energia solar

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Solar Energy 86 (2012) 1920–1928

Study of optimum tilt angles for solar panels in different latitudesfor urban applications

Arbi Gharakhani Siraki ⇑, Pragasen Pillay

P. D. Ziogas Power Electronics Laboratory, Department of Electrical and Computer Engineering, Concordia University, 1455 de Maisonneuve Blvd.,

W. Montreal, Quebec, Canada H3G 1M8

Received 12 August 2011; received in revised form 2 February 2012; accepted 25 February 2012Available online 3 April 2012

Communicated by: Associate Editor Frank Vignola

Abstract

Solar panels are one of the most promising renewable technologies for energizing future buildings. For roof top solar panel installa-tions, knowledge of the optimum tilt angle is important to have the maximum annual or seasonal energy yield. The annual optimum tiltangle is dependent on many factors such as the latitude of the location and the weather condition. In an urban application, the optimumtilt angle can be affected by the surrounding obstacles. Consequently, new concerns such as shading or sky blocking effects have to betaken into consideration. In this paper, a simple method is proposed based on a modified sky model to calculate the optimum angle ofinstallation for the urban applications. The obtained results demonstrate the dependency of the optimum angle of installation on thelatitude, weather condition and surroundings.� 2012 Elsevier Ltd. All rights reserved.

Keywords: Urban buildings; Solar energy; Orientation of photovoltaic panels

1. Introduction

Environmental concerns as well as increasing demandfor cleaner energy are strong motives for further invest-ment and research in renewable resources. Urban environ-ments due to their high density of energy consumption areconsidered to be one of the most promising locations forinstallation of renewable energy technologies. Among dif-ferent types of available technologies, solar panels showpromise for building integrated applications. For solarinstallations, the optimum tilt angle is an important datafor each location which affects the annual energy yield ofthe whole system. The optimum tilt angle is influenced bydifferent factors such as the latitude of the location, clear-

0038-092X/$ - see front matter � 2012 Elsevier Ltd. All rights reserved.

http://dx.doi.org/10.1016/j.solener.2012.02.030

⇑ Corresponding author. Tel.: +1 514 848 2424x7089; fax: +1 514 8482802.

E-mail addresses: [email protected] (A. Gharakhani Siraki),[email protected] (P. Pillay).

ness index, air pollution and distribution of the sunny dayswhich represent the climate condition.

Numerous studies have been done to find out a correla-tion between the annual optimum tilt angle of installationand latitude of the location for different places aroundthe world. Yang and Lu (2000) employed the anisotropicsky model and found that the yearly optimum tilt anglein Hong Kong (with latitude of U = 22.5�) is around 20�(U � 2.5�). Chen et al. (2005) used a genetic algorithmsearching technique to find the optimum tilt angle for Chi-ayi, Taiwan (with latitude of U = 23.5�). They havereported 20� (U � 3.5�) as the optimum angle for that loca-tion. Hussein et al. (2004) utilized a specific simulation pro-gram and found that the optimum angle of installation inCairo, Egypt (U = 30�) can be any angle between 20� and30� with a small change in the outcome. Calabroa (2009)used a simulation tool to study the optimum tilt angle atdifferent latitudes starting from 36� to 46�. They found thatthe annual optimum tilt angle is almost shifted by 10� with


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Calculation of hourly total radiation on a horizontal surface

for monthly average days

Calculation of beam and diffuse components for each hours of

monthly average days on a horizontal surface

Calculation of total insolation level for each hours of monthly average days on a tilted surface

Calculation of the annual insolation level on a tilted surface

Comparison of the annual insolation levels of each tilt angle

to find the optimum one

Take monthly average daily insolation levels on a horizontal surface for the desired location

Fig. 1. Flowchart of the annual optimum tilt angle calculation algorithm.

A. Gharakhani Siraki, P. Pillay / Solar Energy 86 (2012) 1920–1928 1921

respect to the latitude of the location (U � 10�). Chenga etal. (2009) utilized a software package and they have shownthat for places located below the tropic of cancer the opti-mum annual tilt angle is almost equal to the latitude of thelocation. However, for higher latitudes, this angle is foundto be smaller than the location’s latitude and the discrep-ancy is increased by increase in latitude.

All these research studies proposed different correlationsto relate the optimum angle of installation to the latitude ofthe location. However, they all suggest that, for small val-ues of latitude, the annual optimum tilt angle is close to thelocation’s latitude, while for higher ones the optimum tiltangle is smaller.

In the first part of this paper, the cause of this phenom-enon is explored by a simulation based method. In addi-tion, the dependency of the annual optimum angle ofinstallation on clearness index and its annual distributionis also presented.

All the studies found in the literature (Yang and Lu,2000; Chen et al., 2005; Hussein et al., 2004; Calabroa,2009; Chenga et al., 2009; Kacira et al., 2004; Chow andChan, 2004; Chang, 2009, 2008; Tang and Wu, 2004) dealtwith the problem of optimum tilt angle selection withoutconsidering the specific concerns of urban applications.Therefore, a simple model capable of dealing with all theseconcerns is required in order to find out the optimum tiltangle of an urban installation. This is the motive for thesecond part of this paper in which a modification has beenproposed to the HDKR (the Hay, Davies, Klucher, Reindlmodel) anisotropic sky model (Duffie and Beckman, 2006)in order to include the effects of the surrounding obstacles(such as adjacent buildings). Later, this modified model hasbeen used in a typical urban application case to illustratedependency of the optimum tilt angle on the surroundingsituation. The paper is arranged as follows: The methodol-ogy is presented in Section 2. In detail; in Section 2.1 basicsolar equations as well as the HDKR sky model are brieflyreviewed. Then in Section 2.2, a simple modification is pro-posed for HDKR model to make it compatible with urbanapplications. A simple neighborhood surveying approach ispresented in Section 2.3. In Section 3.1 dependency of theoptimum tilt angle to the location’s latitude and theweather condition are explored. Finally in Section 3.2,the modified HDKR model is used to show the effect ofthe surrounding obstacles on the optimum tilt angle. Basedon the achieved results, a number of conclusions are madein Section 4 of the paper.

2. Methodology

In order to find the optimum angle of installation fordifferent latitudes and to investigate the sensitivity of thisangle to each type of radiation as well as the other param-eters, a method with a flowchart shown in Fig. 1 is pro-posed and a program is developed based on theequations from reference (Duffie and Beckman, 2006).Since the readily available data for many locations are

monthly average daily insolation levels on horizontal sur-faces, it is considered to be the input of the developed tool.If data of observed daily sequences, or reference meteoro-logical years are available, they should be used to have abetter accuracy in optimum tilt angle estimation.

In this tool, correlation of Collares Pereira and Rabl(Duffie and Beckman, 2006) is utilized to calculate thehourly insolation values from the average daily insolationlevels on horizontal surfaces. Then correlation proposedby Erbs is used to separate the hourly diffuse radiationfrom the beam component on a horizontal surface. Finally,the HDKR anisotropic sky model (Duffie and Beckman,2006) is employed to estimate the insolation level receivedon a tilted surface for different latitudes and for any desiredinstallation angle. Later in this paper, few simple modifica-tions to this model are proposed and applied in order tomake it compatible with urban applications.

2.1. Basic equations

Following correlation from Duffie and Beckman (2006)is used to find out the hourly insolation values (Ih) (for eachhour of the average day of the month) from the daily inso-lation levels (on horizontal surfaces) (Hh) as shown by:

Ih ¼p24ðaþ b cos xÞ cos x� cos xs

sin xs � xs cos xsH h

a ¼ 0:409þ 0:5016 sin xs �p3

� �b ¼ 0:6609� 0:4767 sin xs �

p3

� � ð1Þ

1922 A. Gharakhani Siraki, P. Pillay / Solar Energy 86 (2012) 1920–1928

in which x is the hour angle of the midpoint for each 1 hperiod (in an average day of each month) and xs is the sun-set hour angle in radians. Eq. (2) gives the sunset hour an-gle of each day based on the declination angle (d) and thelatitude of the location (U).

xs ¼ cos�1ð� tan d � tan /Þ ð2ÞThe extraterrestrial radiation (Iex) on a horizontal surfacecan be calculated by:

Iex ¼ 1:367 1þ 0:033 � cos360n365

� �� ðcos u cos d cos x

þ sin / sin dÞ ð3Þin which n is the number of the day, starting from 1 for thefirst day of January. Clearness index (k) can be calculatedfor each hour of the average day of the month by:

k ¼ Ih

Iexð4Þ

Based on the value of clearness index and the Erbs’s corre-lation (Duffie and Beckman, 2006), it is possible to separatehourly diffuse radiation (Ih,d) from the beam radiation(Ih,b), incident to a horizontal surface as shown by:

Ih;d

Ih¼

1:0� 0:09k for k 6 0:22

0:9511� 0:1604k þ 4:388k2 for 0:22 < k 6 0:8

�16:638k3 þ 12:336k4

0:165 for k > 0:8

8>>><>>>:

ð5ÞKnowing the amount of hourly diffuse and beam radiationson a horizontal plane, it is possible to find out the hourlytotal insolation levels on a tilted plane (IT) using theHDKR model. This model accounts for different type ofradiation including beam (IT,b), diffuse (IT,d) (whichcontains parts of the circumsolar diffuse (IT,cs), isotropicdiffuse (IT,iso) and horizontal brightening (IT,hz) compo-nents) and ground reflectance (IT,ref) as shown by:

IT ¼ IT ;b þ IT ;d þ IT ;refl ð6ÞBeam radiation on a tilted surface can be calculated by:

IT ;b ¼ Rb � ðIh � Ih;dÞ ð7ÞThe value of the coefficient Rb for the northern hemisphereis found from Eq. (8) in which b is the tilt angle of the solarpanel.

Rb ¼cosð/� bÞ cos d cos xþ sinð/� bÞ sin d

cos u cos d cos xþ sin / sin dð8Þ

Circumsolar diffuse (IT,cs), isotropic diffuse (IT,iso) and hor-izontal brightening (IT,hz) components of the diffuse radia-tion can be found considering the anisotropy index asA = Ih,b/Iex and by (Duffie and Beckman, 2006):

IT ;cs ¼ A �Rb � Ih;d ¼ðIh;bÞ

Iex�Rb � Ih;d ¼

ðIh� Ih;dÞIex

�Rb � Ih;d ð9Þ

IT ;iso ¼ ð1�AÞF sky � Ih;d ¼ 1� Ih;b

Iex

� �� F sky � Ih;d

¼ Iex� Ih þ Ih;d

Iex

� �� F sky � Ih;d ð10Þ

IT ;Hz ¼ffiffiffiffiffiffiffiIh;b

Ih

r� ð1 � AÞ � F sky sin3 b

2

� �Ih;d

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 � Ih;d

Ih

s� Iex � Ih þ Ih;d

Iex

� �� F sky sin3 b

2

� �Ih;d ð11Þ

As it can be seen, the anisotropy index which is function ofthe transmittance of the atmosphere for beam radiation de-fines the portion of the contribution of each component ofthe diffuse radiation. In these equations, view factor to thesky is defined by:

F sky ¼1þ cos b

2

� �ð12Þ

The ground reflectance radiation (IT,ref) accounts for the allexisting reflectance from the surrounding environment andit is defined by (Duffie and Beckman, 2006):

IT ;ref ¼ qg �1� cos b

2

� �� Ih ð13Þ

The ground reflectance ratio (qg) changes between 0.2 and0.7 based on the surrounding situation (Kacira et al., 2004).

2.2. Proposed modifications to the HDKR model

for urban application

The HDKR model is an accurate sky model for normalsolar panel applications. However, for urban purposes, it isrequired to apply some modifications to account for theeffects of the surrounding obstacles (surrounding buildingsor trees). There are three different effects caused by theadjacent obstacles. The first one is the shading effect whichoccurs in the hours when the sun is trapped behind anobstacle with respect to the solar panel. As there is nodirect view between the sun and the solar panel, theamount of the beam radiation and the circumsolar radia-tion received by the panel will be equal to zero for this spe-cific period of the time. Based on that, the followingmodifications were applied to the Eq. (7) as well as Eq. (9).

IT ;cs ¼ ð1� KshÞ �ðIh � Ih;dÞ

Iex� Rb � Ih;d ð14Þ

IT ;b ¼ ð1� KshÞ � Rb � ðIh � Ih;dÞ ð15Þ

The shading coefficient (Ksh) is defined as the portion of thecalculation time step in which panel is under full shade. Forinstance for a 1 h calculation time step, 20 min shading willlead to a shading coefficient equal to 0.33 for that specifichour. This coefficient can be estimated for different hoursof an average day (for each month of the year) based onthe outlines of the obstacles plotted in the solar positionplane as is shown in Section 2.3.

The second effect appears as the sky blocking phenome-non. The view to some portions of the sky is blocked due toexistence of the obstacles around the panel. Consequently,isotropic sky and horizontal brightening emitted radiationsfrom the blocked portions of the sky cannot reach the sur-face of the panel. Therefore, it is necessary to modify the

Fig. 3. Locations of the sky trapped behind a tilted panel for different tiltangles.


sky view factor defined by Eq. (12) in order to consider thiseffect. A new sky view factor is proposed by:

F sky ¼ 1� 1

p2

Xn

i¼1

Ai ð16Þ

Ai can be either an area of the sky blocked by an obstaclefound by projection of the obstacle onto the sky as shownin Section 2.3 or an area of the sky blocked behind thetilted panel. In order to find out the portion of the skyblocked behind the tilted panel, an equation is developedbased on the trigonometric relationship that is shown inFig. 2.

In this figure a and c represent the altitude and azimuthangles respectively. Based on Fig. 2, it is possible to findout the locations of the sky (with their altitude and azimuthangles) that are located behind a tilted panel.

tan a ¼ L sin b

� L cos bcos c

� � ð17Þ

a ¼ tan�1ð� tan b � cos cÞ ð18ÞBy using Eq. (18), a group of curves have been plotted fordifferent tilt angles to show the locations of the sky trappedbehind a tilted solar panel. For instance a point of the skywith azimuth angle of c = 150� and altitude angle ofa = 25� shown in Fig. 3 as “P” will be trapped behindthe PV panel that is tilted at an angle of b = 30�.

To be sure that the proposed view factor behaves likethe conventional one in case of no obstacles, a comparisonhas been made in Table 1. Thus, the sky view factors of atilted panel were calculated once with the proposed methodbased on Eq. (16), and another time with the conventionalone using Eq. (12) assuming no obstacle around the panel.In this case Ai in Eq. (16) is the area of the sky blockedbehind the tilted panel.

As it can be seen from Table 1, when there are no obsta-cles around the solar panel, the results of the proposed skyview factor Eq. (16) are very close to those obtained fromEq. (12). This proves the performance of the proposed skyview factor is the same as the sky view factor of the wellaccepted HDKR model in case of no obstacles.

0

90

-90

180

L N

S

W

E

Fig. 2. Trigonometric relationship to find out the portion of sky trappedbehind the panel.

Combination of these curves, solar position curves andoutlines of the obstacles will aid the designer to find outthe required coefficients introduced in the modified skymodel. Later, a typical example is discussed.

Finally, the last effect comes as a result of the reflectanceof adjacent obstacles. Since ground reflectance radiationaccounts for all the reflectance from surrounding environ-ment, it is sensible to increase the ground reflectance ratiobased on the material and reflectance of the obstacles suchas buildings located in front of the panel. The effect ofchanging of the ground reflectance ratio on annual opti-mum tilt angle is discussed in Section 3 of this paper.

2.3. A simple neighborhood surveying approach to

find Ksh and Fsky

A typical neighborhood as shown in Fig. 4 is used todemonstrate the simple approach employed to find theKsh and Fsky values.

In order to find the shading coefficient (Ksh) for an hourof a day, the projection of the surrounding obstacles (build-ings) should be found on the sun path diagram. This can bedone by picking up few important points (like corners) onthe obstacles and calculating their projection in cylindricalcoordinates onto the sun path diagram. This has been donefor two typical points (demonstrated as “a” and “b” inFig. 4) as shown below:Point a:

ca ¼ tan�1 30

40

� �¼ �36:86� ð19Þ

aa ¼ tan�1 40ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi402 þ 302

p !

¼ 38:66� ð20Þ

Point b:

cb ¼ 90þ tan�1 20

40

� �¼ 116:56� ð21Þ

ab ¼ tan�1 20ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi402 þ 202

p !

¼ 24:09� ð22Þ

Table 1Comparison of two sky view factors (Fsky) assuming no obstacle aroundthe panel.

Installationslope

Proposed Eq.(16)

Conventional Eq.(12)

Percentage ofdiscrepancy

0 1.000 1.000 0.0010 0.964 0.992 �2.8220 0.928 0.970 �4.3330 0.890 0.933 �4.6040 0.850 0.883 �3.7450 0.806 0.821 �1.8360 0.756 0.750 0.7670 0.697 0.671 3.8780 0.622 0.587 5.9690 0.500 0.500 0.00

Fig. 5. Sun path diagram for U = 45� as well as outline of the surroundingbuildings for the typical example of Fig. 4.


Using this technique, the projections of the obstacles of theFig. 4 are drawn on the sun path diagram as shown inFig. 5. The sections of the sun path diagram located insidethe territory of each obstacle represent the time when thesun is trapped behind the obstacles. The shading coefficientat a specific hour of a day can be calculated by:

Ksh ¼T sh

60 minð23Þ

where the Tsh is the duration of the time in minutes that thepanel is under the full shade (in that specific hour).

As an instance, based on Fig. 5, for a period of timebetween 9:00 and 10:00 in November, Tsh is approximately30 min and thus the Ksh = 0.50. However, for the time per-iod between 14:00 and 15:00 in October, Tsh is approxi-mately equal to 10 min and thus Ksh = 0.17.

The sky view factor Fsky can be calculated based on Eq.(16) and the projection of the obstacles to the sky as shownin Fig. 6. Fsky is not time dependent, however it changeswith the tilt angle of the solar panel.

Fig. 4. A typical example of

For instance, for a panel with 20� tilt angle, installed inthe location of Fig. 4, the view factor to the sky can be sim-ply derived from Fig. 6 by approximating area of theblocked portion of the sky with help of the shown mesh(the area of each is equal to 0.0152 rad2), as shown below:

F sky ¼ 1� 1

p2

X4

i¼1

Ai ¼ 1� 1

p2ð0:365þ 0:897þ 0:213

þ 0:365� 0:076Þ ¼ 0:821 ð24Þ

� A1 and A4: the area of the sky trapped behind a tiltedpanel with slope of 20�, thus A1 = A4 = 24 � 0.0152rad2 = 0.365 rad2.� A2: The area of the sky blocked because of the large

building, A2 = 59 � 0.0152 rad2 = 0.897 rad2.� A3: The area of the sky blocked because of the small

building, A3 = 14 � 0.0152 rad2 = 0.213 rad2.

an urban neighborhood.

Fig. 6. Blocked portion of the sky for 20� tilted panel installed in thetypical example of Fig. 4.


� Ac: The common area between A3 and A4, Ac = 5 �0.0152 rad2 = 0.076 rad2, is located behind the solarpanel as well as the small building. Thus it should beconsidered only once in the calculation.

The approximate value of the sky view factor calculatedhere is close to the accurate value that is shown in secondcolumn of Table 4.

10

3. Results and discussions

3.1. Dependency of the optimum tilt angle to location’s

latitude and the weather condition

In the first part of this section the dependency of theoptimum tilt angle to latitude of the location is investigatedwithout considering the effect of the weather condition.The contributions of each type of radiation are calculatedbased on the method of Fig. 1 for different locations withdifferent latitudes. This reveals the behavior of each typeof radiation in respect to the locations latitude. Later inthis section, the effect of the weather condition is exploredon the optimum tilt angle. In all the following calculationsthe ground reflectance ratio is considered equal to 0.2.

In order to investigate the sole effect of the latitude onoptimum installation angle, it is necessary to exclude the

Table 4Sky view factors for conditions with and without surrounding buildings.

Installation slope Without buildings With buildings

0 1.000 0.88410 0.964 0.85020 0.928 0.81630 0.890 0.78040 0.850 0.74250 0.806 0.70260 0.756 0.65470 0.697 0.59880 0.622 0.52590 0.500 0.405

effect of the climate conditions from all the calculations.Thus, the clearness index defined by Eq. (4) is consideredto be equal to 0.5 for all months of the year as well as allthe latitudes.

Using the tool developed for optimum angle calculation(based on the flowchart shown in Fig. 1) the discrepancybetween optimum angle of installation and the location’slatitude has been found for the five different latitudes asshown in Fig. 7.

As it can be seen from Fig. 7, the higher the latitude, thelarger the discrepancy between the latitude angle and theoptimum tilt angle. In order to find out the cause of thisphenomenon, the contribution of the different componentsof the total radiation is calculated for different installationangles as shown in Fig. 8 for a latitude of 15� and in Fig. 9for a latitude of 45�(considering k = 0.5).

A comparison of these two figures reveals the fact that,the diffuse radiation always has its maximum value in ahorizontal installation. However, the maximum beamradiation is achieved at an installation angle close to thelatitude angle. Since the combination of these two compo-nents creates the dominant part of the total radiation, itmakes the optimum angle smaller than the latitude.Besides, for higher latitudes the difference between thesetwo angles becomes high enough to cause a larger discrep-ancy between the value of the optimum angle and the loca-tion’s latitude. In order to see the effect of the climate onthe results, the monthly average daily insolation levels onhorizontal surfaces have been obtained from NASA infor-mation center (Surface meteorology and Solar Energy) forthe same five latitudes located on the meridian of 100�West as shown in Table 2.

The results of the calculations based on the real climateconditions are shown in Fig. 10. Even with using real climateconditions, the same trend is still observable. However, theresult for the latitude of 15� seems to be an exception to thisrule, where the optimum installation angle found to bearound 20� which is 5� more than the location’s latitude.

In order to understand the reason behind this observa-tion, the average monthly clearness indexes of the latitudesof 15�, 35� and 55� are shown in Fig. 11.

15 20 25 30 35 40 45 50 550

2

4

6

8

Latitude, φ

β −

φ

Fig. 7. The discrepancy between optimum angle of installation and thelatitude of location considering k = 0.5.

0 10 20 30 40 50 60 70 80 900

200

400

600

800

1000

1200

1400

1600

1800

Angle of Installation, β

Inso

latio

n (k

Wh/

m2/

year

)

Beam Diffuse Gorund Ref. Total

Optimum Angle

Fig. 8. The contribution of different radiation types in total radiation forU = 15�, k = 0.5.

0 10 20 30 40 50 60 70 80 900

200

400

600

800

1000

1200

1400

1600


Inso

latio

n (K

wh/

m2/

year

)

Beam Diffuse Ground Ref. Total

Optimum Angle

Fig. 9. The contribution of different radiation types in total radiation forU = 45�, k = 0.5.

Table 2Monthly average daily insolation levels on horizontal surfaces for latitudeslocated on the meridian of 100 west.

Month Latitude

15 25 35 45 55

January 5.64 3.88 3.01 1.89 0.84February 6.48 4.75 3.60 2.62 1.74March 7.23 6.05 4.93 3.75 3.11April 7.35 6.38 6.25 5.20 4.66May 6.70 6.77 6.51 6.00 5.53June 5.62 6.89 7.04 6.74 5.83July 5.86 6.65 7.11 6.66 5.50August 5.74 6.69 6.30 5.74 4.65September 5.05 5.81 5.28 4.43 3.02October 5.67 5.19 4.40 3.07 1.81November 5.62 4.39 3.17 1.76 0.99December 5.33 3.65 2.65 1.58 0.55

15 20 25 30 35 40 45 50 5515

20

25

30

35

40

45

50

55

Latitude, φ

Opt

imum

Tilt

Ang

le,

β opt

β=φ

Fig. 10. Optimum angle of installation for 5 different latitudes on thesame meridian (100� west).

1 2 3 4 5 6 7 8 9 10 11 120

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Months

Ave

rage

Mon

thly

Cle

arne

ss I

ndex

Latitude of 15 Latitude of 35 Latitude of 55

Fig. 11. The clearness indexes of latitudes 15, 35, and 55 on the meridian(100� west).


As it can be seen from Fig. 11, latitudes of 35� and 55�(on the meridian 100� west) have flat clearness index curvesthat shows an almost similar climate conditions for allmonths of the year in that location. However, for latitudeof 15�, a huge difference is observed between winter monthsand the summer months (4–9). The lower clearness indexesin summer months lead to the lower portion of the extrater-restrial radiation to reach the earth’s surface in that loca-tion. Consequently, this gives a privilege to the wintermonths and thus the optimum angle is declined towardswinter months. The contribution of the different compo-nents of the total radiation is shown in Fig. 12 for the lat-itude of 15� and longitude of 100� west, using the realclimate condition.

In conclusion, for the locations with almost the sameweather condition (the same clearness index) during awhole year, it is possible to claim that for small values oflatitude, the optimum angle is close to the location’s

latitude, while for the higher ones the optimum angle issmaller. However, the climate condition with significantlyvariable monthly clearness index values can considerablyinfluence this rule as shown for the place located on

0 10 20 30 40 50 60 70 80 900

500

1000

1500

2000

2500


Inso

latio

n (k

Wh/

m2/

year

)

Beam Diffuse Ground Ref. Total

Optimum Angle

Fig. 12. The contribution of different radiation types in total radiation forU = 15� and real climate condition.


latitude of 15� and longitude of 100� west. Therefore thelatitude of the location as well as the weather conditionduring the calculation period should be considered forselection of the optimum tilt angle.

3.2. Dependency of the optimum tilt angle to surrounding

situation in an urban environment

The same neighborhood as shown in Fig. 4 is used hereto show the effects of the surrounding obstacles on the opti-mum installation angle of a solar panel. The location ofthis neighborhood is chosen to be on the latitude of 45�north and longitude of 100� west.

Table 3Monthly average shading coefficients for the shown typical neighborhood.

Time January February March April May June

0:00–1:00 0 0 0 0 0 01:00–2:00 0 0 0 0 0 02:00–3:00 0 0 0 0 0 03:00–4:00 0 0 0 0 0 04:00–5:00 0 0 0 0 0 05:00–6:00 0 0 0 0 0 06:00–7:00 0 0 0 0 0 07:00–8:00 0 0 0 0 0 08:00–9:00 0 0 0 0 0 09:00–10:00 0.54 0.27 0 0 0 010:00–11:00 1.00 1.00 0.93 0 0 011:00–12:00 1.00 1.00 1.00 0 0 012:00–13:00 1.00 1.00 1.00 0 0 013:00–14:00 1.00 1.00 0.94 0 0 014:00–15:00 0.55 0.28 0 0 0 015:00–16:00 0 0 0 0 0 016:00–17:00 0 0 0 0 0.26 017:00–18:00 0 0 0 0.63 1.00 0.9618:00–19:00 0 0 0 0.64 1.00 1.0019:00–20:00 0 0 0 0 0.33 0.6920:00–21:00 0 0 0 0 0 021:00–22:00 0 0 0 0 0 022:00–23:00 0 0 0 0 0 023:00–24:00 0 0 0 0 0 0

In order to use the proposed method to calculate theoptimum tilt angle for the panel installed in the illustratedsituation in Fig. 4, it is required to find out the shadingcoefficients for each hour of the average day (during awhole year) as well as the sky view factors for different tiltangles. The values of the shading coefficients are calculatedbased on the comparison of the sun path diagram and therough outline of the buildings as shown in Fig. 5 and theyare tabulated in Table 3.

The sky view factors are calculated accurately with aprogram based on the accurate computation of the blockedareas using the method discussed in Section 2.3. The resultsare shown for 10 different installation angles and they arecompared with the cases where the effects of surroundingbuildings are ignored. This is done to observe the magni-tude of the error caused by ignoring the effects of the obsta-cles. In these calculations, the ground reflectance ratio isconsidered to be equal to 0.2 for both cases.

As it can be seen from Table 4, the proposed methodcreates the opportunity to find out the sky view factors ina more accurate manner in case of an urban application.These factors can be significantly smaller in comparisonto the standard view factors (introduced by Eq. (12)) wherebuilding effects are not considered. Employing the shadingcoefficients, the new sky view factors and the proposedmethod, the optimum angle of installation was calculatedfor the neighborhood. The comparison between applica-tion without any obstacle (shown with index 1) and theone that contains buildings (shown with the subscript“2”) has been shown in Fig. 13.

As it can be seen, due to the effects of the surroundingbuildings both beam and diffuse components were declined.

July August September October November December

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0.16 0.47 0.620 0 0 1.00 1.00 1.000 0 0 1.00 1.00 1.000 0 0 1.00 1.00 1.000 0 0 1.00 1.00 1.000 0 0 0.17 0.48 0.630 0 0 0 0 00.09 0 0 0 0 01.00 0.92 0.14 0 0 01.00 0.93 0.15 0 0 00.53 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0

0 10 20 30 40 50 60 70 80 900

200

400

600

800

1000

1200

1400

1600

1800


Inso

ltion

( k

Wh/

m2 /y

ear)

Beam1

Diffuse1

Gound Ref.1

Total1

Beam2

Diffuse2

Ground Ref.2

Total2

βop1β

op2

Fig. 13. Effect of the obstacles on optimum tilt angle for U = 45� andmeridian of 100� west.

Fig. 14. Effect of the ground reflectance ratio on optimum angle ofinstallation for U = 45� and meridian of 100� west.


Besides, the installation angle in which the maximum valueof the beam radiation was achieved is shifted from 45� to30�. In addition, the optimum angle of installation is foundto be around bopt2 = 33.1� (instead of bopt1 = 37.7�). Thus,due to the existence of the surrounding buildings, the solarpanel should be tilted towards a flatter angle to have ahigher annual energy yield.

To observe the effect of the ground reflectance ratio onthe annual optimum tilt angle, calculations have been donefor the same surrounding conditions with different groundreflectance ratios starting from 0.2 to 0.7. The results havebeen shown in the Fig. 14.

As was expected, higher ground reflectance ratio leadsto higher optimum tilt angle. This means, for instance, ifa building with glazed facade is placed in front of the solarpanel, the annual optimum tilt angle will be increased.

4. Conclusions

Knowledge of the optimum tilt angle is important toobtain the highest possible annual or seasonal energy yield.

Researchers claim that for small values of latitude, theoptimum annual tilt angle is closer to the location’s lati-tude, while for the higher ones, it is smaller in comparisonto the latitude. In this paper the reason behind these obser-vations has been explored, based on the contribution of thedifferent components of the total radiation while excludingthe climate conditions’ effect. It was shown that where themonthly clearness indexes are almost the same during ayear, the above mentioned claim is always valid. However,climate conditions with significantly variable monthlyclearness indexes can considerably influence this rule asshown for a specific place located on the latitude of 15�north and longitude of 100� west. Therefore both the lati-tude and the climate condition should be considered forcalculation of the optimum tilt angle. In addition, a simplemodification is proposed to the HDKR model to make itcompatible for urban applications. The proposed methodhas been applied to a typical urban case and the resultreveals the dependency of the optimum tilt angle to thesurrounding obstacles. In conclusion, in addition to the lat-itudes and the weather conditions, the influence of thesurrounding obstacles on optimum tilt angle should beconsidered when choosing the installation angles. It shouldbe observed that not only the optimum tilt angle isinfluenced by the obstructions but also the optimumazimuth.

References

Calabroa, E., 2009. Determining optimum tilt angles of photovoltaicpanels at typical north-tropical latitudes. J. Renew. Sust. Energy 1.

Chang, T.P., 2008. Study on the optimal tilt angle of solar collectoraccording to different radiation types. Int. J. Appl. Sci. Eng. 6.

Chang, T.P., 2009. The Sun’s apparent position and the optimal tilt angleof a solar collector in the northern hemisphere. J. Sol. Energy 83,1274–1284.

Chen, Y., Lee, C., Wu, H., 2005. Calculation of the optimum installationangle for fixed solar-cell panels based on the genetic algorithm and thesimulated-annealing method. IEEE Trans. Energy Convers. 20, 467–473.

Chenga, C.L., Jimenez, C.S.S., Lee, M., 2009. Research of BIPV optimaltilted angle, use of latitude concept for south orientated plans. J.Renew. Energy 34, 1644–1650.

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Kacira, M., Simsek, M., Babur, Y., Demirkol, S., 2004. Determiningoptimum tilt angles and orientations of photovoltaic panels inSanliurfa, Turkey. J. Renew. Energy 29, 1265–1275.

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