evaluation of slope irradiance and illuminance models against measured hong kong data

Evaluation of slope irradiance and illuminance models againstmeasured Hong Kong data

Danny H.W. Li, Joseph C. Lam*

Building Energy Research Group, Department of Building and Construction, City University of Hong Kong, Kowloon, Hong Kong

Received 14 September 1998; received in revised form 17 May 1999; accepted 5 July 1999

Abstract

This paper presents the evaluation of three inclined irradiance and illuminance anisotropic models (Klucher, Hay and Perez)

for Hong Kong using two-year (1996±1997) measured data. A statistical analysis of the three models has shown that predictionsfor solar irradiance tend to be more accurate than daylight illuminance. All three models produce large errors for north-facingsurfaces. Predictions for south-facing surfaces show reasonably good agreement with measured data, indicating good estimationsof anisotropic di�use component at sunlit surfaces. In general, Perez's model shows the best predictions for vertical solar

irradiance and daylight illuminance followed by, Hay's, then Klucher's models. 7 2000 Elsevier Science Ltd. All rights reserved.

1. Introduction

Solar radiation data are essential to architects, en-gineers and scientists for both energy-e�cient buildingdesigns and the development of active energy systems.In subtropical Hong Kong, solar heat gain accountsfor nearly half of the total building envelope coolingload [1]. More solar radiation means more total solarheat gain, and hence more cooling requirements. Anunderstanding of the availability of more comprehen-sive solar radiation data would be invaluable to the re-duction of solar radiation cooling load in building.Furthermore, it is common knowledge that arti®ciallighting load can be reduced by using daylight. Day-lighting is recognised as an important and useful strat-egy in energy-e�cient building designs. It isparticularly attractive in hot climates since it reduceselectricity use, not only for arti®cial lighting but alsofor air-conditioning, due to less heat dissipation fromelectric light ®ttings [2,3]. Direct and di�use com-ponents of solar radiation and outdoor illuminancedata are essential to determine the amount of solar

heat gain and daylight entering a building. Horizontal

solar data can be used in studying horizontal fenestra-

tion such as skylight. For energy-e�cient building de-

signs, accurately knowing the amount of solar

radiation and daylight level on vertical surfaces is im-

portant, particularly for high-rise buildings. The global

radiation on a horizontal surface has been measured in

most meteorological stations around the world but

there are only a few stations which measure solar com-

ponents on inclined surfaces [4]. Therefore, di�erent

mathematical models [5±8] have been developed to

predict the solar irradiance and daylight illuminance

on tilted surfaces from the measurements on a hori-

zontal surface. The prediction models are mainly

developed in North America and some European

countries. Recently, Muneer [9] has evaluated Perez's

model [8] using Japanese data. However, very little

work has been done for the subtropical region near

Southern China. This paper describes the work on

modelling irradiances and illuminances on inclined sur-

faces based on measured horizontal data in Hong

Kong. A comparative study on several inclined models

is presented and the implications for the determination

of vertical data in Hong Kong are discussed.

Building and Environment 35 (2000) 501±509

0360-1323/00/$ - see front matter 7 2000 Elsevier Science Ltd. All rights reserved.

PII: S0360-1323(99 )00043 -8

www.elsevier.com/locate/buildenv

* Corresponding author.

E-mail address: [email protected] (J.C. Lam).

2. Local climate

Hong Kong has a total area of about 1000 km2 situ-ated along the southern coast of China within the sub-tropical region, with a latitude of 22.38N and alongitude of 114.28E. It has a variety of weather fromseason to season due to its location on the south-eastcoast of the Asiatic continent opposite a vast expanseof the ocean. Winter months are between Decemberand February. Mean temperature is around 15±188Cwith the monsoon coming from the North and North-east. Spring is from March until early May, and isusually cloudy with periods of light rain. The summerseason spans from late May until September. Themonsoon blows from the South and South-east withan average temperature of 27±298C. It is hot andhumid with occasional showers and thunderstorms. Attimes, typhoons strike Hong Kong and bring heavyrain and strong winds. Autumn is short and normallyruns from October to November. Sunny bright skiesdominate with dry conditions and a mean temperatureof 258C.

3. Measuring station and data corrections

The measuring station is located on the roof-top ofthe City University of Hong Kong. The instrumenta-tion at the station consists of six pyranometers and sixilluminance sensors for the horizontal and vertical

solar radiation and daylight illuminance data measure-ments. All sensors are installed on the roof-top in aposition relatively free from any external obstructions,and readily accessible for inspection and general clean-ing. Data collection starts before sunrise and ®nishesafter sunset. All measurements are referred to truesolar time. Two pyranometers (CM11) manufacturedby Kipp and Zonen, The Netherlands, are used tocarry out the measurement of solar radiation on ahorizontal plane. The di�use radiation pyranometer is®tted with a shadow-ring (CM121) 55 mm wide and620 mm in diameter, which shades the thermopilefrom direct radiation. The shadow-ring is paintedblack to minimise the e�ect of multiple re¯ections. Thetwo pyranometers are connected to an integrator(CC12) which calculates the radiation (integral of irra-diance) at an intervals of 10 min. The data from theintegrator are sent to a micro-computer for storagethrough a RS232C cable. The global solar radiationon vertical planes facing north, east, south and westare measured by other four pyranometers. Each pyran-ometer is connected to its own integrator (CC20)which measures the actual value (irradiance). The irra-diance data are captured simultaneously twice per sec-ond, averaged over 10-min intervals and stored intotwo computers which are connected to the four inte-grators.

The measurements of global and di�use outdoorilluminance on a horizontal plane are made by themeans of two illuminance meters. The two silicon

Nomenclature

a1, a2 the solid angles occupied by the circumsolarregion, weighted by its average incidence onthe slope and the horizontal, respectively (sr)

Dh di�use irradiance on a horizontal surface(W/m2)

Dvh sky-di�use irradiance on the vertical surface(W/m2)

Dbh sky-di�use irradiance on a plane with surfaceinclination b (W/m2)

F1, F2 the coe�cients of circumsolar and horizonbrightness, respectively (dimensionless)

F ' 1-(Dh/Gh)2 (dimensionless)

F0 Hh/Goh (dimensionless)fij coe�cients used to calculate values of F1 and

F2 ( fi1 and fi2 dimensionless, fi3 in degreeÿ1)Gh global irradiance on a horizontal surface (W/

m2)Goh extraterrestrial irradiance on a horizontal

surface (W/m2)Gon extraterrestrial irradiance at normal inci-

dence (W/m2)

Gvh global irradiance on the vertical surface (W/m2)

Hh direct beam irradiance on a horizontal sur-face (W/m2)

Hn direct normal beam irradiance (W/m2)Hvh direct beam irradiance on the vertical surface

(W/m2)m air mass (dimensionless)Rvh re¯ected irradiance on the vertical surface

(W/m2)Z zenith angle of the sun (degrees)

Greek symbolsa solar altitude (degrees)b surface inclination (degrees)y incidence angle of sun on a tilted surface

(degrees)r ground re¯ectivity (dimensionless)E sky clearness (dimensionless)D sky brightness (dimensionless)

D.H.W. Li, J.C. Lam / Building and Environment 35 (2000) 501±509502

photovoltaic cells contain cosine and colour correction.Again, a shadow-ring is used to block the direct com-ponent for the di�use illuminance measurement. Themeasured results are converted into digital signals byan analogue/digital converter (12 bits A/D-D/A card)before feeding into a micro-computer for storage.Data are collected every 15 s, averaged over 5-minintervals. For the measurements of the global illumi-nance at the four cardinal orientations, four illumi-nance sensors of the same model type are used. Thefour meters are connected to another analogue/digitalconverter and computer for data conversion and log-ging. The data collection method is the same as thehorizontal illuminance measurement.

Quality control tests were carried out to eliminatespurious data and inaccurate measurements resultingfrom the cosine response error of the pyranometersand illuminance sensors. These tests are as follows:

1. rejecting all di�use data which are greater than thecorresponding global values;

2. rejecting all global data that are greater than thecorresponding extraterrestrial solar component;

3. rejecting all data with a solar altitude, a, of lessthan 58;

4. rejecting all data when the direct normal values [i.e.(global±di�use)/sina ] exceed the correspondingextraterrestrial solar component;

5. rejecting all di�use data which are greater than halfof the corresponding extraterrestrial solar com-ponent (i.e. Shadow-ring is not properly adjusted);

6. applying a shade ring correction to the observed dif-fuse horizontal data according to the methoddescribed by LeBaron et al. [10].

Data that passed the quality control tests were con-verted into hourly values. Hourly direct horizontalirradiance and illuminance data were obtained by sub-tracting the corrected di�use components from the cor-responding global values.

4. Inclined models

Measured solar radiation data are more readilyavailable than measured daylight data. Inclined surfacemodel development is, therefore, mainly carried outfor solar radiation. In the following equations, the re-spective solar irradiance quantities can be transposedwith their illuminance counterparts. For solar irradi-ance on a vertical surface, it is evaluated as the sum ofdirect beam, sky-di�use and ground-re¯ected com-ponents. Mathematically, it can be expressed as:

Gvh � Hvh �Dvh � Rvh �1�where

Gvh is the global irradiance on the vertical surface(W/m2);Hvh is the direct beam irradiance on the vertical sur-face (W/m2);Dvh is the sky-di�use irradiance on the vertical sur-face (W/m2) andRvh is the re¯ected irradiance on the vertical surface(W/m2).

Di�erent models have been developed to determinethe total irradiance on an inclined surface frommeasured horizontal data. The main di�erence amongthe models is the handling of the sky-di�use com-ponent [11,12]. The direct beam and ground-re¯ectedirradiance are expressed as follows:

Hvh � �Hh= sin a� � cos y �2�

Rvh � 0:5rGh �3�where

Gh is global irradiance on a horizontal surface (W/m2);Hh is the direct beam irradiance on a horizontalsurface (W/m2);a is the solar altitude (degrees);y is the incidence angle of sun on a tilted surface(degrees); andr is the ground re¯ectivity (dimensionless).

Some researchers [5,13] assumed isotropicity of sky-di�use component while others [11,14] adopted anapproach of non-isotropicity, particularly because ofthe strong forward scattering e�ect of aerosols [15].The approaches are purely geometrical and they havealso been applied for daylight calculations [16,17]. Forthe present study, three anisotropic models (Klucher[6], Hay [7] and Perez [8]) were compared with themeasured vertical data. These models are simple touse, and are studied and assessed widely by a numberof researchers [11,18±20]. It should be emphasised thatthe simple nature of the models makes them good can-didates for general applications.

4.1. Klucher's model

Klucher's model [6] is based on the principle thatovercast skies tend to be isotropic while clear skiesshow strong horizon brightening. Mathematically, theincident di�use component, Dbh, on a plane with sur-face inclination b, is given in the following form:

Dbh � Dhf��1� cos b�=2� � �1� F 0

sin 3�b=2�� 1� F 0�cos 2 y��sin 3 Z ��g�4�

D.H.W. Li, J.C. Lam / Building and Environment 35 (2000) 501±509 503

F 0 � 1ÿ �Dh=Gh�2 �5�where

Z is the zenith angle of the sun (degrees); andDh is the di�use irradiance on a horizontal surface(W/m2).

Under overcast skies, F ' becomes zero, thereby redu-cing the model to that of isotropic; while under clearskies, F ' approaches unity, approximating a clear skymodel [21].

4.2. Hay's model

Hay [7] established a model which assumes the sky-di�use component to be composed of a circumsolarcomponent coming from the direction of the sun and auniform di�use component from the rest of the sky.Hay assumed linearity on the isotropic and circumso-lar contributions to di�use component on a tiltedplane. The two components are mixed via a modulat-ing function F0 as follows:

Dbh � DhfF 00 � �cos y= sin a� � ��1ÿ F 00 � � �1�

cos b�=2�g�6�

F 00 � Hh=Goh �7�where Goh is the extraterrestrial irradiance on a hori-zontal surface (W/m2).

The ®rst term corresponds to the circumsolar zoneand the second is the isotropic contribution. Eq. 6reduces to the isotropic model for overcast sky whenF0=0.

4.3. Perez's model

Perez's model [8] is based on a three-zone treatmentof the sky-di�use component. It is expressed as:

Dbh � Dh�0:5�1� cos b��1ÿ F1� � F1�a1=a2�

� F2 sin b� �8�

where

a1, a2 are the solid angles occupied by the circumso-lar region, weighted by its average incidence on theslope and the horizontal, respectively (sr); andF1, F2 are the coe�cients of circumsolar and hor-izon brightness, respectively (dimensionless).

For a vertical surface (b=908), Eq. 8 becomes:

Dvh � Dh�0:5�1ÿ F1� � F1�a1=a2� � F2� �9�For the point-source simpli®ed version, the expressionsfor a1 and a2 are taken as:

a1 � max�0, cos y� �10�

a2 � max� cos 85 ^ , cos Z � �11�Coe�cients F1 and F2 are functions of three variablesdescribing the sky conditions: i.e. (a) the solar zenithangle, Z; (b) the sky's clearness, E, expressed as:

E � ��Dh �Hn�=Dh � 5:535 � 10ÿ6Z 3�=�1� 5:535

� 10ÿ6Z 3� �12�

where Hn is the direct normal beam irradiance (W/m2);and (c) the sky's brightness, D, given by:

D � mDh=Gon �13�where

m is the air mass (dimensionless); andGon is the extraterrestrial irradiance at normal inci-dence (W/m2).

The Perez model considers a set of categories orbins for E, and each of them can be expressed as linearfunctions of Z and D as follows:

F1 � maxf0, � f11 � f12D� f13Z�p=180��g �14�

F2 � � f21 � f22D� f23Z�p=180�� 15�The required coe�cients fij are obtained from Perez etal. [8].

5. Evaluation of the models

The evaluation was carried out for both the solarirradiance and the outdoor illuminance data. The glo-bal component on a vertical surface was determinedfrom the two-year (1996±1997) measured horizontaldata using the three models (Klucher, Hay and Perez)and compared with the measured vertical data of thesame period. The ground re¯ectance was estimated tobe 0.22. The comparison was conducted on an hourlybasis. The performance of each model was assessedstatistically in terms of the mean-bias-error (MBE), ameasure of the underlying trend, and the root-mean-square-error (RMSE), an indication of data scattering.In order to examine the seasonal variations, MBE andRMSE were computed for each month as well as thewhole year.

5.1. Vertical solar irradiance

Tables 1 to 4 show the MBE and RMSE results ofeach model for all 12 months and the whole year atthe four cardinal orientations. It can be observed from


the Tables that both the lowest and highest RMSE

occur in the south-facing surface for Hay's model, ran-

ging from 21.3 W/m2 in March to 52.7 W/m2 in

December; Perez's model, ranging from 17.5 W/m2 in

May to 65 W/m2 in December. Klucher's model ranges

from 25.1 W/m2 in February for the south-facing sur-

face to 74.3 W/m2 in October for the north-facing sur-

face. For each model, the RMSE values in W/m2 for

all vertical surfaces do not di�er a great deal. How-

ever, in terms of percentage, north-facing surface has

the highest RMSE values and this can be explained by

the fact that solar irradiance received by the north-

facing surface is basically di�use component of low

measured mean value.

Klucher's model performs apparently the worst

among the three models, particularly for the north-

facing surface, even though the anisotropic e�ect has

been considered. For completely overcast skies, Klu-

cher's model is reduced to isotropic and the vertical

irradiance is half of the corresponding horizontal com-

ponent. For non-overcast skies, the predicted solar

irradiance values of Klucher's model are always

greater than those of the other two models. Its predic-

tive ability is, thus, comparable to that of the other

two models with general accurate results only in winter

Table 1

Summary of MBE and RMSE for vertical global solar irradiance (north-facing surface)

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Yr

Klucher

MBE (W/m2) 35.4 28.5 43.2 47.4 37.9 39.4 41.4 47.9 51.9 57.9 42.4 41.2 42.6

%MBE 37.0 37.2 49.8 47.4 38.7 33.6 36.2 44.8 47.4 58.8 44.8 45.4 42.8

RMSE (W/m2) 46.4 40.5 58.5 63.1 54.3 55.6 59.8 66.6 66.0 74.3 54.7 50.4 58.2

%RMSE 48.6 52.8 67.5 63.1 55.4 47.3 52.2 62.3 60.3 75.5 57.9 55.6 58.5

Hay

MBE (W/m2) 3.1 10.1 15.8 24.3 17.0 14.8 14.9 14.8 17.1 20.9 6.7 11.8 14.5

%MBE 3.3 13.2 18.2 24.3 17.4 12.6 13.0 13.9 15.6 21.3 7.1 13.0 14.6

RMSE (W/m2) 27.1 26.2 27.1 37.7 32.7 32.8 33.5 33.4 31.5 37.9 30.7 30.0 32.1

%RMSE 28.3 34.1 31.3 37.7 33.4 27.9 29.3 31.3 28.8 38.5 32.5 33.0 32.3

Perez

MBE (W/m2) 11.0 9.8 15.8 17.0 12.5 14.7 13.8 12.2 14.9 24.9 16.5 17.6 15.0

%MBE 11.5 12.7 18.3 16.9 12.8 12.5 12.1 11.4 13.6 25.3 17.4 19.4 15.0

RMSE (W/m2) 22.3 18.8 23.6 29.2 26.3 31.6 30.7 25.1 23.6 36.5 27.3 26.2 27.4

%RMSE 23.2 24.5 27.2 29.0 26.9 26.9 26.8 23.5 21.6 37.1 28.8 28.9 27.6

Table 2

Summary of MBE and RMSE for vertical global solar irradiance (east-facing surface)


Klucher

MBE (W/m2) 15.8 16.8 28.7 38.2 33.3 38.8 41.6 40.2 37.1 33.6 19.5 17.2 30.9

%MBE 9.1 13.8 21.8 29.0 27.3 26.8 27.7 24.3 19.6 18.1 10.2 10.2 20.1

RMSE (W/m2) 39.8 35.4 52.9 66.5 57.6 60.6 64.2 66.0 56.0 59.3 47.4 44.2 55.9

%RMSE 23.0 28.9 40.2 50.5 47.1 41.7 42.8 39.8 29.7 31.9 24.9 26.2 36.4

Hay

MBE (W/m2) ÿ7.4 3.4 5.8 19.3 14.9 17.1 18.9 14.1 11.6 5.6 ÿ5.6 ÿ4.1 8.8

%MBE ÿ4.3 2.7 4.4 14.7 12.2 11.8 12.6 8.5 6.1 3.0 ÿ3.0 ÿ2.4 5.7

RMSE (W/m2) 37.9 32.3 35.7 47.0 36.7 41.2 39.0 38.6 32.4 38.6 39.6 41.8 38.7

%RMSE 21.9 26.5 27.1 35.6 30.1 28.4 26.0 23.3 17.2 20.8 20.8 24.7 25.2

Perez

MBE (W/m2) 5.3 6.8 11.9 19.2 15.3 22.3 24.5 20.6 20.1 18.0 10.9 6.3 15.7

%MBE 3.0 5.6 9.1 14.6 12.6 15.4 16.4 12.4 10.7 9.7 5.7 3.7 10.2

RMSE (W/m2) 28.6 24.5 31.2 43.1 32.5 43.6 43.6 39.4 32.7 37.4 35.3 34.5 36.5

%RMSE 16.5 20.1 23.7 32.7 26.6 30.0 29.1 23.8 17.3 20.1 18.6 20.4 23.8


months on the east, south and west vertical planes

when facing the direct sun at low solar altitude. It is

signi®cantly inferior for the rest of the year. For

shaded vertical surfaces, particularly during non-over-

cast conditions, the anisotropic correction factors in

Klucher's model always tend to over-estimate the solar

irradiance, causing the MBE and RMSE values to be

higher than those of the other two models.

Hay's model performs better in vertical solar irradi-

ance estimation than Klucher's model, particularly for

the south- and west-facing surfaces. Hay's model can

predict the vertical solar irradiance from less than half

of horizontal di�use irradiance for shaded surfaces to

over half of horizontal solar irradiance for sunlit sur-

faces under clear sky. The MBE values range from an

under-estimation of 4.3% in January for the east-

facing surface to an over-estimation of 24.3% in April

for the north-facing surface. For the whole year, the

RMSE ranges from 16.5% for the south-facing surface

to 32.3% for the north-facing surface.

Perez's model tends to over-estimate the solar irradi-

ance for all vertical surfaces with the annual average

MBE values not more than 15%. Overall, Perez's

model most accurately predicts the vertical irradiance,

particularly for the north-facing surface when the solar

irradiance received consists of di�use component only

Table 3

Summary of MBE and RMSE for vertical global solar irradiance (south-facing surface)


Klucher

MBE (W/m2) 0.9 9.6 19.0 33.1 34.4 40.1 38.6 37.5 31.8 8.7 ÿ4.0 1.2 22.7

%MBE 0.3 5.5 12.7 26.7 34.6 42.4 38.0 27.8 18.7 3.3 ÿ1.2 0.4 12.6

RMSE (W/m2) 31.8 25.1 30.8 46.6 49.3 56.4 53.2 54.3 43.0 27.5 33.3 51.0 43.8

%RMSE 10.3 14.4 20.6 37.7 49.5 59.7 52.3 40.3 25.2 10.5 9.8 16.4 24.2

Hay

MBE (W/m2) ÿ3.5 4.3 7.1 13.6 12.2 14.2 8.8 7.6 9.5 ÿ6.7 ÿ10.3 ÿ3.1 5.0

%MBE ÿ1.1 2.5 4.7 11.0 12.3 15.1 8.6 5.7 5.6 ÿ2.6 ÿ3.0 ÿ1.0 2.8

RMSE (W/m2) 33.1 23.2 21.3 27.5 27.7 29.2 25.1 28.7 25.5 26.0 34.4 52.7 29.8

%RMSE 10.7 13.3 14.3 22.3 27.8 30.9 24.7 21.3 14.9 9.9 10.1 17.0 16.5

Perez

MBE (W/m2) 19.2 11.6 13.2 10.3 7.6 9.1 6.6 8.7 14.8 10.8 15.0 15.7 11.4

%MBE 6.2 6.7 8.8 8.3 7.6 9.6 6.5 6.5 8.7 4.1 4.4 5.1 6.3

RMSE (W/m2) 47.4 30.2 28.8 21.1 17.5 19.3 18.1 22.0 26.8 33.6 44.4 65.0 32.3

%RMSE 15.3 17.4 19.2 17.0 17.6 20.4 17.8 16.3 15.7 12.9 13.1 21.0 17.9

Table 4

Summary of MBE and RMSE for vertical global solar irradiance (west-facing surface)


Klucher

MBE (W/m2) 25.1 20.3 26.9 33.1 27.0 32.7 26.4 34.0 36.3 31.8 25.2 26.0 29.0

%MBE 14.9 15.9 19.4 22.6 18.8 23.2 16.3 20.1 22.3 16.7 12.7 15.6 18.2

RMSE (W/m2) 36.1 32.8 45.9 52.5 54.8 54.6 52.5 59.9 59.6 58.5 42.9 41.5 51.1

%RMSE 21.4 25.7 33.1 35.9 38.0 38.8 32.3 35.4 36.6 30.8 21.7 25.0 32.0

Hay

MBE (W/m2) 4.2 8.2 11.6 16.3 10.7 13.0 4.0 7.4 10.6 5.2 1.4 5.2 8.2

%MBE 2.5 6.4 8.3 11.2 7.4 9.2 2.5 4.3 6.5 2.7 0.7 3.1 5.2

RMSE (W/m2) 32.4 28.0 28.8 31.9 38.3 31.1 30.8 32.4 35.7 38.7 33.0 34.5 33.3

%RMSE 19.2 21.8 20.7 21.8 26.6 22.0 18.9 19.1 21.9 20.4 16.7 20.8 20.9

Perez

MBE (W/m2) 15.8 11.0 13.6 13.4 9.7 11.3 6.1 9.3 10.4 13.1 17.1 14.5 11.7

%MBE 9.4 8.6 9.8 9.2 6.7 8.0 3.8 5.5 6.4 6.9 8.7 8.7 7.4

RMSE (W/m2) 43.9 35.1 37.7 32.2 39.9 33.9 32.8 34.6 37.0 46.4 45.1 44.0 38.5

%RMSE 26.0 27.4 27.1 22.0 27.7 24.1 20.2 20.4 22.7 24.4 22.9 26.4 24.1


or includes only a small portion of direct componentwhen the solar altitude is high. This indicates that Per-ez's model is good for estimating solar irradiance forshaded surfaces. For the north- and east-facing sur-faces, the performance of Perez's model is better thanKlucher's and Hay's models. For other vertical sur-faces, its predictive ability is close to Hay's model. Theannual average RMSE values for all vertical surfacesare less than 28%.

5.2. Vertical outdoor illuminance

In view of the fact that solar irradiance and outdoorilluminance have similar characteristics in nature,

models developed for solar irradiance prediction on a

vertical surface can also be employed in daylight calcu-

lations [22]. Similar to the method of predicting verti-

cal solar irradiance, the vertical outdoor illuminance

for the four principle orientations were determined by

the same inclined models (i.e. Klucher, Hay and Perez)

and compared with the measured data. The MBE and

RMSE for each month and the whole year are sum-

marised in Tables 5 to 8. The three models have trends

similar to what have been observed in the irradiance

data. Again, the statistical results show that in most

cases, Perez's model predicts the daylight level the

best, followed by Hay's then, Klucher's models. All

three models provide good results for the south-facing

Table 5

Summary of MBE and RMSE for vertical global outdoor illuminance (north-facing surface)


Klucher

MBE (klux) 3.93 3.43 4.36 4.99 4.77 4.62 5.41 5.13 5.46 7.01 5.81 6.30 5.18

%MBE 42.1 45.0 46.3 48.3 45.0 36.7 43.6 41.5 38.2 63.4 56.4 65.5 47.1

RMSE (klux) 4.90 4.70 6.04 6.85 6.62 6.30 7.28 7.35 7.80 8.95 7.43 7.56 7.01

%RMSE 52.6 61.8 64.1 66.3 62.5 50.1 58.6 59.5 54.5 80.9 72.2 78.5 63.8

Hay

MBE (klux) 0.21 1.48 1.68 2.61 2.37 1.74 2.17 1.25 1.28 2.80 1.66 2.54 1.93

%MBE 2.3 19.5 17.9 25.2 22.4 13.9 17.5 10.1 8.9 25.3 16.1 26.5 17.5

RMSE (klux) 2.31 2.82 2.98 4.08 4.11 3.58 3.79 4.48 4.31 4.60 4.22 4.18 3.93

%RMSE 24.8 37.0 31.6 39.5 38.9 28.4 30.5 36.2 30.1 41.6 41.0 43.4 35.8

Perez

MBE (klux) 1.38 1.40 0.84 1.03 0.97 1.03 0.98 ÿ0.09 ÿ1.20 1.82 1.60 2.23 1.01

%MBE 14.8 18.4 8.9 9.9 9.2 8.2 7.9 ÿ0.8 ÿ8.4 16.5 15.6 23.2 9.2

RMSE (klux) 2.14 2.22 2.21 2.64 2.66 2.87 3.19 3.35 3.13 3.10 2.55 2.87 2.82

%RMSE 22.9 29.2 23.5 25.4 25.2 22.8 25.7 27.2 21.9 28.0 24.8 29.9 25.6

Table 6

Summary of MBE and RMSE for vertical global outdoor illuminance (east-facing surface)


Klucher

MBE (klux) 1.44 2.10 3.22 4.63 4.48 4.41 5.33 4.42 2.90 4.46 3.66 4.95 4.10

%MBE 7.8 17.1 24.9 35.7 34.7 28.6 33.1 24.1 13.1 23.3 19.1 30.5 25.5

RMSE (klux) 4.13 3.94 5.35 7.55 7.31 6.98 8.09 7.36 8.47 7.54 6.54 7.36 7.02

%RMSE 22.2 32.0 41.4 58.3 56.6 45.4 50.3 40.1 38.2 39.4 34.2 45.4 43.7

Hay

MBE (klux) ÿ1.27 0.67 0.95 2.66 2.43 1.87 2.65 1.50 ÿ0.28 1.30 0.75 2.30 1.56

%MBE ÿ6.8 5.5 7.4 20.6 18.8 12.2 16.5 8.2 ÿ1.3 6.8 3.9 14.2 9.7

RMSE (klux) 4.01 3.20 3.30 5.02 4.65 4.52 4.89 4.72 6.52 4.81 4.70 5.68 4.73

%RMSE 21.5 26.1 25.6 38.7 36.0 29.4 30.4 25.7 29.4 25.2 24.5 35.0 29.4

Perez

MBE (klux) 0.47 1.03 0.92 2.11 1.83 2.03 2.43 1.50 ÿ0.73 1.76 1.81 2.68 1.65

%MBE 2.5 8.4 7.1 16.3 14.2 13.2 15.1 8.2 ÿ3.3 9.2 9.5 16.5 10.3

RMSE (klux) 2.95 2.57 2.84 4.52 3.87 4.74 5.21 4.71 5.11 4.34 4.28 5.60 4.40

%RMSE 15.8 20.9 22.0 34.9 30.0 30.8 32.4 25.7 23.1 22.7 22.4 34.6 27.4


surface but the daylight level is not estimated accu-

rately for the north-facing surface. As expected, except

for Perez's model for the north-facing surface, the%RMSE is generally higher than that for vertical irra-

diance since the models developed are initially for

solar irradiance purpose.

The three models display a consistent over-predic-

tion for the north-facing surface for almost all months

(except in August and September for Perez's model),

indicating that shaded vertical surfaces receive less

than half of the corresponding horizontal daylight.

Many studies have also found that the ratio of vertical

to horizontal di�use illuminance for shaded surfaces is

about 0.4 [22,23]. It means that the measured horizon-

tal and vertical illuminance data in 1996 and 1997 are

in good agreement with this ®nding. For the south-

facing surface, all three models are in close agreement

throughout the year showing the importance of aniso-

tropy on sunlit surfaces. Their di�erences in RMSE

for the whole year are around 6%. The statistical

errors for the east-facing surface are much larger than

those for the west-facing surface indicating the aniso-

tropic e�ect of di�use component under clear sky may

occur more frequently in the afternoon than in the

morning. For the annual RMSE, Perez's model ranges

from 19.1% to 27.4%, Hay's model from 18.2% to

35.8%, and Klucher's model from 24.3% to 63.8%.

Table 7

Summary of MBE and RMSE for vertical global outdoor illuminance (south-facing surface)


Klucher

MBE (klux) 0.7 1.57 1.44 3.47 3.67 3.36 3.79 3.56 2.36 0.51 ÿ0.48 0.67 2.27

%MBE 2.1 9.3 8.3 26.7 32.1 28.5 31.1 22.8 12.0 1.8 ÿ1.4 2.0 11.7

RMSE (klux) 2.46 2.63 2.89 5.30 5.34 5.34 5.59 5.36 4.80 3.34 3.65 5.09 4.69

%RMSE 7.3 15.6 16.7 40.7 46.8 45.3 45.8 34.2 24.4 11.9 15.5 15.5 24.3

Hay

MBE (klux) 0.26 0.92 0.10 1.41 1.13 0.05 0.30 0.09 ÿ0.07 ÿ1.10 ÿ1.08 0.41 0.18

%MBE 0.8 5.4 0.6 10.9 9.9 0.4 2.4 0.6 ÿ0.4 ÿ3.9 ÿ3.0 1.2 0.9

RMSE (klux) 2.36 2.04 2.26 2.86 3.11 4.11 3.41 3.43 4.04 3.60 3.81 5.32 3.53

%RMSE 7.0 12.1 13.0 22.0 27.2 34.8 27.9 21.9 20.6 12.9 10.7 16.2 18.2

Perez

MBE (klux) 3.14 1.61 0.59 0.47 ÿ0.26 ÿ1.04 ÿ1.08 ÿ0.67 ÿ0.10 0.48 1.62 2.30 0.30

%MBE 9.3 9.6 3.4 3.6 ÿ2.3 ÿ8.8 ÿ8.8 ÿ4.3 ÿ0.5 1.7 4.6 7.0 1.5

RMSE (klux) 4.98 3.44 2.67 2.32 2.53 3.63 3.51 2.80 3.78 3.70 4.56 6.55 3.72

%RMSE 14.8 20.4 15.4 17.8 22.1 30.8 28.7 17.9 19.2 13.2 12.8 19.9 19.1

Table 8

Summary of MBE and RMSE for vertical global outdoor illuminance (west-facing surface)


Klucher

MBE (klux) 3.17 2.69 1.62 2.84 3.20 3.92 3.59 3.69 3.57 3.10 2.53 3.44 3.16

%MBE 17.8 20.6 9.6 18.2 20.7 23.1 21.2 19.,8 19.6 14.9 11.7 19.0 17.9

RMSE (klux) 4.06 4.02 4.60 5.20 5.84 6.29 6.38 6.82 6.23 6.77 5.03 5.02 5.79

%RMSE 22.8 30.8 27.1 33.3 37.7 37.1 37.7 36.7 34.3 32.7 23.2 27.6 32.8

Hay

MBE (klux) 0.75 1.42 ÿ0.07 1.09 1.36 1.68 1.01 0.65 0.82 0.10 ÿ0.21 0.84 0.79

%MBE 4.2 10.9 ÿ0.4 7.0 8.8 9.9 6.0 3.5 4.5 0.5 ÿ1.0 4.6 4.5

RMSE (klux) 3.34 3.02 3.93 3.38 3.70 4.18 3.59 4.22 4.18 5.14 4.35 4.23 4.04

%RMSE 18.8 23.1 23.2 21.7 23.9 24.6 21.2 22.7 23.0 24.8 20.1 23.3 22.9

Perez

MBE (klux) 2.46 1.71 ÿ0.03 0.34 0.83 1.25 0.83 0.37 0.54 0.33 0.70 1.07 0.75

%MBE 13.8 13.1 ÿ0.2 2.2 5.4 7.3 4.9 2.0 2.9 1.6 3.2 5.9 4.2

RMSE (klux) 5.08 4.27 4.37 3.40 3.92 4.60 3.98 4.13 4.53 4.66 4.34 4.28 4.27

%RMSE 28.5 32.7 25.7 21.8 25.4 27.0 23.5 22.3 24.8 22.5 20.0 23.6 24.2


6. Conclusions

An evaluation of the predicted vertical solar irradi-ance and daylight illuminance based on three inclinedsurface models (Klucher, Hay and Perez) and two-year(1996±1997) measured horizontal solar data in HongKong has been conducted. It has been observed thatrelatively high MBE and RMSE values are found forthe north-facing surface which only receives basicallydi�use components; and the vertical components pre-dicted for south-facing surface are the most accuratewhen the solar irradiance and daylight illuminancereceived have a high portion of direct components.There is a consistent trend of over-estimation by Klu-cher's model, particularly when the solar irradianceand daylight illuminance received, consists of mainlydi�use components. Hay's model performs better thanKlucher's model and gives the most accurate predic-tion for the south- and the west-facing surfaces. TheRMSE values for the whole year range from 16.5 to32.3% for solar irradiance prediction and 18.2 to35.8% for daylight illuminance estimation. Perez'smodel tends to have good prediction for shaded verti-cal surfaces and surfaces facing the sun at high solaraltitude. The annual RMSE values for solar irradianceand daylight illuminance are both less than 28%. Ingeneral, Perez's model produces the best agreementwith the measured vertical data for both solar irradi-ance and daylight illuminance.

Acknowledgements

This work was funded by an RGC Competitive Ear-marked Research Grants (Project No.9040139).

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evaluation of slope irradiance and illuminance models against measured hong kong data

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