radiation data derived from imaging data of the geostationary

satellite METEOSAT2, for 1985 and 1986 [2].

2.1. Pyranometer-measured data

Measured data of monthly average daily radiation on a horizontal plane for 20stations in Zimbabwe was provided by the Met. Dept. The records containedmonthly average data for periods ranging from 12 to 23 years. Before adoptingthese data some consistency checks were made. Global radiation data whichexceeded calculated extra-terrestrial radiation on a horizontal plane, for thelocation, for the month, were discarded. In addition, data records which deviatedby more than 20% from the long-term monthly average were discarded. Wheretwo or more stations located very near to each other recorded substantiallydi�erent monthly values, the lower (conservative) record was adopted. Threepercent of the data records were discarded through this screening criteria. Theremaining clean data were averaged over the period of recording. The coe�cientof variation, year-to-year of monthly means, for the data varied from 4 to 9%.These ground-measured radiation data were adopted as the most reliable of thethree data sources, and were used to derive or correct the other two data sources.(They are marked `Pyranometer' in Table 2).

2.2. Incorporating sunshine data

All the 20 ground stations in section 2.1 also record sunshine data. Howevernine other stations record sunshine data but not radiation data. The correlationdeveloped by Angstrom [1] and later modi®ed by Page [4], (see also [7]) relatingthe monthly average number of sunshine hours and monthly average clearness


index can be applied to generate hemispherical radiation from sunshine data. Therelationship is commonly expressed in the following form:

�Kh � a� b� �n= �N � �1�where n is the monthly average duration of sunshine, N is the day length and a, bare correlation coe�cients which can be determined from a least squares ®t onmeasured sunshine duration and clearness index. The regression coe�cients a andb are determined for grid square sub-regions (pixels), covering Zimbabwe. Foreach pixel a representative station measuring both radiation and sunshine is usedto determine a and b. The coe�cients are tabulated in Table 1.

The monthly average hemispherical radiation is then determined as:

�H h � �Kh�Eq: 1� � �Ho �2�where Ho is monthly average extra-terrestrial radiation on a horizontal plane.

The values of hemispherical radiation generated this way are incorporated inthe hemispherical radiation database.

2.3. Incorporating satellite-measured data

Monthly average hemispherical radiation data for Africa based onmeasurements of the European geostationary meteorological satelliteMETEOSAT2 was presented by Raschke et al. [2]. The measurements were madefor 1985 and 1986 and were used to derive radiation data on a grid of resolution2.5 � 2.58 longitude and latitude. Six grid points fall within Zimbabwe.

The monthly values of radiation obtained from the two methods, forcorresponding locations and months, di�er signi®cantly because of the inherentdi�erences between the satellite and the ground measuring methods. In order tomake values from the two measuring methods compatible with each other, thetwo methods have to be calibrated against each other. In this paper, the groundmeasurements are taken as the datum, and the satellite-measurements are`cultured' to a ground-measurement basis. The procedure used is as follows.

Using spatial interpolation, satellite-measured values Hsat, at each ground-measuring station for each of the 24 months of 1985 and 1986, can be deduced.These values can be paired with actual ground measurements, Hground, for

Table 1

Values of Angstrom regression coe�cients for Zimbabwe for square pixels centered at given longitude

and latitude

Pixel 1 2 3 4 5 6 7 8

Longitude (8E) 33.75 31.25 31.25 31.25 28.75 28.75 28.75 26.25

Latitude (8S) 18.75 16.25 18.75 21.25 21.25 18.75 17.25 18.75

a 0.32 0.36 0.27 0.24 0.29 0.32 0.35 0.25

b 0.45 0.43 0.48 0.49 0.46 0.47 0.48 0.58


corresponding locations and months. Applying least-squares regression, acalibration equation relating Hground and Hsat is determined. Six representativestations are used in this paper, to develop the correlation shown in Fig. 1. Theequation is:

Hground,1985=86 � 6:17� 0:75Hsat,1985=86 �4�The coe�cient of correlation R= 0.85 (R 2=0.73) is considered high enough tojustify the use of Eq. (4). Now, Hground,1985/86 of Eq. (4) is the ground `cultured'satellite record for 1985 and 1986. To relate this 1985/86 record to the long-termaverage radiation, Hground,1985/86 is multiplied by a `time series factor', TSF,accounting for the ratio between the long-term average radiation and the 1985/86average, i.e.:

TSF � �long-term average ground-measured data�=�1985=86

average ground-measured data�

The long-term average radiation derived from satellite data, Hground,long-term isthen given by:

Hground,long-term � TSF�Hground,1985=86 �5�

Fig. 1. Correlation of ground and satellite measured data: Zimbabwe 1985/86.


TSF for a station is obtained by averaging all the 1985/86 monthly radiationvalues for the station and dividing this into the long-term annual average for thatstation. TSF for the country is then obtained as the average of TSFs for a numberof representative stations (seven were used in this study). For the set of stationsused, TSF is incidentally equal to 1.0 with a coe�cient of variation of 2%.

Eq. (5) is used to calibrate satellite-measured data at each grid point, for eachmonth and the resulting values are incorporated into the hemispherical databasewhich will now comprise of records for 35 stations.

The consolidated global radiation database is shown in Table 2.

3. Deriving monthly average di�use radiation from hemispherical radiation

In Zimbabwe, hourly di�use radiation is recorded at two stations, Harare andBulawayo, using the common instrument set-up of a pyranometer shaded with ashadow band. In this study, only the Bulawayo records are used. The datasourced comprise monthly average hourly records of hemispherical and di�useradiation for 13 years. The di�use radiation data have already been corrected forthe shade ring e�ectÐthe e�ect that the shade ring, by obstructing part of the skydome from the pyranometer, not only cuts o� beam radiation (for which it isintended), but also a signi®cant part of di�use radiation. In this study the lowcorrection factors used by the Met. Dept are replaced by a more conservativecorrection method [8]. The Met. Dept's corrected data is therefore ®rst`uncorrected' and then re-corrected.

3.1. Determining measured di�use radiation data from corrected data

The correlation factors used by the Met. Dept to obtain true di�use radiationfrom measured data are shown in columns 3, 5 and 7 in Table 3. The correctionfactors are based on Drummond's isotropic correction factor [9] (see also Ref.[10]) which is multiplied by anisotropic factors of 1.03, 1.05 and 1.07 if the day'ssunshine duration n is n < 4.1 h, 4.1 h < n < 8 h or n>8 h, respectively. Thereare, however a number of short-comings in the correction factors of Table 3.

First, there should be at least two di�erent correction factors between n = 0and n = 4.1 h (0 < Kh < 0.45) since the onset of isotropic sky conditions occursat a Kh value greater than 0; at about Kh=0.26 [8]. Second, the application of justa single correction factor for all hours of the day cannot be justi®ed since anin®nite number of combinations of diurnal cloudiness can result in a particularnumber of sunshine hours per day. Certainly a di�erent correction factor shouldapply for each of the combinations. Third, the shade-ring correction factor wasalso shown to be dependent on the sun's position in the sky (as represented by thesun's altitude angle) [8,11]. This suggests an hourly variation of the correctionfactor, which is not taken into account by the Met. Dept's correction method.

In order to determine the average correction factors used by the Met. Dept overthe whole 13-year period, it is necessary to determine the frequency of occurrence


Table

2

Hem

isphericalradiationdatabase

forZim

babwe(Pyranometer

records:1971±1993;Sunshinerecords:1971±1993;Satellitedata:1985±1986)

Station

Latitude

(8)

Longitude

(8)

Monthly

averagedailyglobalradiation

(MJ/m

2)

Data

source

July

Aug

Sep

Oct

Nov

Dec

Jan

Feb

Mar

April

May

Jun

Chisengu

ÿ19.9

32.9

15.8

18.8

21.9

23.4

24.8

23.3

23.7

23.1

21.4

18.2

17.2

14.9

Sunshinedata

Chipinge

ÿ20.0

32.6

17.0

20.0

22.9

24.3

24.5

23.6

24.5

23.2

22.4

20.5

18.4

16.2

Sunshinedata

Banket

ÿ17.3

30.4

18.4

21.5

24.4

25.3

24.6

23.3

23.9

23.3

22.6

20.8

18.4

16.5

Sunshinedata

Chivhu

ÿ19.0

30.9

17.0

19.9

22.8

23.6

23.2

21.9

23.3

22.2

21.5

19.6

17.5

16.0

Sunshinedata

Henderson

ÿ17.6

31.0

17.2

20.3

22.9

24.0

22.7

21.2

21.4

20.6

20.8

19.3

17.6

16.0

Sunshinedata

Chisumbanje

ÿ20.8

32.3

16.8

19.1

21.9

23.5

24.3

23.8

25.1

23.5

22.3

19.7

17.6

15.6

Sunshinedata

Gokwe

ÿ18.0

29.9

19.2

22.2

24.7

25.9

24.9

23.2

23.9

23.6

23.0

21.5

19.3

17.8

Sunshinedata

Kwekwe

ÿ18.9

29.8

18.6

21.6

24.4

25.6

24.6

23.6

24.5

23.9

23.5

21.5

19.3

17.7

Sunshinedata

Hwange

ÿ18.6

27.0

19.6

22.9

25.5

26.1

25.8

23.8

24.2

23.6

23.4

22.0

19.8

18.3

Sunshinedata

WestNicholsonÿ2

1.2

29.4

16.8

19.5

21.7

22.5

23.6

23.5

24.3

23.0

21.9

20.0

18.1

16.0

Pyranometer

Bulawayo

ÿ20.2

28.6

17.6

20.2

22.4

23.4

23.3

22.9

22.8

22.6

21.6

20.0

18.0

16.8

Pyranometer

Mt.Darw

inÿ1

6.8

31.7

18.6

21.2

23.1

24.2

24.4

22.6

23.2

22.4

22.7

21.4

19.9

18.1

Pyranometer

BeitBridge

ÿ22.2

30.0

16.0

18.9

21.5

23.2

24.2

25.0

25.3

24.4

22.6

20.2

17.3

15.1

Pyranometer

Bu�alo

Range

ÿ21.0

31.7

15.0

17.4

20.6

22.5

24.2

26.1

27.0

25.7

23.0

21.9

19.8

17.4

Pyranometer

Marondera

ÿ18.3

31.5

18.7

21.3

23.3

23.2

23.3

21.9

22.4

22.2

21.7

20.9

19.2

17.7

Pyranometer

Masvingo

ÿ20.1

30.9

17.0

20.1

22.3

22.9

23.4

22.9

24.3

22.7

22.0

20.3

18.4

16.3

Pyranometer

Nyanga

ÿ18.4

32.7

18.9

21.8

23.8

23.6

22.9

20.5

21.7

21.0

21.6

21.1

19.3

18.3

Pyranometer

Kariba

ÿ16.5

28.9

19.5

22.3

24.1

25.7

25.8

23.3

23.0

22.8

22.6

22.2

20.5

18.8

Pyranometer


Table

2(continued

)

Station

Latitude

(8)

Longitude

(8)

Monthly

averagedailyglobalradiation

(MJ/m

2)

Data

source

July

Aug

Sep

Oct

Nov

Dec

Jan

Feb

Mar

April

May

Jun

Harare

ÿ17.8

31.1

17.5

20.5

22.9

24.0

23.4

21.3

22.4

20.8

20.3

19.6

18.1

16.9

Pyranometer

VictoriaFalls

ÿ18.1

25.9

20.5

23.4

25.3

25.6

25.3

24.1

24.2

24.0

23.9

23.3

21.5

19.7

Pyranometer

Binga

ÿ17.7

27.4

19.4

21.9

23.6

25.3

25.6

23.2

23.2

22.6

23.3

22.4

20.5

18.4

Pyranometer

GrandReef

ÿ19.0

32.7

17.5

20.8

23.1

24.0

24.0

22.2

24.0

21.7

21.9

20.1

18.5

16.6

Pyranometer

Gweru

ÿ19.7

29.9

19.2

22.2

24.4

24.4

24.0

22.8

23.8

23.1

22.7

21.4

19.7

18.4

Pyranometer

Kadoma

ÿ18.3

29.9

19.1

22.0

24.0

24.8

24.4

22.3

23.6

23.1

22.4

21.6

19.7

18.8

Pyranometer

Karoi

ÿ16.8

29.6

20.0

22.8

24.5

25.2

24.5

21.8

22.5

21.6

22.5

21.9

20.3

19.1

Pyranometer

Makoholi

ÿ19.8

30.8

18.3

21.3

22.9

23.3

23.7

23.1

24.4

22.9

21.1

20.9

19.2

17.8

Pyranometer

Matopos

ÿ20.6

28.7

17.3

20.5

23.3

23.2

24.6

23.6

24.1

23.4

23.0

20.5

17.9

16.7

Pyranometer

Tsholotsho

ÿ19.8

27.7

18.1

20.5

22.6

23.2

22.8

22.8

23.9

21.2

21.1

20.9

18.5

17.2

Pyranometer

Mukandi

ÿ18.7

32.9

17.7

20.7

23.2

23.2

22.9

21.1

21.8

20.9

21.4

19.9

18.9

17.0

Pyranometer

GridPoint1

ÿ16.3

31.3

19.7

22.3

24.3

23.9

23.7

21.3

21.4

21.6

22.0

20.2

19.4

18.4

Satellite

GridPoint2

ÿ18.8

26.3

19.4

21.6

23.8

23.5

24.9

23.4

23.1

23.8

22.7

20.7

19.4

19.0

Satellite

GridPoint3

ÿ18.8

28.8

19.5

21.9

24.1

24.2

24.6

22.7

22.4

23.5

23.7

20.7

19.1

18.8

Satellite

GridPoint4

ÿ18.8

31.3

19.2

21.8

24.2

24.2

24.2

21.8

21.8

22.8

22.6

20.6

18.7

18.2

Satellite

GridPoint5

ÿ21.3

28.8

18.5

20.9

23.1

23.1

24.6

23.4

23.3

23.5

22.5

19.7

18.4

17.5

Satellite

GridPoint6

ÿ21.3

31.3

18.0

20.4

22.5

22.7

23.9

22.6

22.4

22.7

22.4

19.4

17.5

17.6

Satellite


of the three sunshine duration ranges. The sunshine duration ranges are ®rstconverted to clearness index ranges using the Angstrom coe�cients (Table 1) forthe pixel containing Bulawayo (pixel 5). The assumption here is that thecorrelation of monthly average sunshine duration and monthly average clearnessindex can be generalised to apply to daily values. With a knowledge of the long-term monthly average clearness index, Kh, the frequency of occurrence of eachclearness index range (and therefore the frequency of application of each of thethree correction factors), for each month, is determined [3]. The averagecorrection factor, Fbar, which was used for a particular month is then thefrequency-weighted sum of the three correction factors given for that month. Theprocedure is illustrated in Table 3. The average correction factor found asdescribed above is then used to `uncorrect' the Met. Dept's corrected di�useradiation data.

3.2. Re-correcting di�use radiation data

Of the models encountered in the literature which take into account thevariability of the shade-ring correction with both clearness index and the sun's

Table 3

Determining the average shade-ring correction factor used by the Met. Dept. (Bulawayo data)

Month Khc n< 4.1 hrsb Kh < 0.45a 4.1 < n< 8 hrsb

0.45 < Kh < 0.60an>8 hrsb Kh>0.60a FMet

f Frecorg

Correctiond Frequencye Correctiond Frequencye Correctiond Frequencye

January 0.55 1.265 0.301 1.289 0.211 1.314 0.488 1.294 1.400

February 0.56 1.291 0.289 1.317 0.203 1.342 0.508 1.322 1.437

March 0.59 1.292 0.247 1.316 0.183 1.341 0.570 1.324 1.433

April 0.64 1.246 0.165 1.271 0.145 1.295 0.690 1.283 1.375

May 0.68 1.193 0.089 1.216 0.122 1.239 0.789 1.232 1.328

June 0.70 1.165 0.055 1.188 0.106 1.210 0.839 1.205 1.244

July 0.70 1.177 0.055 1.200 0.106 1.223 0.839 1.218 1.303

August 0.70 1.225 0.055 1.248 0.106 1.272 0.839 1.267 1.396

September 0.65 1.278 0.149 1.303 0.137 1.323 0.714 1.314 1.465

October 0.61 1.279 0.218 1.322 0.167 1.347 0.615 1.328 1.494

November 0.56 1.274 0.289 1.299 0.203 1.324 0.508 1.304 1.465

December 0.54 1.254 0.313 1.278 0.219 1.302 0.468 1.282 1.442

a Kh=daily clearness index.b n= number of sunshine hours per day.c Kh=the long-term monthly average clearness index.d Correction=the total shade-ring correction factor used by the Met. Dept when the sunshine dur-

ation is in the given category.e Frequency=the frequency of occurrence of the corresponding clearness index range and monthly

average clearness index [3].f FMet=the predicted monthly average shadow band correction factor used by the Met. Dept.g Frecor=the average monthly recorrection factor applied to the data [8].


altitude, Siren's model was adopted because it yields the most conservativecorrection factors from the point of view of designers of solar energy systems [8].

The correction factor of Siren is presented as a family of curves plotting theratio of the correction factor to be applied, f, to the isotropic [9] correction factor,fi, against clearness index, kh, for di�erent values of the sun's altitude angle.However, there are no analytical expressions, given in Siren's paper, representingthe curves. For convenience in correcting a large amount of data, analyticalexpressions which closely approximate the correction factor are derived in thisstudy. For a shade ring with width-to-radius ratio, b/r, of 0.175Ðthese are thedimension for the Bulawayo instrumentÐthe expression for the ratio of theanisotropic to the isotropic factor, f/fi, is:

f=fi � 1� �0:225 sin a� 0:043� sin�p�kh ÿ 0:26�=0:72� �6�where: a is the sun's altitude angle and kh is the hourly clearness index.

Monthly averaged hourly clearness index data was obtained for Bulawayo froma 13-year database (Met. Dept). The isotropic correction factor is determinedfrom Drummond's equation. The correction factor determined this way was usedto re-correct monthly average hourly data for Bulawayo before its use in thefollowing section.

3.3. Correlation of Hd/Hh with monthly average clearness index for Bulawayo

The corrected monthly average di�use radiation data are plotted againstcorresponding hemispherical data and least-squares regression equations ®tted asshown in Fig. 2.

Two least-squares regression polynomial equations are ®tted: a fourth-order anda ®rst-order (linear) polynomial. The coe�cients of correlation are 0.91 and 0.90for the fourth- and ®rst-order polynomial equations, respectively. There is littlegain in goodness-of-®t in moving from a ®rst-order to a fourth-order polynomialto justify the adoption of the more complicated fourth-order equation. The ®rst-order polynomial is adopted for use in this study. The correlation is:

�H d= �H h � 1:0294ÿ 1:144 �Kh for 0:47< �Kh<0:75

�H d= �H h � 0:175 for �Kh > 0:75 �7�This correlation agrees well with the Page correlation [4] which was developed forlatitudes 408N±408S. The Page correlation is:

�H d= �H h � 1ÿ 1:13 �Kh �8�For high values of clearness index (close to 0.8), the fourth-order correlationcurves upwards which is physically not reasonable. For this region, a linetangential to the lowest point of the fourth-order polynomial is constructed and


found to intersect the ®rst-order polynomial curve at Kh=0.75 and Hd/Hh=0.175;thus the appearance of these limits in Eq. (7). There was no data for Bulawayoyielding Kh values lower than 0.47. However, the linear correlation equation isextrapolated in the few cases that Kh values less than 0.47 are encountered in thecountry's database.

When di�use radiation values generated from global radiation values using thiscorrelation are compared with those calculated from the Erbs et al. correlation [6],for all stations and months, the maximum deviation is found to be close to 20%.For predicting the insolation available to solar energy devices which use asigni®cant proportion of di�use radiation, for example ¯at-plate collectors, thisderivation translates to a deviation in energy available to the collector of only 1±2%. However, for predicting the radiation available to collectors which acceptonly beam radiation, e.g. tracking concentrating collectors, the deviation resultingfrom using the di�erent di�use-fraction correlation is quite high; up to about 20%for a 20% deviation in di�use radiation (see section 4). This fact underscores theimportance of developing a local correlation rather than relying on correlationsdeveloped elsewhere.

The spatial and temporal average of the deviation between di�use radiationgenerated in this study and those of the Erbs et al. correlation [6] is con®ned towithin 10%. This is considered to be the margin of error that can be expected ofthe di�use radiation values generated in this study.

Fig. 2. Monthly average Hd/Hh vs monthly average clearness index: from Bulawayo data 1980±92.


4. Estimating normal beam radiation

In this section beam radiation is calculated using two di�erent methods and theresults are compared.

4.1. Using rh and rd factors

Long-term beam radiation at normal incidence for an hour, �Ibn, can beestimated from a knowledge of hourly hemispherical radiation on a horizontalplane, �Ih, and hourly di�use radiation on horizontal plane, �Id, using therelationship:

�I bn � � �Ih ÿ �Id�= cos yhour �9�

where yhour is the angle of incidence of beam radiation on a horizontal surface atthe mid-point of the hour in question.

The hourly values for hemispherical and di�use radiation, �Ih and �Id, can bedetermined using Collares-Pereira and Rabl rh and rd conversion factors forresolving average daily insolation values to average hourly values [5]

�I h�t� � rh�t� �Hh �10a�

and

�I d�t� � rd�t� �Hd �10b�

The hourly conversion factors, rh and rd calculated from the formulae given byCollares-Pereira and Rabl, for di�erent locations and months, deviate by up toabout 1% from the condition that their sum over all daylight hours should beequal to 1. They are corrected to make them satisfy the requirement thatX

day

rd�t� �Hd � �H d

and Xday

rh�t� �Hh � �H h

A conservative measure adopted in calculating beam radiation in this study, wasto disregard all beam radiation occurring at yhour>808 (cosyhour < 0.17). The rhand rd factors vary with latitude, but for Zimbabwe which is contained within 78of latitude, the variation is found to be insigni®cant. It is considered adequatelyaccurate to calculate rh and rd factors at a central latitude (198S), and to use thesevalues for all locations in the country.


4.2. Comparative study using INSEL

For comparison with beam data generated as described above, and forconsistency checking, the software tool INSEL [12], was used to estimate beamradiation on a two-axis tracked surface on the basis of generated time-series data.

The INSEL tool allows the generation of realistic hourly global radiation dataIh from monthly averages [13]. The INSEL data generator is based on theprocedure proposed by Gordon and Reddy that generates hourly clearness indexdata with a given autocorrelation [14].

In this study, monthly mean global radiation data of six stations is used togenerate a set of hourly data Ih which sums up exactly to the given monthly®gure. The INSEL standard autocorrelation coe�cients of r (1 day)=0.3 and r (2days)=0.17 are used in this study.

The correlation of Erbs et al. is used to generate hourly data Id of di�useradiation [6]. This data is corrected with a monthly correction factor so that themonthly average of generated data coincides with the monthly average of di�usedata as obtained earlier in this study. In general, the di�use radiation data has tobe corrected down to about 10% lower values in order to obtain the prescribedmonthly averages.

With the given hourly global and di�use radiation data, hourly direct normalbeam radiation data Ibn are generated by applying Eq. (9) with I replacing �I ,whereby hours with Id>Ih are discarded and only hours with cos(yhour)>0.17(yhour < 808) are considered.

Results for six stations in Zimbabwe are shown in Table 4 together with datafrom section 4.1.

Deviations of monthly average normal beam radiation as derived by the twodi�erent methods are given in %. Generally, the INSEL method results in slightlylower beam radiation data. The deviation of annual averages is con®ned to lessthan 3% for all stations whereas monthly mean values may deviate in extremecases by up to 13.8%. However, generally the deviation is less than 7% which isconsidered to be the margin of error of the monthly beam radiation ®gures asgenerated with the method of section 4.1, which is adopted here for generatingbeam radiation values for the attached maps.

5. Results

The results of this study are presented as a set of isoline maps of long-termaverage radiation. A sample of the maps is shown in Figs. 4±12. This gives avisual picture of the spatial distribution of radiation over Zimbabwe which can beconveniently read by potential users.

For the estimation of radiation available to tilted apertures which accept bothbeam and di�use radiation the monthly averages of both hemispherical (global)and di�use radiation are necessary inputs [15]. However, for estimating radiation


Table

4

Comparison

between

beam

radiation

values

from

studywith

INSEL

values

(INSEL=beam

radiation

values

generated

bytheIN

SEL

software;SECT.

4.1=

beam

radiationvalues

generatedbythemethodofsection4.1;DEV(%

)=[INSELÐ

SECT.4.1]/SECT.4.1)

Station

Month

January

February

March

April

May

June

July

August

September

October

Novem

ber

Decem

ber

BeitBridge

INSEL

22.4

21.9

21.4

22.4

22.5

19.4

21.8

22.8

22.0

20.9

19.9

21.6

21.6

SECT4.1

22.5

22.7

20.8

22.5

23.3

21.4

22.3

22.8

21.3

21.8

20.9

21.5

22.0

DEV(%

)8.4

3.5

ÿ2.6

0.7

3.8

10.3

2.5ÿ0

.2ÿ3

.34.6

5.3

ÿ0.7

1.9

Bulawayo

INSEL

18.4

18.5

19.1

21.1

23.2

24.0

24.7

24.9

22.8

20.8

19.2

18.3

21.2

SECT4.1

18.4

19.3

18.6

21.0

23.5

24.3

24.7

24.5

22.3

22.0

19.5

18.4

21.4

DEV(%

)ÿ0

.14.3

ÿ2.8

ÿ0.2

1.3

1.3

0.3ÿ1

.4ÿ2

.56.1

1.7

0.5

0.7

Gokwe

INSEL

20.4

20.4

21.2

23.4

24.9

24.7

26.9

28.7

27.5

25.6

21.6

18.8

23.7

SECT4.1

20.7

21.3

20.6

23.3

25.1

24.9

27.1

28.3

26.5

27.3

23.0

19.2

23.9

DEV(%

)1.7

4.3

ÿ2.6

ÿ0.1

0.7

0.7

0.8ÿ1

.6ÿ3

.46.7

6.5

2.4

1.2

Nyanga

INSEL

17.6

16.9

19.4

22.7

24.2

23.5

24.9

26.5

25.0

20.7

18.3

14.8

21.2

SECT4.1

16.7

16.6

18.3

22.5

25.1

26.7

26.6

27.4

24.7

22.4

19.3

14.8

21.8

DEV(%

)ÿ4

.6ÿ1

.4ÿ5

.8ÿ0

.74.1

13.8

6.8

3.6

ÿ1.1

7.9

5.1

0.3

2.7

VictoriaFalls

INSEL

20.3

21.0

23.6

27.5

31.3

30.2

30.1

32.0

28.3

24.4

23.1

20.5

26.0

SECT4.1

21.2

22.1

22.4

27.6

29.8

29.4

29.7

30.0

27.9

26.5

23.8

21.0

25.9

DEV(%

)4.8

6.4

ÿ5.1

0.3

ÿ4.7ÿ2

.6±

ÿ6.3

ÿ1.5

8.5

3.3

2.1

ÿ0.3

Harare

INSEL

18.1

16.2

16.2

19.2

21.4

22.1

22.2

23.8

23.3

22.0

19.2

15.8

20.0

SECT4.1

18.1

16.2

16.0

19.1

21.7

22.3

22.2

23.7

22.6

23.1

20.2

16.0

20.1

DEV(%

)0.2

0.1

ÿ1.3

ÿ0.6

1.3

0.7

0.3ÿ0

.2ÿ3

.05.0

5.4

1.6

0.8


Fig. 3. Sensitivity of beam radiation to change in di�use radiation: data from 35 locations in

Zimbabwe.

Fig. 4. Global radiationÐAugust (MJ/m2).


Fig. 6. Global radiationÐAnnual (MJ/m2).

Fig. 5. Global radiationÐDecember (MJ/m2).


Fig. 8. Di�use radiationÐDecember (MJ/m2).

Fig. 7. Di�use radiationÐAugust (MJ/m2).


Fig. 10. Beam radiationÐAugust (MJ/m2).

Fig. 9. Di�use radiationÐAnnual (MJ/m2).


Fig. 12. Beam radiationÐAnnual (MJ/m2).

Fig. 11. Beam radiationÐDecember (MJ/m2).


available to apertures which accept only beam radiation the only necessary inputis the annual average of normal beam radiation.

5.1. Sensitivity of beam radiation to change in di�use radiation

The accuracy of beam radiation values calculated using the method outlinedabove depends to some extent on the quality of the correlation of di�use fractionto clearness index used. The sensitivity of beam radiation to the deviation indi�use radiation from the di�use radiation ®gures used in this study is shown inFig. 3. The points in Fig. 3 are based on data for all the 35 stations shown inTable 2 and the curves are the envelopes for the highest and lowest sensitivityover the year. Although a number of factors, e.g. the latitude of the location, thesun's declination and the diurnal variation in radiation (as represented by the rhand rd factors), all a�ect sensitivity, the monthly average clearness index, Kh hasthe strongest in¯uence. As shown by the sensitivity envelopes in Fig. 3 there is aclear decrease of sensitivity with increase in clearness index.

6. Conclusion

In this study, ®gures for annual normal beam radiation for Zimbabwe areobtained using all data on global and di�use radiation presently available.

Two independent methods are applied to obtain monthly beam radiationaverages from monthly global and di�use radiation data. Results for monthlyaverages di�er by typically less than 7%. Results for annual averages di�er by lessthan 3% for all stations.

A sensitivity analysis reveals that uncertainties in monthly di�use data mayresult in up to equal relative uncertainties of the estimated normal beam radiation,depending on clearness index.

Given the 10% uncertainty of the used di�use radiation data, the ®gures forannual beam radiation data presented here should be considered with an errormargin of up to210%.

Generally the beam radiation in Zimbabwe can be expected to average about 20MJ/m2 (2000 kWh/m2/year) with a peak of 26 MJ/m2 (2600 kWh/m2/year) aroundVictoria Falls. The upward gradient towards the western end of the country isbased on observations from one station only (Victoria Falls). More extendedmeasurements should be carried out to con®rm or correct this.

Acknowledgements

The authors are indebted to Mr T.C. Temba from the Department ofMeteorological Services of the Government of Zimbabwe who provided the rawdata for this study.


References

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[11] LeBaron BA, Michalsky JJ, Perez R. A simple procedure for correcting shadowband data for all

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[12] Schumacher J. INSEL manual. Carl von Ossietzky University, Oldenberg, Germany, 1997.

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Documents

Mapping global, diffuse and beam solar radiation over Zimbabwe