research article an algorithm to determine the optimum tilt...

Hindawi Publishing CorporationJournal of Renewable EnergyVolume 2013 Article ID 307547 12 pageshttpdxdoiorg1011552013307547

Research ArticleAn Algorithm to Determine the Optimum Tilt Angle ofa Solar Panel from Global Horizontal Solar Radiation

Emanuele Calabrograve

Department of Physics University of Messina Viale DrsquoAlcontres 31 98166 Messina Italy

Correspondence should be addressed to Emanuele Calabro ecalabroyahoocom

Received 19 December 2012 Revised 24 February 2013 Accepted 1 March 2013

Academic Editor Jayanta Deb Mondol

Copyright copy 2013 Emanuele Calabro This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper proposes an algorithm to calculate the optimum tilt angle of solar panels by means of global horizontal solar radiationdata provided from Earth-based meteorological stations This mathematical modeling is based on the maximization of thetheoretical expression of the global solar irradiation impinging on an inclined surface with respect to the slope and orientationof the panel and to the solar hour angle A set of transcendent equations resulted whose solutions give the optimum tilt andorientation of a solar panel A simulation was carried out using global horizontal solar radiation data from the European SolarRadiation Atlas and some empirical models of diffuse solar radiation The optimum tilt angle resulted was related to latitude bya linear regression with significant correlation coefficients The standard error of the mean values resulted increased significantlywith latitude suggesting that unreliable values can be provided at high latitudes

1 Introduction

Most countries in the world have realized the need for reduc-tion of gases emission to contrast the adverse global climaticchange encouraging the use of renewable and sustainablesources of energy Indeed large quantities of carbon dioxidenitrogen and sulfur oxides are emitted in the world byconventional energy sources which are released to the earthrsquosatmosphere contributing to climate change

Furthermore the world will soon run out of its conven-tional energy resources because of the rapid depletion of fossilfuel reserves This future scenario and the risks associatedwith CO

2emissions and global warming have increased the

interest in renewable energyThe major renewable energy systems include photo-

voltaics (PVs) solar thermal wind biomass hydroelectricand geothermal However among various renewable energysources the photovoltaic technology for power generationis considered well-suited technology particularly for dis-tributed power generation

Solar panel is the energy conversion fundamental com-ponent of PV systems or solar collectors Solar panels uselight energy from the sun to generate electricity through the

photovoltaic effect whereas solar thermal systems generateheat

The amount of electrical power produced from PVsystems is related to the amount of solar irradiation projectingon the modules Hence the global solar irradiation on tiltedsurfaces facing in different directions should be consideredto estimate thermal and electrical power obtained in archi-tectural planning

The literature provides that solar power supplied by themodules depends on many extrinsic factors such as insola-tion levels temperature load conditions and orientation ofthe panel

The solar radiation is also a function of the nature andextent of cloud cover and of the atmospherersquos water vapourcontent because solar radiation entering Earthrsquos atmosphereis scattered by atmospheric gases aerosols and clouds

Indeed meteorological parameters used as predictorsinclude the amount and distribution of clouds or other obser-vations such as the fractional sunshine and water content[1 2]

Aerosols can either absorb or scatter the radiation andalter the energy balance of Earth especially under clear skies[3]

2 Journal of Renewable Energy

These parameters obviously cannot be modifiedwhereas other variables can be changed to maximize thesolar energy acquired by the panel In fact the design ofa solar energy module involves complex tradeoffs due tothe interaction of several factors such as the characteristicsof the solar cells power supply requirements and powermanagement features of the embedded system applicationbehavior inclination and orientation of the panel

Hence it is essential to understand and exploit thesefactors in order to maximize the energy efficiency of a solarmodule

In particular for fixed absorber surfaces solar energygain is strictly related to the slope and azimuth angles of asolar panel

The global solar radiation for inclined surfaces can becalculated by the values of direct and diffuse solar radiationon the corresponding horizontal surfaceMeteorological datafrom all parts of the world are needed to know the horizontalglobal solar radiation and for some regions measureddata may only be applied within a radius of about 50 kmfrom weather stationsThis circumstance leads to interpolateparameters between stations

Furthermore spectral irradiance is usually not measuredroutinely so that the energy instantaneous production of asolar plant is to rely on appropriate models

Otherwise the users need accurate computations of theslope and orientation of solar panels in order tomaximize thesolar energy that is collected by fixed solar panels

Advantages of fixed solar panels are mainly related totheir tolerance to misalignment as approximately 20 of theincident solar radiation is diffuse light available at any angleof misalignment with the direct sun

In contrast the main advantage of tracking systems is tocollect solar energy for the longest period of the day withthe most accurate alignment as sunrsquos position shifts with theseasons Indeed daily solar energy collected was calculatedto be 19ndash24 higher by a solar PV panel with one axis east-west tracking system than by a fixed system [4]

Nevertheless since solar tracking systems have high oper-ation and maintenance costs and are not always applicable itis often convenient to set the solar collector at a fixed value ofan optimum tilt angle [5]

However some algorithms for the minimization of theenergy loss generated by the driving actuator were recentlyproposed [6 7]

Otherwise the increase of diffuse solar radiation at lowlatitudes at some locations with respect to high latitudesmakes fixed solar panels a competitive alternative to otherenergy sources mainly in those locations [8]

Hence the following main question regarding the designof fixed solar panels arises

What tilt and orientation of a solar panel have to bechosen to maximize the solar radiation

Solar radiation impinging on an inclined surface can bedivided into direct diffuse and ground reflected radiationHence a method to determine these quantities for eachlatitude has to be studied to answer to that fundamentalquestion

The diffuse solar component is the most difficult quantityto determine because the distribution of the sky-diffuseradiance strictly depends on the local condition of the sky

One way to estimate the diffuse solar component is tostudy the regression between the global and the diffuse solarradiation at locations where appropriate data are availableestablishing models which may be used to predict the diffusesolar radiation

Liu and Jordan performed the first studies on this subjectdetermining a relationship between daily diffuse and globalradiation on a horizontal surface assuming an isotropicdiffusion of solar radiance in the sky [9]

Erbs et al used a database acquired from four US weatherstations composed of hourly direct normal radiation andglobal radiation to develop an estimation of the diffusefraction of hourly daily andmonthly average global radiation[10]

Moreover [11 12] used also two predictors for theircorrelations the clearness index and the solar elevation

Garrison proposed a model to represent the dependencyof the diffuse fraction on the surface albedo atmosphericprecipitable water atmospheric turbidity solar elevation andglobal horizontal radiation [13]

Reindl et al considered two more significant predictorsthe ambient temperature and relative humidity [14] reducingthe standard error of Liu and Jordan-type models

On the other side [15] developed an exponential modelfor the estimation of the direct normal beam radiation fromthe global radiation named the Disc model This model wasthen improved by [16]

However accurate mathematical modeling of global anddiffuse solar radiation that was used for the simulation in thisstudy is proposed in the following section

Various optimum tilt angle values were provided in theliterature for fixed solar panels

For instance Qiu and Riffat suggested the tilt angle of thesolar collector set within the optimum tilt angle of plusmn10∘ as anacceptable practice [17]

Other computations led to the values of 120573 = 120593 plusmn 20∘ [18]120573 = 120593 plusmn 8

∘ [19] 120573 = 120593 plusmn 5∘ [20] and 120573 = 120593 plusmn 15∘ [21] where120593 represents the geographical latitude and the signs + and minusrefer to winter and summer months respectively

The disagreement among these values may be due to twomain reasons

(1) firstly the different methods of calculation that wereused for the determination of the optimum slopevalue of a solar panel

(2) secondly the different empirical models that wereconsidered for the determination of diffuse solarradiance and its link with the amount of global solarradiation

The aim of this study was to propose an algorithm for thedetermination of the optimum tilt and orientation of a solarpanel using a mathematical model based on the orientationof a generic surface with respect to the position of the sun inthe sky The other physical variables and the empirical modelof diffuse solar radiation are considered successively in thealgorithm linking it to a specific location

Journal of Renewable Energy 3

2 Mathematical Modeling of Global andDiffuse Solar Radiation

The literature provides several methods to estimate the globaland diffuse solar radiation using climatologic parameters

A typical empirical model is the regression equation ofthe Angstrom type [22]

119867119892

119867o= 119886 + 119887 (

119899

119873

) (1)

where 119867119892

is the monthly average of daily global solarradiation impinging on a horizontal surface at a location119867

119900

is themonthlymean of daily radiation on a horizontal surfacein the absence of atmosphere 119899 is the monthly mean of dailynumber hour of observed sunshine 119873 is the monthly meanvalue of day length of the interested location and ldquo119886rdquo and ldquo119887rdquoare the regression constants determined from climatologicaldataThe ratio 119899119873 is often called the ldquopossible sunshine hourpercentagerdquo

Regression coefficients ldquo119886rdquo and ldquo119887rdquo can be obtained fromsome relationships as proposed by [23 24]

A statistically worldwide 53 average decrease of globalsolar radiation with largest decline between 45∘ and 30∘Nwas found by analyzing the data collected in 45 actinometricstations during the years 1958 1965 1975 and 1985 [25] Asignificant decrease in mean yearly global solar radiationbetween the years 1964 and 1990 under completely overcastskies was found in some locations in Germany [26]

The observed changes in clear-sky radiation could berelated to the recovery by the volcanic eruptions effectssubmicron aerosol particles with simultaneous reduction ofaerosol mass concentration and increasing absorption byurban aerosol

Hence global horizontal solar irradiation data should beupdated and have to be preferred with respect to empiricalexpression such as (1)

Otherwise daily extraterrestrial radiation on a horizontalsurface named 119867

119900 can be computed for the day 119899 from the

following equation [27]

119867119900= 86400 lowast

119866119904119888

120587

times (1 + 0033 cos(2120587 119899

365

)) cos120593 cos 120575

lowast (plusmnradic(1 minus tan2120593 tan2120575))

+ cosminus1 (minus tan120593 tan 120575) lowast sin120593 sin 120575

(2)

where 119866sc is the solar constant (1367Wm2) 119899 is the numberof the day 120575 is the solar declination and 120593 is the geographicallatitudeThe hour angle of sunrise 120596

119904has been expressed as a

function of 120593 and 120575 by Cooperrsquos equationThe ratio of solar radiation at the surface of the Earth to

extraterrestrial radiation is called the clearness index withthe monthly average clearness index 119870

119879 defined as

119870119879=

119867119892

119867119900

(3)

where119867119892is the monthly average of daily solar radiation on a

horizontal surfaceThe global horizontal solar irradiation119867

119892 provided from

meteorological stations includes the horizontal direct beamirradiation119867

119887and the horizontal diffuse sky irradiation119867

119889

Some models can convert global horizontal irradiationto direct beam irradiation and diffuse sky irradiation on thehorizontal plane by means of empirical relationships

Heliosat is an algorithm which was developed to estimateground level global horizontal irradiance by using Meteosatsatellites images taken in the visible band

Some predictive models for estimating global solar irra-diation have to be used to obtain the solar radiation on atilted surface so that the relationship between the global solarirradiation on horizontal planes and that on tilted planes canbe evaluated [9 28ndash31]

Unfortunately no theoretic relationship between the hor-izontal sky diffuse irradiation 119867

119889and the horizontal global

solar irradiation 119867119892can be determined rigorously Indeed

a double integral like equation (5) of [32] should be solvedwhich cannot be computed even in the simplest case ofuniform and isotropic sky diffuse solar radiation because thedistribution of solar irradiance through the sky is difficult tobe represented adequately

The direct and diffuse components of solar radiation canbe estimated using empirical relationships 119867

119889= 119891(119867

119892) by

means of the clearness index 119870119879

Someof these algorithms requiring the direct normal anddiffuse radiation on a horizontal surface as input can providevery different estimated results in different locations [33]

Indeed the most considerable cause of error in thecomputation of the optimum tilt and orientation of a solarpanel depends on the model of diffuse solar radiation whichis used

Muneer [34] recommends themodel proposed by [35] forthe desert and tropical locations

119867119889

119867119892

= 135 minus 161119870119879 (4)

For temperate climates and locations out with the tropicsequation (5) given by [36] may be used

119867119889

119867119892

= 100 minus 113119870119879 (5)

The daily horizontal diffuse irradiation 119867119889and direct beam

irradiation119867119887= 119867119892minus 119867119889can be obtained by (4) and (5) as

well

3 The Algorithm to Maximize the Global SolarRadiation on a Fixed Sloped Surface

31 The Case of Isotropic Diffusion Global solar radiation ona tilted surface 119868

119879consists of daily direct solar radiation 119868

119887

diffuse solar radiation 119868119889 and ground reflected radiation 119868

119903

Daily solar radiation on a tilted surface for a given monthcan be estimated as follows [9]

119868119879= 119868119887+ 119868119889+ 119868119903 (6)


Thedaily direct radiation on a tilted surface 119868119887can be obtained

by means of 119877119887 the ratio of the average daily direct radiation

on a tilted plane to that on a horizontal plane and theparameters to it correlated [37]

119868119887=

cos 120579cos 120579119911

= 119867119887119877119887 (7)

119877119887= cos120573

minus sin120573 (sin 120575 cos120593 cos 120574

minus cos 120575 sin120593 cos120596 cos 120574 minus cos 120575 sin 120574 sin120596)

times (sin 120575 sin120593 + cos 120575 cos120593 cos120596)minus1(8)

where 120573 is the slope of the panel as to the horizontal plane 120574is the azimuth 120596 is the solar hour angle 120593 is the latitude and120579 and 120579

119911are the solar incidence angle on the considered plane

and the solar zenith angle respectivelyIf we consider a uniform and isotropic distribution of

diffuse solar radiation over the sky hemisphere 119868119889would be

easily obtained from the simple approximation of [9]

119868119889(iso) =

119867119889(1 + cos120573)2

(9)

The evaluation of the ground-reflected diffuse radiationdepends on 119868

119903 Most studies consider that the ground reflec-

tion process is ideally isotropic in which a specific case 119868119903can

be simplified as follows

119868119903=

119867119892120588 (1 minus cos120573)

2

(10)

where 120588 represents the diffuse reflectance of the ground (alsocalled ground albedo)

Finally the daily global solar irradiation on slopes 119868119879can

be expressed as the sum of (7) (9) and (10)The global solar radiation incident on a sloped surface

depends on the position of the sun along its daily trajectory(represented by the solar angle 120596) and on the orientation ofthe panel (represented by the slope 120573 and the azimuth 120574)

The other quantities appearing in (6) (7) (8) (9) and (10)depend on the local conditions and can be considered fixedparameters

The algorithm proposed here is based on the assumptionthat the daily solar irradiation impinging on a collectingsurface is maximum with respect to the angles 120573 120574 and 120596where 120573 is the slope of the panel as to the horizontal plane 120574is the azimuth and 120596 is the solar hour angle respectively

This physical condition for the maximization of the solarradiation acquired by a solar panel can be represented by themathematical expressions [8]

120597119868119879

120597120573

= 0

120597119868119879

120597120574

= 0

120597119868119879

120597120596

= 0

(11)

Their application to the isotropic model of [9] provides thefollowing expressions [8]

120597119868119879

120597120573

= sin120573(119867119900+119867119889

2

minus

119867119892120588

2

)

minus cos120573119867119900

times (cos 120575 sin120593 cos120596 cos 120574 minus sin 120575 cos120593 cos 120574

+ cos 120575 sin 120574 sin120596)

times (sin 120575 sin120593 + cos 120575 cos120593 cos120596)minus1 = 0

120597119868119879

120597120574

= sin 120574 (cos 120575 sin120593 cos120596 minus sin 120575 cos120593)

minus cos 120574 cos 120575 sin120596 = 0

120597119868119879

120597120596

= sin120596 sin 120575 cos 120574 minus cos 120575 cos120593 sin 120574

minus cos120596 sin 120575 sin120593 sin 120574 = 0

(12)

This set of equations can be solved with respect to the angles120573 120574 and 120596 [8]

120573 = tanminus1 119867119900


+ cos 120575 sin 120574 sin120596)

times [(119867119900+119867119889

2

minus

119867119892120588

2

)

times(sin 120575 sin120593 + cos 120575 cos120593 cos120596) ]minus1

(13)


120574 = tanminus1 cos 120575 sin120596(cos 120575 sin120593 cos120596 minus sin 120575 cos120593)

(14)

120596 = sinminus1 (sin 120575 cos 120575 sin 120574 cos 120574 cos120593

plusmn [sin4120575 (sin2120593sin2120574cos2120574 + sin4120593sin4120574)

minussin2120575 cos2120575sin2120593cos2120593sin4120574]12

)

times(sin2120575cos2120574 + sin2120575sin2120593sin2120574)minus1

(15)

The transcendent equations (13) (14) and (15) can be solvedby iterative methods with respect to the angles 120573 120574 and 120596which provide respectively the optimum tilt and orientationof the solar panel and the angular position of the sun in thesky where the maximization of the solar radiation on thepanel occurs

32 The Case of Anisotropic Diffusion The diffuse solar radi-ance influences the performance of most of the solar energytechnologies In fact the scattering atmospheric processesredistribute solar energy out of the direct beam into thediffuse radiation The PV systems like flat-plate collectorshave the peculiarity to use both direct and diffuse forms ofradiation whereas solar concentrator systems can use onlydirect radiation

Unfortunately the planetary network of diffuse solarradiationmeasurements stations is poor whereas global solarradiation data are available for many locations

Nevertheless empirical correlations between diffuse andglobal ratios and diffuse and direct solar radiation can be usedto reduce errors in computing hemispherical radiation fromestimates of direct and diffuse radiation especially regardingnonclear sky conditions

Besides the isotropic model of diffuse solar radiationdescribed in [9] several models have been proposed to repre-sent the anisotropy of the diffuse component 119868

119889by means of

empirical relations which should modify the expressions (12)and the relative solutions (13) (14) and (15)

The assumption that the diffuse radiation originatesentirely from the solar disk gives the relation

119868119889(disk) =

119867119889cos 120579

cos 120579119911

(16)

which is the modeling opposite to the isotropic one of [9]Le Quere [38] andHay andDavies [39] proposed amodel

for the diffuse solar component as a combination of the twocomponents that is (9) and (16)

119868119889= 119865 lowast 119868

119889(iso) + (1 minus 119865) lowast 119868

119889(disk) (17)

where (1 minus 119865) expresses the anisotropy degreeThe value 119865 = 08 was suggested by [38] while [39]

assumed the ratio of terrestrial direct radiation to extrater-restrial radiation as the degree of anisotropy (1 minus 119865)

Klucher [40] proposed a model in which the isotropiccomponent 119868

119889(iso) is multiplied by two factors that represent

both circumsolar and horizon brightening

119868119889= 119868119889(iso) (1 + 119870sin3 (

120573

2

)) (1 + 119870cos2120579sin3120579119911) (18)

where the parameter 119870 expresses the degree of anisotropyas a modulating function of the amount of direct radiationreceived by the surface119870 = 1minus119867

119889119867119892(thismodel reduces to

the Liu-Jordan isotropic model if the ratio of diffuse to globalradiation119867

119889119867119892is close to the unity)

Further models have been proposed in the literatureto perform the anisotropy of diffuse solar irradiance bymeans of some coefficients derived from statistical analysesof empirical data for specific locations [41ndash43]

Nevertheless the anisotropic models of [38ndash40] havebeen taken into account here because of their easy applica-bility to every location In addition they are extensively vali-datedmodels that convert hemispherical data on a horizontalsurface to hemispherical data on a tilted surface computingdiffuse solar radiation

The mathematical conditions (11) applied to the modelsof [38 39] provided the following solutions

120573 = tanminus1 [ (119867119900+ (1 minus 119865)119867

119889)

times (cos 120575 sin120593 cos120596 cos 120574

minus sin 120575 cos120593 cos 120574 + cos 120575 sin 120574 sin120596) ]

times [(119867119900+119865119867119889

2

+ (1 minus 119865)119867119889minus

119867119892120588

2

)

times (sin 120575 sin120593 + cos 120575 cos120593 cos120596) ]minus1

(19)

120574 = tanminus1 [ (119867119900+ (1 minus 119865)119867

119889) cos 120575 sin120596]

times [(119867119900+ (1 minus 119865)119867

119889)

times(cos 120575 sin120593 cos120596 minus sin 120575 cos120593)]minus1

(20)

120596 = sinminus1 [ (119867119900+ (1 minus 119865)119867

119889) sin 120575 cos 120575 sin 120574 cos 120574 cos120593

plusmn [[(119867119900+ (1 minus 119865)119867

119889) sin 120575]4

lowast (sin2120593sin2120574cos2120574 + sin4120593sin4120574)

minus [(119867119900+ (1 minus 119865)119867

119889) sin 120575 cos 120575]2

timessin2120593cos2120593sin4120574]12

]

times [[(119867119900+ (1 minus 119865)119867

119889) sin 120575]2

times (cos2120574 + sin2120593sin2120574)]minus1

(21)


Table 1 Optimum tilt angle values andmonthlymean of the daily global solar radiation values computed bymeans of the algorithm proposedin Section 2 applied to themonthly averages of the daily global horizontal solar irradiation data acquired from 1985 to 1989 at the site of Tripoli(latitude +32∘571015840N longitude +13∘121015840E)

Solar declin (deg) Isotropic model Le Querersquos model Hay and Daviesrsquo model Klucherrsquos model SEM values120573 (deg) 119864 (MJm2) 120573 (deg) 119864 (MJm2) 120573 (deg) 119864 (MJm2) 120573 (deg) 119864 (MJm2) 120573 (deg) 119864 (MJm2)

1198891= minus21269 507 1244 553 1305 533 1273 570 1854 117 126

1198892= minus13289 436 1713 475 1796 460 1759 487 2156 095 087

1198893= minus2819 337 2254 370 2361 358 2318 374 2406 072 028

1198894= 9415 219 2708 243 2824 235 2779 238 2601 045 042

1198895= 18792 129 2926 143 3016 138 2983 136 2756 025 050

1198896= 23314 86 3092 95 3153 92 3134 89 2931 017 044

1198897= 21517 106 3192 114 3255 112 3239 109 3000 015 050

1198898= 13784 184 3071 198 3158 194 3135 194 2877 025 055

1198899= 2217 295 2574 319 2677 312 2643 320 2590 050 022

11988910= minus9599 409 2016 440 2099 430 2069 449 2373 075 069

11988911= minus19148 485 1326 532 1394 511 1358 548 1879 118 113

11988912= minus23335 518 1064 572 1121 545 1087 589 1654 135 123

Table 2 Optimum tilt angle values andmonthlymean of the daily global solar radiation values computed bymeans of the algorithmproposedin Section 2 applied to themonthly averages of the daily global horizontal solar irradiation data acquired from 1985 to 1989 at the site of Athens(latitude +37∘581015840N longitude +23∘431015840E)


1198891= minus21269 501 658 605 742 535 673 602 989 222 066

1198892= minus13289 423 942 527 1066 458 968 522 1216 219 054

1198893= minus2819 330 1316 424 1494 362 1359 409 1465 187 037

1198894= 9415 247 2047 302 2259 271 2124 288 2013 103 047

1198895= 18792 171 2475 205 2665 187 2556 189 2323 061 062

1198896= 23314 136 2803 158 2958 148 2882 146 2603 039 066

1198897= 21517 157 2927 177 3082 169 3015 168 2697 036 073

1198898= 13784 229 2703 259 2883 247 2803 253 2533 055 065

1198899= 2217 336 2173 378 2349 360 2262 382 2239 090 031

11988910= minus9599 432 1421 495 1564 462 1477 506 1748 145 062

11988911= minus19148 482 738 585 832 516 756 582 1063 219 065

11988912= minus23335 523 592 625 667 556 606 622 935 218 069

Table 3 Optimum tilt angle values andmonthlymean of the daily global solar radiation values computed bymeans of the algorithmproposedin Section 2 applied to the monthly averages of the daily global horizontal solar irradiation data acquired from 1985 to 1989 at the site of Rome(latitude +41∘541015840N longitude +12∘271015840E)


1198891= minus21269 575 693 646 793 604 721 658 1147 166 091

1198892= minus13289 495 1056 569 1204 527 1102 581 1438 171 074

1198893= minus2819 401 1605 467 1815 431 1682 474 1812 147 044

1198894= 9415 28 2002 343 2267 307 2095 333 1977 122 056

1198895= 18792 202 2407 246 2666 222 2507 23 2246 079 076

1198896= 23314 168 2714 20 2945 184 2817 185 2490 057 083

1198897= 21517 19 2882 219 3108 207 2998 209 2621 052 090

1198898= 13784 265 2682 301 2919 286 2807 297 2499 070 078

1198899= 2217 365 1981 418 2206 392 2080 423 2075 115 040

11988910= minus9599 49 1529 536 1683 516 1604 553 1935 117 076

11988911= minus19148 557 801 626 913 586 835 639 1256 163 090

11988912= minus23335 615 705 667 794 64 738 684 1276 131 115


Table 4 Optimum tilt angle values andmonthlymean of the daily global solar radiation values computed bymeans of the algorithmproposedin Section 2 applied to the monthly averages of the daily global horizontal solar irradiation data acquired from 1985 to 1989 at the site of Paris(latitude +48∘511015840N longitude +2∘201015840E)


1198891= minus21269 567 398 681 494 599 410 665 654 233 051

1198892= minus13289 469 605 604 753 504 625 58 821 274 044

1198893= minus2819 383 1030 502 1266 417 1073 483 1194 241 047

1198894= 9415 287 1574 381 1895 317 1647 357 1592 180 064

1198895= 18792 204 1766 284 2100 228 1836 245 1694 145 076

1198896= 23314 183 2216 239 2546 204 2311 212 2059 100 088

1198897= 21517 211 2483 259 2813 233 2605 242 2276 087 097

1198898= 13784 278 2173 339 2506 305 2292 329 2072 117 080

1198899= 2217 39 1748 457 2024 42 1850 462 1867 146 049

11988910= minus9599 487 1022 571 1214 52 1075 579 1332 188 060

11988911= minus19148 581 593 663 712 611 621 669 967 183 073

11988912= minus23335 626 431 703 522 653 450 706 802 170 074

Equations (19) (20) and (21) can be solved by iterative meth-ods with respect to the angles 120573 120574 and 120596 The mathematicalconditions (11) applied to the model of [40] represented by(18) provided the following expressions

120597119868119879

120597120573

= minus 119867119900sin120573

minus 119867119900cos120573 [ (sin 120575 cos120593 cos 120574 minus cos 120575 sin120593 cos120596 cos 120574

minus cos 120575 sin 120574 sin120596)

times (sin 120575 sin120593 + cos 120575 cos120593 cos120596)minus1]

minus119867119889sin120573

(1 + 119865sin3 (1205732))2

+ 3sin2 (120573

2

)119867119889119865

(1 + cos120573)2

minus119867119889119865 sin120573119902

111990221199023

2

+ 3119865sin2 (120573

2

) 11990211199022119867119889119865

(1 + cos120573)2

+

119867119892120588 sin1205732

+ 119867119889119865 (1 + cos120573) (radic1199021)

lowast [cos120573 (cos 120575 sin120593 cos120596 cos 120574 + cos 120575 sin 120574 sin120596minus sin 120575 cos120593 cos 120574)

minus sin120573 (sin 120575 sin120593 + cos 120575 cos120593 cos120596)] 11990221199023= 0

(22)

120597119868119879

120597120574

= sin120573 (119867119900sin 120575 cos120593 sin 120574 minus 119867

119900cos 120575 sin120593 cos120596 sin 120574

+119867119900cos 120575 cos 120574 sin120596)

times (sin 120575 sin120593 + cos 120575 cos120593 cos120596)minus1

+ (1 + 119865sin3 (120573

2

))119867119889119865 (1 + cos120573) 119902

2(radic1199021)

times [sin120573 (sin 120575 cos120593 sin 120574 + cos 120575 cos 120574 sin120596

minus cos 120575 sin120593 cos120596 sin 120574) ] = 0(23)

120597119868119879

120597120596

=

119867119900sin120573 (cos 120575 cos120596 sin 120574 minus cos 120575 sin120593 sin120596 cos 120574)

sin 120575 sin120593 + cos 120575 cos120593 cos120596

+ (1 + 119865sin3 (120573

2

))119867119889119865 (1 + cos120573) 119902

2radic1199021

times [sin120573 (cos120596 cos 120575 sin 120574 minus sin120596 cos 120575 sin120593 cos 120574)

minus cos120573 cos 120575 cos120593 sin120596]

+ (1 + 119865sin3 (120573

2

))119867119889119865 (1 + cos120573) (3

4

) 1199021

times [1 minus (sin 120575 sin120593 + cos 120575 cos120593 cos120596)2]12

times (sin120596 cos 120575 cos120593) = 0(24)

where

1199021= [cos120573 (sin 120575 sin120593 + cos 120575 cos120593 cos120596)

+ sin120573 (cos 120575 sin120593 cos120596 cos 120574

+ cos 120575 sin120596 sin 120574 minus sin 120575 cos120593 cos 120574)]2

1199022= [1 minus (sin 120575 sin120593 + cos 120575 cos120593 cos120596)2]

32

1199023= [1 + 119865sin3 (

120573

2

)] (25)

This set of equation can be solved only using iterativemethods

4 Applying the Maximization Algorithmto a Data Set of Global Horizontal SolarRadiation

The data collected at the European Solar Radiation Atlas(available at the Internet site HelioClim) were used to test thealgorithm proposed here


Table 5 Optimum tilt angle values andmonthlymean of the daily global solar radiation values computed bymeans of the algorithmproposedin Section 2 applied to themonthly averages of the daily global horizontal solar irradiation data acquired from 1985 to 1989 at the site of London(latitude +51∘321015840N longitude 0∘051015840W)


1198891= minus21269 618 224 738 310 647 232 706 400 236 036

1198892= minus13289 521 439 661 595 555 456 63 626 280 041

1198893= minus2819 414 779 559 1046 449 814 526 926 290 052

1198894= 9415 32 1331 44 1736 353 1402 408 1370 233 080

1198895= 18792 258 1888 346 2365 286 1996 317 1786 145 109

1198896= 23314 207 1787 299 2255 231 1871 252 1684 169 107

1198897= 21517 217 1661 317 2120 242 1737 267 1583 184 103

1198898= 13784 282 1454 396 1885 312 1529 358 1444 217 090

1198899= 2217 391 1102 511 1437 426 1163 493 1224 243 113

11988910= minus9599 549 865 629 1084 58 923 638 1186 182 113

11988911= minus19148 631 368 720 484 659 387 713 653 185 106

11988912= minus23335 642 173 758 241 668 178 722 331 226 032

Table 6 Optimum tilt angle values andmonthlymean of the daily global solar radiation values computed bymeans of the algorithmproposedin Section 2 applied to the monthly averages of the daily global horizontal solar irradiation data acquired from 1985 to 1989 at the site ofStockholm (latitude +59∘171015840N longitude +18∘031015840E) The monthly mean of the daily global solar radiation was not acquired in the periodsrelative to the solar declinations 119889

1 11988911 and 119889

12


1198891= minus21269 nc nc nc nc nc nc

1198892= minus13289 671 518 732 699 695 565 738 879 137 070

1198893= minus2819 547 1025 631 1377 579 1115 638 1256 188 068

1198894= 9415 431 1864 513 2432 464 2037 518 1832 180 119

1198895= 18792 330 2199 418 2910 362 2388 407 1995 177 170

1198896= 23314 295 2539 373 3292 324 2754 359 2219 152 195

1198897= 21517 298 2184 390 2912 329 2359 370 1968 178 175

1198898= 13784 338 1417 465 2012 372 1515 434 1398 250 125

1198899= 2217 463 1025 580 1445 498 1102 569 1152 244 079

11988910= minus9599 605 565 695 793 635 609 693 825 192 056



In particular the global horizontal solar radiation 119867119892

relative to Tripoli Athens Rome Paris London and Stock-holm located from 32∘ to 59∘ north latitude was consideredby averaging the values on a monthly basis from 1985 to 1989and was used for this simulation

The calculations of 120573 120574 and 120596 were carried out at everymonth because their time variation cannot be neglected [44]

The solar declination values 120575 were computed at themiddle of every month by Cooperrsquos equation which wasalso used to calculate the sunset hour angle 120596

119904[37] Daily

extraterrestrial radiation on a horizontal surface 119867119900 was

computed at the same day by [27] equation (5) from [39] wasused as a relationship between 119867

119889and 119867

119892 Ground albedo

was fixed at the typical value 120588 = 020

The isotropic model of [9] and the anisotropic models of[38ndash40] were used to calculate the diffuse solar component119867119889The solutions of (14) (15) (20) (21) (23) and (24) gave

the values 120574 = 0∘ and 120596 = 0∘ confirming that the optimumorientation of a solar panel is toward South where in fact thesolar hour angle at solar noon must be zero as it is defined

Nevertheless other models of the solar diffuse compo-nent could provide different values

The optimum tilt angle values provided by the solutionsof (13) (19) (22) were reported on Tables 1 2 3 4 5 and 6

The corresponding monthly average of the daily globalsolar radiation (measured in MJm2) collected by a surfaceinclined at each 120573 value was indicated in Tables 1ndash6 as


Table 7 The average 120573 of the optimum tilt angle values computed using the diffuse solar radiation models relative to the latitude valuesused above A linear regression with 95 confidence interval provided the coefficients 119886

1and 119886

2and the correlation coefficients ranging from

0944 to 0993 reported in the last column

120573 (1198971)

Coefficients120573 (1198972) 120573 (119897

3) 120573 (119897

4) 120573 (119897

5) 120573 (119897

6) 119886

11198862

Correlation 119903

5407 5608 6207 6280 6772 mdash 3133 068 09534645 4825 5430 5393 5917 7090 1625 086 09443597 3813 4432 4462 4870 5987 680 084 09572337 2770 3157 3355 3803 4815 minus607 087 09751365 1880 2250 2403 3017 3792 minus1495 087 0978905 1470 1842 2095 2473 3377 minus1927 087 09851103 1678 2062 2362 2608 3467 minus1565 083 09881925 2470 2872 3127 3370 4022 minus423 075 09893115 3640 3995 4322 4553 5275 642 077 09934320 4737 5237 5393 5990 6570 1584 083 09835190 5413 6020 6310 6807 mdash 2361 084 09775560 5815 6515 6720 6975 mdash 3056 076 0968

Table 8 The average standard error of the mean (SEM) of thetilt angles and of the monthly average of the daily global solarradiation values of diffuse solar radiation computed separately forsummer months (solar declinations 119889

4ndash1198899) and for winter months

(solar declinations 1198891ndash1198893and 119889

10ndash11988912) as a function of latitude (the

correlation coefficient 119903 relative to winter months was computedexcluding the values relative to the latitude of Stockholm)

Latitude (deg)Average values of SEM values

Summer months Winter months120573 119864 120573 119864

3295 030 044 101 0913796 064 057 201 0594190 082 071 150 0824885 126 075 215 0585153 202 092 233 0465928 196 144 nc ncCorrelationcoefficient 119903 0956 0935 0835 minus0789

well Such a value can be considered representative of theconversion efficiency of a PV module because of a linearrelationship with solar radiation rate [45]

The results led to the following considerations

(1) The optimum tilt angle values of solar panels deter-mined for winter months were confirmed to be verydifferent from the values recommended for summermonths

(2) The value 120574 = 0∘ was confirmed to be the optimumorientation value for a solar panel

(3) The disagreement among the models of diffuse solarirradiance resultedwas relevant for thewintermonths(represented by the solar declinations 119889

1ndash3 and 11988910ndash12corresponding to the first three and the last three rows

of Tables 1ndash6) and resulted in increase with increasinglatitude

Furthermore the optimum tilt angle values were averagedfor each monthly solar declination and were reported as afunction of latitude in Table 7The average tilt angles resultedwere significantly related to latitudes A correlation betweenthese two sets of variables was studied applying a linearrelationship to predict the value of the optimum tilt angle fora given geographical latitude

In particular the regression line was performed by themethod of least squares to make the sum of the squaresof the differences between the ordinates of the points andthose on the straight line as small as possible The correlationcoefficient ldquo119903rdquo of the variables was calculated defined as theircovariance (that measures how the two variables are linearlyrelated) divided by the product of their individual standarddeviations In addition a confidence interval was consideredto know how accurate is the regression

Linear regressions with 95 confidence interval providedcorrelation coefficients ranging from 0944 to 0993 reportedin the last column of Table 7 (larger correlation values werefound for summer months) Two typical fits for two solardeclinations are shown in Figures 1(a) and 1(b)

This result shows that the optimum tilt angle of a solarpanel could be easily obtained as a function of geographicallatitude by means of the coefficients 119886

1and 119886

2of a linear

regression 120573 = 1198861+1198862lowast120593The coefficients 119886

1and 1198862 obtained

by the regression depend on the solar declination that wasused

Furthermore the application of the algorithm showedthat the differences among the tilt angle values computedusing variousmodels of diffuse solar irradiance increase withincreasing of the geographical latitude suggesting that fur-ther empirical correlations between diffuse and global ratiosand diffuse and direct solar radiation should be investigatedespecially at high latitudes


30 35 40 45 50 55 60

30

35

40

45

50

55O

ptim

um ti

lt an

gle (

deg)

Latitude (deg)

(a)

30 35 40 45 50 55 60

35

40

45

50

55

60

Opt

imum

tilt

angl

e (de

g)

Latitude (deg)

(b)

Figure 1 Average values of optimum tilt angles computed using diffuse solar radiation models as a function of latitude relative to the solardeclination 119889

9(a) and to the solar declination 119889

3(b) Linear regressions with 95 confidence interval provided the correlation coefficients

119903 = 0993 and 119903 = 0957 respectively

30 35 40 45 50 55 60

SEM

of t

ilt an

gle v

alue

s

Latitude (deg)

02

04

06

08

1

12

14

16

18

2

22

(a)

30 35 40 45 50 55 60

SEM

of m

onth

ly av

erag

e of d

aily

sola

r rad

iatio

nva

lues

Latitude (deg)

04

06

08

1

12

14

16

(b)

Figure 2The standard error of themean (SEM) of the tilt angle values (a) and of themonthly average of the daily global solar radiation values(b) computed for the summer months (solar declinations 119889

4ndash1198899) as a function of latitudeThe linear fits provided the correlation coefficients


Finally the standard error of the mean (SEM) of thetilt angles and of the monthly average of the daily globalsolar radiation values was computed and reported in the lastcolumn of Tables 1ndash6 to quantify the disagreement amongthe various models of diffuse solar irradiance used here

The average values of these SEM values were computedseparately for summer months and for winter months andreported in Table 8

A significant correlation between the averages of theseSEM values and the latitudes was found for summer monthsIn fact a linear regression with 95 confidence interval

provided the correlation coefficients 119903 = 0956 and 119903 = 0935for the tilt angles and the monthly average of the daily globalsolar radiation values respectively These linear regressionswere represented in Figures 2(a) and 2(b)

Regarding the winter months lower correlations werefound (see Table 8) Nevertheless the values obtained forthe winter months can be considered less reliable becausethe decrease in intensity of direct and diffuse solar radiationduring winter months produces an increase in the relativeerror for estimating the tilt angles and the monthly averageof the daily global solar radiation values


5 Conclusions

The method of maximization of global solar radiation pro-posed here led to take into account a set of equations whichcan be solved with respect to the slope 120573 and the azimuth120574 of a collecting surface and to the solar hour angle 120596 Theproposed algorithm was applied to some models of diffusesolar irradiance using global horizontal solar radiation datafrom the European Solar Radiation Atlas The relative equa-tions provided a set of angles 120573 120574 and 120596 as a function of theother physical parameters The better solar panel orientationtoward South and the large time dependence of the optimumtilt angle were confirmed suggesting the use of semifixedpanels on building structures

The solutions of the set of equations provided tilt anglesand monthly average of daily global solar radiation valuesstrictly related to the model of diffuse solar irradiation whichwas used

The computed tilt angles resulted were significantlyrelated to the latitude value because linear regressions with95 confidence interval provided correlation coefficientsranging from 0944 to 0993This result allows to estimate theoptimum tilt angle of a solar panel as a function of latitude bymeans of a linear regression whose coefficients depend onglobal horizontal solar radiation data

Nevertheless relevant differences among the monthlyaverage of the daily global solar radiation and the tilt anglevalues obtained using different models of diffuse solarradiation were highlighted by the simulations performedseparately for summer and winter months

The standard error of the mean of such values resultedin significant increase with increasing latitude for summermonths leading to the conclusion that values unreliable forthe optimum tilt angle of a solar panel can be provided by themodels of diffuse solar radiation at high latitudes

Hence further research is needed to better estimate thediffuse solar radiation

References

[1] ASHRAE Handbook HVAC Applications ASHRAE AtlantaGa USA 1999

[2] M Iqbal ldquoEstimation of the monthly average of the diffusecomponent of total insolation on a horizontal surfacerdquo SolarEnergy vol 20 no 1 pp 101ndash105 1978

[3] C A Gueymard ldquoImportance of atmospheric turbidity andassociated uncertainties in solar radiation and luminous effi-cacy modellingrdquo Energy vol 30 no 9 pp 1603ndash1621 2005

[4] O C Vilela N Fraidenraich and C Tiba ldquoPhotovoltaicpumping systems driven by tracking collectors Experimentsand simulationrdquo Solar Energy vol 74 no 1 pp 45ndash52 2003

[5] H K Elminir A E Ghitas F El-Hussainy R Hamid MM Beheary and K M Abdel-Moneim ldquoOptimum solar flat-plate collector slope case study for Helwan Egyptrdquo EnergyConversion and Management vol 47 no 5 pp 624ndash637 2006

[6] M A Ionita and C Alexandru ldquoDynamic optimization of thetracking system for a pseudo-azimuthal photovoltaic platformrdquoJournal of Renewable and Sustainable Energy vol 4 Article ID053117 15 pages 2012

[7] S Seme and G Stumberger ldquoA novel prediction algorithm forsolar angles using solar radiation and Differential Evolution fordual-axis sun tracking purposesrdquo Solar Energy vol 85 no 11pp 2757ndash2770 2011

[8] E Calabro ldquoThe disagreement between anisotropic-isotropicdiffuse solar radiationmodels as a function of solar declinationcomputing the optimum tilt angle of solar panels in the area ofsouthern-Italyrdquo Smart Grid and Renewable Energy vol 3 no 4pp 253ndash259 2012

[9] B Y H Liu and R C Jordan ldquoThe interrelationship andcharacteristic distribution of direct diffuse and total solarradiationrdquo Solar Energy vol 4 no 3 pp 1ndash19 1960

[10] D G Erbs S A Klein and J A Duffie ldquoEstimation of thediffuse radiation fraction for hourly daily andmonthly-averageglobal radiationrdquo Solar Energy vol 28 no 4 pp 293ndash302 1982

[11] M Iqbal ldquoPrediction of hourly diffuse solar radiation frommeasured hourly global radiation on a horizontal surfacerdquo SolarEnergy vol 24 no 5 pp 491ndash503 1980

[12] A Skartveit and J A Olseth ldquoAmodel for the diffuse fraction ofhourly global radiationrdquo Solar Energy vol 38 no 4 pp 271ndash2741987

[13] J D Garrison ldquoA study of the division of global irradiance intodirect and diffuse irradiance at thirty-three US sitesrdquo SolarEnergy vol 35 no 4 pp 341ndash351 1985

[14] D T Reindl W A Beckman and J A Duffie ldquoDiffuse fractioncorrelationsrdquo Solar Energy vol 45 no 1 pp 1ndash7 1990

[15] E L Maxwell ldquoA quasi-physical model for converting hourlyglobal horizontal to direct normal insolationrdquo Tech RepSERITR-215-3087 Solar Energy Institute Golden Colo USA1987

[16] R R Perez P Ineichen E L Maxwell R D Seals andA Zelenka ldquoDynamic global-to-direct irradiance conversionmodelsrdquo in Proceedings of ISES Solar World Congress pp 951ndash956 Denver Colo USA 1992

[17] G Qiu and S B Riffat ldquoOptimum tilt angle of solar collectorsand its impact on performancerdquo International Journal of Ambi-ent Energy vol 24 no 1 pp 13ndash20 2003

[18] J I Yellott ldquoUtilization of sun and sky radiation for heatingcooling of buildingsrdquo ASHRAE Journal vol 15 no 12 pp 31ndash42 1973

[19] G Lewis ldquoOptimum tilt of a solar collectorrdquo Solar and WindTechnology vol 4 no 3 pp 407ndash410 1987

[20] H P GargAdvances in Solar Energy Technology Reidel BostonMass USA 1987

[21] P J Lunde SolarThermal Engineering JohnWiley amp Sons NewYork NY USA 1980

[22] AAngstrom ldquoSolar and terrestrial radiationrdquoQuarterly Journalof the Royal Meteorological Society vol 50 pp 121ndash126 1924

[23] G N Tiwari and S Suneja Solar Thermal Engineering SystemNarosa Publishing New Dehli India 1997

[24] I Ulfat F Ahmad and I Siddiqui ldquoDetermination of Angstromcoefficient for the prediction of monthly average daily GobalSolar Radiation Horizontal surface at Karachi PakistanrdquoKarachi University Journal of Science vol 33 pp 7ndash11 2005

[25] G Stanhill and S Moreshet ldquoGlobal radiation climate changesthe world networkrdquo Climatic Change vol 21 no 1 pp 57ndash751992

[26] B G Liepert and G J Kukla ldquoDecline in global solar radiationwith increased horizontal visibility in Germany between 1964and 1990rdquo Journal of Climate vol 10 no 9 pp 2391ndash2401 1997


[27] S A Klein ldquoCalculation of monthly average insolation on tiltedsurfacesrdquo Solar Energy vol 19 no 4 pp 325ndash329 1977

[28] R Perez P Ineichen R Seals J Michalsky and R StewartldquoModeling daylight availability and irradiance componentsfrom direct and global irradiancerdquo Solar Energy vol 44 no 5pp 271ndash289 1990

[29] L Robledo and A Soler ldquoModelling irradiance on inclinedplaneswith an anisotropicmodelrdquoEnergy vol 23 no 3 pp 193ndash201 1998

[30] F J Olmo J Vida I Foyo Y Castro-Diez and L Alados-Arboledas ldquoPrediction of global irradiance on inclined surfacesfrom horizontal global irradiancerdquo Energy vol 24 no 8 pp689ndash704 1999

[31] L Wenxian G Wenfeng P Shaoxuan and L Enrong ldquoRatiosof global radiation on a tilted to horizontal surface for YunnanProvince Chinardquo Energy vol 20 no 8 pp 723ndash728 1995

[32] D H W Li and T N T Lam ldquoDetermining the optimumtilt angle and orientation for solar energy collection basedon measured solar radiance datardquo International Journal ofPhotoenergy vol 2007 Article ID 85402 2007

[33] H D Kambezidis B E Psiloglou and C Gueymard ldquoMeasure-ments and models for total solar irradiance on inclined surfacein Athens Greecerdquo Solar Energy vol 53 no 2 pp 177ndash185 1994

[34] T Muneer Solar Radiation and Daylight Models ElsevierBerlin Germany 2nd edition 2004

[35] T Muneer and M M Hawas ldquoCorrelation between dailydiffuse and global radiation for Indiardquo Energy Conversion andManagement vol 24 no 2 pp 151ndash154 1984

[36] J K Page The Estimation of Monthly Mean Value of DailyShort Wave Irradiation on Vertical and Inclined Surfaces fromSunshine Records for Latitudes 60∘N to 40∘S BS32 Departmentof Building Science University of Sheffield Sheffield UK 1977

[37] J A Duffie and W A Beckman Solar Engineering of ThermalProcess John Wiley amp Sons New York NY USA 1991

[38] J LeQuere ldquoRapport sur la comparaison des methods de calculdes besoins de chauffage des logementsrdquo Tech Rep CSTB TEA-S 87 CSTB Paris France 1980

[39] J E Hay and J A Davies ldquoCalculation of the solar radiationincident on an inclined surfacerdquo in Proceedings of the 1stCanadian Solar Radiation Data Workshop J E Hay and T KWon Eds pp 59ndash72 Toronto Canada 1980

[40] T M Klucher ldquoEvaluation of models to predict insolation ontilted surfacesrdquo Solar Energy vol 23 no 2 pp 111ndash114 1979

[41] R R Perez J T Scott and R Stewart ldquoAn anisotropicmodel fordiffuse radiation incident on slopes of different orientations andpossible applications to CPCsrdquo in Proceedings of American SolarEnergy Society (ASES rsquo83) pp 883ndash888 Minneapolis MinnUSA 1983

[42] R Perez R Seals P Ineichen R Stewart and D Menicucci ldquoAnew simplified version of the perez diffuse irradiance model fortilted surfacesrdquo Solar Energy vol 39 no 3 pp 221ndash231 1987

[43] T Muneer Solar Radiation and Daylight Models for the EnergyEfficient Design of Buildings Architectural Press Oxford UK1997

[44] E Calabro ldquoDetermining optimum tilt angles of photovoltaicpanels at typical north-tropical latitudesrdquo Journal of Renewableand Sustainable Energy vol 1 no 3 Article ID 033104 6 pages2009

[45] S Naihong N Kameda Y Kishida and H Sonoda ldquoExperi-mental andTheoretical study on the optimal tilt of photovoltaicpanelsrdquo Journal of Asian Architecture and Building Engineeringvol 5 pp 399ndash405 2006

TribologyAdvances in

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Submit your manuscripts athttpwwwhindawicom


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EnergyJournal of


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Journal ofEngineeringVolume 2014

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International Journal ofPhotoenergy


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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


These parameters obviously cannot be modifiedwhereas other variables can be changed to maximize thesolar energy acquired by the panel In fact the design ofa solar energy module involves complex tradeoffs due tothe interaction of several factors such as the characteristicsof the solar cells power supply requirements and powermanagement features of the embedded system applicationbehavior inclination and orientation of the panel

Hence it is essential to understand and exploit thesefactors in order to maximize the energy efficiency of a solarmodule

In particular for fixed absorber surfaces solar energygain is strictly related to the slope and azimuth angles of asolar panel

The global solar radiation for inclined surfaces can becalculated by the values of direct and diffuse solar radiationon the corresponding horizontal surfaceMeteorological datafrom all parts of the world are needed to know the horizontalglobal solar radiation and for some regions measureddata may only be applied within a radius of about 50 kmfrom weather stationsThis circumstance leads to interpolateparameters between stations

Furthermore spectral irradiance is usually not measuredroutinely so that the energy instantaneous production of asolar plant is to rely on appropriate models

Otherwise the users need accurate computations of theslope and orientation of solar panels in order tomaximize thesolar energy that is collected by fixed solar panels

Advantages of fixed solar panels are mainly related totheir tolerance to misalignment as approximately 20 of theincident solar radiation is diffuse light available at any angleof misalignment with the direct sun

In contrast the main advantage of tracking systems is tocollect solar energy for the longest period of the day withthe most accurate alignment as sunrsquos position shifts with theseasons Indeed daily solar energy collected was calculatedto be 19ndash24 higher by a solar PV panel with one axis east-west tracking system than by a fixed system [4]

Nevertheless since solar tracking systems have high oper-ation and maintenance costs and are not always applicable itis often convenient to set the solar collector at a fixed value ofan optimum tilt angle [5]

However some algorithms for the minimization of theenergy loss generated by the driving actuator were recentlyproposed [6 7]

Otherwise the increase of diffuse solar radiation at lowlatitudes at some locations with respect to high latitudesmakes fixed solar panels a competitive alternative to otherenergy sources mainly in those locations [8]

Hence the following main question regarding the designof fixed solar panels arises

What tilt and orientation of a solar panel have to bechosen to maximize the solar radiation

Solar radiation impinging on an inclined surface can bedivided into direct diffuse and ground reflected radiationHence a method to determine these quantities for eachlatitude has to be studied to answer to that fundamentalquestion

The diffuse solar component is the most difficult quantityto determine because the distribution of the sky-diffuseradiance strictly depends on the local condition of the sky

One way to estimate the diffuse solar component is tostudy the regression between the global and the diffuse solarradiation at locations where appropriate data are availableestablishing models which may be used to predict the diffusesolar radiation

Liu and Jordan performed the first studies on this subjectdetermining a relationship between daily diffuse and globalradiation on a horizontal surface assuming an isotropicdiffusion of solar radiance in the sky [9]

Erbs et al used a database acquired from four US weatherstations composed of hourly direct normal radiation andglobal radiation to develop an estimation of the diffusefraction of hourly daily andmonthly average global radiation[10]

Moreover [11 12] used also two predictors for theircorrelations the clearness index and the solar elevation

Garrison proposed a model to represent the dependencyof the diffuse fraction on the surface albedo atmosphericprecipitable water atmospheric turbidity solar elevation andglobal horizontal radiation [13]

Reindl et al considered two more significant predictorsthe ambient temperature and relative humidity [14] reducingthe standard error of Liu and Jordan-type models

On the other side [15] developed an exponential modelfor the estimation of the direct normal beam radiation fromthe global radiation named the Disc model This model wasthen improved by [16]

However accurate mathematical modeling of global anddiffuse solar radiation that was used for the simulation in thisstudy is proposed in the following section

Various optimum tilt angle values were provided in theliterature for fixed solar panels

For instance Qiu and Riffat suggested the tilt angle of thesolar collector set within the optimum tilt angle of plusmn10∘ as anacceptable practice [17]

Other computations led to the values of 120573 = 120593 plusmn 20∘ [18]120573 = 120593 plusmn 8

∘ [19] 120573 = 120593 plusmn 5∘ [20] and 120573 = 120593 plusmn 15∘ [21] where120593 represents the geographical latitude and the signs + and minusrefer to winter and summer months respectively

The disagreement among these values may be due to twomain reasons

(1) firstly the different methods of calculation that wereused for the determination of the optimum slopevalue of a solar panel

(2) secondly the different empirical models that wereconsidered for the determination of diffuse solarradiance and its link with the amount of global solarradiation

The aim of this study was to propose an algorithm for thedetermination of the optimum tilt and orientation of a solarpanel using a mathematical model based on the orientationof a generic surface with respect to the position of the sun inthe sky The other physical variables and the empirical modelof diffuse solar radiation are considered successively in thealgorithm linking it to a specific location





119867119892

119867o= 119886 + 119887 (

119899

119873

) (1)

where 119867119892


119900









119867119900= 86400 lowast

119866119904119888

120587

times (1 + 0033 cos(2120587 119899

365

)) cos120593 cos 120575



(2)





119879 defined as

119870119879=

119867119892

119867119900

(3)






119889









119889= 119891(119867

119892) by





119867119889

119867119892

= 135 minus 161119870119879 (4)


119867119889

119867119892

= 100 minus 113119870119879 (5)



well




119887


119903


119868119879= 119868119887+ 119868119889+ 119868119903 (6)





119868119887=

cos 120579cos 120579119911

= 119867119887119877119887 (7)

119877119887= cos120573









119868119889(iso) =

119867119889(1 + cos120573)2

(9)





119868119903=

119867119892120588 (1 minus cos120573)

2

(10)








120597119868119879

120597120573

= 0

120597119868119879

120597120574

= 0

120597119868119879

120597120596

= 0

(11)


120597119868119879

120597120573

= sin120573(119867119900+119867119889

2

minus

119867119892120588

2

)

minus cos120573119867119900


+ cos 120575 sin 120574 sin120596)


120597119868119879

120597120574


minus cos 120574 cos 120575 sin120596 = 0

120597119868119879

120597120596



(12)


120573 = tanminus1 119867119900


+ cos 120575 sin 120574 sin120596)

times [(119867119900+119867119889

2

minus

119867119892120588

2

)


(13)



(14)




)


(15)






119889by means of



119868119889(disk) =

119867119889cos 120579

cos 120579119911

(16)



119868119889= 119865 lowast 119868

119889(iso) + (1 minus 119865) lowast 119868

119889(disk) (17)






119868119889= 119868119889(iso) (1 + 119870sin3 (

120573

2

)) (1 + 119870cos2120579sin3120579119911) (18)








120573 = tanminus1 [ (119867119900+ (1 minus 119865)119867

119889)



times [(119867119900+119865119867119889

2

+ (1 minus 119865)119867119889minus

119867119892120588

2

)


(19)

120574 = tanminus1 [ (119867119900+ (1 minus 119865)119867

119889) cos 120575 sin120596]

times [(119867119900+ (1 minus 119865)119867

119889)


(20)

120596 = sinminus1 [ (119867119900+ (1 minus 119865)119867


plusmn [[(119867119900+ (1 minus 119865)119867

119889) sin 120575]4


minus [(119867119900+ (1 minus 119865)119867

119889) sin 120575 cos 120575]2

timessin2120593cos2120593sin4120574]12

]

times [[(119867119900+ (1 minus 119865)119867

119889) sin 120575]2


(21)




1198891= minus21269 507 1244 553 1305 533 1273 570 1854 117 126

1198892= minus13289 436 1713 475 1796 460 1759 487 2156 095 087

1198893= minus2819 337 2254 370 2361 358 2318 374 2406 072 028

1198894= 9415 219 2708 243 2824 235 2779 238 2601 045 042

1198895= 18792 129 2926 143 3016 138 2983 136 2756 025 050

1198896= 23314 86 3092 95 3153 92 3134 89 2931 017 044

1198897= 21517 106 3192 114 3255 112 3239 109 3000 015 050

1198898= 13784 184 3071 198 3158 194 3135 194 2877 025 055

1198899= 2217 295 2574 319 2677 312 2643 320 2590 050 022

11988910= minus9599 409 2016 440 2099 430 2069 449 2373 075 069

11988911= minus19148 485 1326 532 1394 511 1358 548 1879 118 113

11988912= minus23335 518 1064 572 1121 545 1087 589 1654 135 123



1198891= minus21269 501 658 605 742 535 673 602 989 222 066

1198892= minus13289 423 942 527 1066 458 968 522 1216 219 054

1198893= minus2819 330 1316 424 1494 362 1359 409 1465 187 037

1198894= 9415 247 2047 302 2259 271 2124 288 2013 103 047

1198895= 18792 171 2475 205 2665 187 2556 189 2323 061 062

1198896= 23314 136 2803 158 2958 148 2882 146 2603 039 066

1198897= 21517 157 2927 177 3082 169 3015 168 2697 036 073

1198898= 13784 229 2703 259 2883 247 2803 253 2533 055 065

1198899= 2217 336 2173 378 2349 360 2262 382 2239 090 031

11988910= minus9599 432 1421 495 1564 462 1477 506 1748 145 062

11988911= minus19148 482 738 585 832 516 756 582 1063 219 065

11988912= minus23335 523 592 625 667 556 606 622 935 218 069



1198891= minus21269 575 693 646 793 604 721 658 1147 166 091

1198892= minus13289 495 1056 569 1204 527 1102 581 1438 171 074

1198893= minus2819 401 1605 467 1815 431 1682 474 1812 147 044

1198894= 9415 28 2002 343 2267 307 2095 333 1977 122 056

1198895= 18792 202 2407 246 2666 222 2507 23 2246 079 076

1198896= 23314 168 2714 20 2945 184 2817 185 2490 057 083

1198897= 21517 19 2882 219 3108 207 2998 209 2621 052 090

1198898= 13784 265 2682 301 2919 286 2807 297 2499 070 078

1198899= 2217 365 1981 418 2206 392 2080 423 2075 115 040

11988910= minus9599 49 1529 536 1683 516 1604 553 1935 117 076

11988911= minus19148 557 801 626 913 586 835 639 1256 163 090

11988912= minus23335 615 705 667 794 64 738 684 1276 131 115




1198891= minus21269 567 398 681 494 599 410 665 654 233 051

1198892= minus13289 469 605 604 753 504 625 58 821 274 044

1198893= minus2819 383 1030 502 1266 417 1073 483 1194 241 047

1198894= 9415 287 1574 381 1895 317 1647 357 1592 180 064

1198895= 18792 204 1766 284 2100 228 1836 245 1694 145 076

1198896= 23314 183 2216 239 2546 204 2311 212 2059 100 088

1198897= 21517 211 2483 259 2813 233 2605 242 2276 087 097

1198898= 13784 278 2173 339 2506 305 2292 329 2072 117 080

1198899= 2217 39 1748 457 2024 42 1850 462 1867 146 049

11988910= minus9599 487 1022 571 1214 52 1075 579 1332 188 060

11988911= minus19148 581 593 663 712 611 621 669 967 183 073

11988912= minus23335 626 431 703 522 653 450 706 802 170 074


120597119868119879

120597120573

= minus 119867119900sin120573


minus cos 120575 sin 120574 sin120596)


minus119867119889sin120573

(1 + 119865sin3 (1205732))2

+ 3sin2 (120573

2

)119867119889119865

(1 + cos120573)2

minus119867119889119865 sin120573119902

111990221199023

2

+ 3119865sin2 (120573

2

) 11990211199022119867119889119865

(1 + cos120573)2

+

119867119892120588 sin1205732

+ 119867119889119865 (1 + cos120573) (radic1199021)



(22)

120597119868119879

120597120574


119900cos 120575 sin120593 cos120596 sin 120574

+119867119900cos 120575 cos 120574 sin120596)


+ (1 + 119865sin3 (120573

2

))119867119889119865 (1 + cos120573) 119902

2(radic1199021)



120597119868119879

120597120596

=



+ (1 + 119865sin3 (120573

2

))119867119889119865 (1 + cos120573) 119902

2radic1199021



+ (1 + 119865sin3 (120573

2

))119867119889119865 (1 + cos120573) (3

4

) 1199021


times (sin120596 cos 120575 cos120593) = 0(24)

where





32

1199023= [1 + 119865sin3 (

120573

2

)] (25)







1198891= minus21269 618 224 738 310 647 232 706 400 236 036

1198892= minus13289 521 439 661 595 555 456 63 626 280 041

1198893= minus2819 414 779 559 1046 449 814 526 926 290 052

1198894= 9415 32 1331 44 1736 353 1402 408 1370 233 080

1198895= 18792 258 1888 346 2365 286 1996 317 1786 145 109

1198896= 23314 207 1787 299 2255 231 1871 252 1684 169 107

1198897= 21517 217 1661 317 2120 242 1737 267 1583 184 103

1198898= 13784 282 1454 396 1885 312 1529 358 1444 217 090

1198899= 2217 391 1102 511 1437 426 1163 493 1224 243 113

11988910= minus9599 549 865 629 1084 58 923 638 1186 182 113

11988911= minus19148 631 368 720 484 659 387 713 653 185 106

11988912= minus23335 642 173 758 241 668 178 722 331 226 032


1 11988911 and 119889

12



1198892= minus13289 671 518 732 699 695 565 738 879 137 070

1198893= minus2819 547 1025 631 1377 579 1115 638 1256 188 068

1198894= 9415 431 1864 513 2432 464 2037 518 1832 180 119

1198895= 18792 330 2199 418 2910 362 2388 407 1995 177 170

1198896= 23314 295 2539 373 3292 324 2754 359 2219 152 195

1198897= 21517 298 2184 390 2912 329 2359 370 1968 178 175

1198898= 13784 338 1417 465 2012 372 1515 434 1398 250 125

1198899= 2217 463 1025 580 1445 498 1102 569 1152 244 079

11988910= minus9599 605 565 695 793 635 609 693 825 192 056







119904[37] Daily



119889and 119867










1and 119886



120573 (1198971)

Coefficients120573 (1198972) 120573 (119897

3) 120573 (119897

4) 120573 (119897

5) 120573 (119897

6) 119886

11198862

Correlation 119903





















1and 119886

2of a linear






30 35 40 45 50 55 60

30

35

40

45

50

55O

ptim

um ti

lt an

gle (

deg)

Latitude (deg)

(a)

30 35 40 45 50 55 60

35

40

45

50

55

60

Opt

imum

tilt

angl

e (de

g)

Latitude (deg)

(b)





30 35 40 45 50 55 60

SEM

of t

ilt an

gle v

alue

s

Latitude (deg)

02

04

06

08

1

12

14

16

18

2

22

(a)

30 35 40 45 50 55 60

SEM

of m

onth

ly av

erag

e of d

aily

sola

r rad

iatio

nva

lues

Latitude (deg)

04

06

08

1

12

14

16

(b)










5 Conclusions







References



















































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of





















119867119892

119867o= 119886 + 119887 (

119899

119873

) (1)

where 119867119892


119900









119867119900= 86400 lowast

119866119904119888

120587

times (1 + 0033 cos(2120587 119899

365

)) cos120593 cos 120575



(2)





119879 defined as

119870119879=

119867119892

119867119900

(3)






119889









119889= 119891(119867

119892) by





119867119889

119867119892

= 135 minus 161119870119879 (4)


119867119889

119867119892

= 100 minus 113119870119879 (5)



well




119887


119903


119868119879= 119868119887+ 119868119889+ 119868119903 (6)





119868119887=

cos 120579cos 120579119911

= 119867119887119877119887 (7)

119877119887= cos120573









119868119889(iso) =

119867119889(1 + cos120573)2

(9)





119868119903=

119867119892120588 (1 minus cos120573)

2

(10)








120597119868119879

120597120573

= 0

120597119868119879

120597120574

= 0

120597119868119879

120597120596

= 0

(11)


120597119868119879

120597120573

= sin120573(119867119900+119867119889

2

minus

119867119892120588

2

)

minus cos120573119867119900


+ cos 120575 sin 120574 sin120596)


120597119868119879

120597120574


minus cos 120574 cos 120575 sin120596 = 0

120597119868119879

120597120596



(12)


120573 = tanminus1 119867119900


+ cos 120575 sin 120574 sin120596)

times [(119867119900+119867119889

2

minus

119867119892120588

2

)


(13)



(14)




)


(15)






119889by means of



119868119889(disk) =

119867119889cos 120579

cos 120579119911

(16)



119868119889= 119865 lowast 119868

119889(iso) + (1 minus 119865) lowast 119868

119889(disk) (17)






119868119889= 119868119889(iso) (1 + 119870sin3 (

120573

2

)) (1 + 119870cos2120579sin3120579119911) (18)








120573 = tanminus1 [ (119867119900+ (1 minus 119865)119867

119889)



times [(119867119900+119865119867119889

2

+ (1 minus 119865)119867119889minus

119867119892120588

2

)


(19)

120574 = tanminus1 [ (119867119900+ (1 minus 119865)119867

119889) cos 120575 sin120596]

times [(119867119900+ (1 minus 119865)119867

119889)


(20)

120596 = sinminus1 [ (119867119900+ (1 minus 119865)119867


plusmn [[(119867119900+ (1 minus 119865)119867

119889) sin 120575]4


minus [(119867119900+ (1 minus 119865)119867

119889) sin 120575 cos 120575]2

timessin2120593cos2120593sin4120574]12

]

times [[(119867119900+ (1 minus 119865)119867

119889) sin 120575]2


(21)




1198891= minus21269 507 1244 553 1305 533 1273 570 1854 117 126

1198892= minus13289 436 1713 475 1796 460 1759 487 2156 095 087

1198893= minus2819 337 2254 370 2361 358 2318 374 2406 072 028

1198894= 9415 219 2708 243 2824 235 2779 238 2601 045 042

1198895= 18792 129 2926 143 3016 138 2983 136 2756 025 050

1198896= 23314 86 3092 95 3153 92 3134 89 2931 017 044

1198897= 21517 106 3192 114 3255 112 3239 109 3000 015 050

1198898= 13784 184 3071 198 3158 194 3135 194 2877 025 055

1198899= 2217 295 2574 319 2677 312 2643 320 2590 050 022

11988910= minus9599 409 2016 440 2099 430 2069 449 2373 075 069

11988911= minus19148 485 1326 532 1394 511 1358 548 1879 118 113

11988912= minus23335 518 1064 572 1121 545 1087 589 1654 135 123



1198891= minus21269 501 658 605 742 535 673 602 989 222 066

1198892= minus13289 423 942 527 1066 458 968 522 1216 219 054

1198893= minus2819 330 1316 424 1494 362 1359 409 1465 187 037

1198894= 9415 247 2047 302 2259 271 2124 288 2013 103 047

1198895= 18792 171 2475 205 2665 187 2556 189 2323 061 062

1198896= 23314 136 2803 158 2958 148 2882 146 2603 039 066

1198897= 21517 157 2927 177 3082 169 3015 168 2697 036 073

1198898= 13784 229 2703 259 2883 247 2803 253 2533 055 065

1198899= 2217 336 2173 378 2349 360 2262 382 2239 090 031

11988910= minus9599 432 1421 495 1564 462 1477 506 1748 145 062

11988911= minus19148 482 738 585 832 516 756 582 1063 219 065

11988912= minus23335 523 592 625 667 556 606 622 935 218 069



1198891= minus21269 575 693 646 793 604 721 658 1147 166 091

1198892= minus13289 495 1056 569 1204 527 1102 581 1438 171 074

1198893= minus2819 401 1605 467 1815 431 1682 474 1812 147 044

1198894= 9415 28 2002 343 2267 307 2095 333 1977 122 056

1198895= 18792 202 2407 246 2666 222 2507 23 2246 079 076

1198896= 23314 168 2714 20 2945 184 2817 185 2490 057 083

1198897= 21517 19 2882 219 3108 207 2998 209 2621 052 090

1198898= 13784 265 2682 301 2919 286 2807 297 2499 070 078

1198899= 2217 365 1981 418 2206 392 2080 423 2075 115 040

11988910= minus9599 49 1529 536 1683 516 1604 553 1935 117 076

11988911= minus19148 557 801 626 913 586 835 639 1256 163 090

11988912= minus23335 615 705 667 794 64 738 684 1276 131 115




1198891= minus21269 567 398 681 494 599 410 665 654 233 051

1198892= minus13289 469 605 604 753 504 625 58 821 274 044

1198893= minus2819 383 1030 502 1266 417 1073 483 1194 241 047

1198894= 9415 287 1574 381 1895 317 1647 357 1592 180 064

1198895= 18792 204 1766 284 2100 228 1836 245 1694 145 076

1198896= 23314 183 2216 239 2546 204 2311 212 2059 100 088

1198897= 21517 211 2483 259 2813 233 2605 242 2276 087 097

1198898= 13784 278 2173 339 2506 305 2292 329 2072 117 080

1198899= 2217 39 1748 457 2024 42 1850 462 1867 146 049

11988910= minus9599 487 1022 571 1214 52 1075 579 1332 188 060

11988911= minus19148 581 593 663 712 611 621 669 967 183 073

11988912= minus23335 626 431 703 522 653 450 706 802 170 074


120597119868119879

120597120573

= minus 119867119900sin120573


minus cos 120575 sin 120574 sin120596)


minus119867119889sin120573

(1 + 119865sin3 (1205732))2

+ 3sin2 (120573

2

)119867119889119865

(1 + cos120573)2

minus119867119889119865 sin120573119902

111990221199023

2

+ 3119865sin2 (120573

2

) 11990211199022119867119889119865

(1 + cos120573)2

+

119867119892120588 sin1205732

+ 119867119889119865 (1 + cos120573) (radic1199021)



(22)

120597119868119879

120597120574


119900cos 120575 sin120593 cos120596 sin 120574

+119867119900cos 120575 cos 120574 sin120596)


+ (1 + 119865sin3 (120573

2

))119867119889119865 (1 + cos120573) 119902

2(radic1199021)



120597119868119879

120597120596

=



+ (1 + 119865sin3 (120573

2

))119867119889119865 (1 + cos120573) 119902

2radic1199021



+ (1 + 119865sin3 (120573

2

))119867119889119865 (1 + cos120573) (3

4

) 1199021


times (sin120596 cos 120575 cos120593) = 0(24)

where





32

1199023= [1 + 119865sin3 (

120573

2

)] (25)







1198891= minus21269 618 224 738 310 647 232 706 400 236 036

1198892= minus13289 521 439 661 595 555 456 63 626 280 041

1198893= minus2819 414 779 559 1046 449 814 526 926 290 052

1198894= 9415 32 1331 44 1736 353 1402 408 1370 233 080

1198895= 18792 258 1888 346 2365 286 1996 317 1786 145 109

1198896= 23314 207 1787 299 2255 231 1871 252 1684 169 107

1198897= 21517 217 1661 317 2120 242 1737 267 1583 184 103

1198898= 13784 282 1454 396 1885 312 1529 358 1444 217 090

1198899= 2217 391 1102 511 1437 426 1163 493 1224 243 113

11988910= minus9599 549 865 629 1084 58 923 638 1186 182 113

11988911= minus19148 631 368 720 484 659 387 713 653 185 106

11988912= minus23335 642 173 758 241 668 178 722 331 226 032


1 11988911 and 119889

12



1198892= minus13289 671 518 732 699 695 565 738 879 137 070

1198893= minus2819 547 1025 631 1377 579 1115 638 1256 188 068

1198894= 9415 431 1864 513 2432 464 2037 518 1832 180 119

1198895= 18792 330 2199 418 2910 362 2388 407 1995 177 170

1198896= 23314 295 2539 373 3292 324 2754 359 2219 152 195

1198897= 21517 298 2184 390 2912 329 2359 370 1968 178 175

1198898= 13784 338 1417 465 2012 372 1515 434 1398 250 125

1198899= 2217 463 1025 580 1445 498 1102 569 1152 244 079

11988910= minus9599 605 565 695 793 635 609 693 825 192 056







119904[37] Daily



119889and 119867










1and 119886



120573 (1198971)

Coefficients120573 (1198972) 120573 (119897

3) 120573 (119897

4) 120573 (119897

5) 120573 (119897

6) 119886

11198862

Correlation 119903





















1and 119886

2of a linear






30 35 40 45 50 55 60

30

35

40

45

50

55O

ptim

um ti

lt an

gle (

deg)

Latitude (deg)

(a)

30 35 40 45 50 55 60

35

40

45

50

55

60

Opt

imum

tilt

angl

e (de

g)

Latitude (deg)

(b)





30 35 40 45 50 55 60

SEM

of t

ilt an

gle v

alue

s

Latitude (deg)

02

04

06

08

1

12

14

16

18

2

22

(a)

30 35 40 45 50 55 60

SEM

of m

onth

ly av

erag

e of d

aily

sola

r rad

iatio

nva

lues

Latitude (deg)

04

06

08

1

12

14

16

(b)










5 Conclusions







References



















































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of





















119868119887=

cos 120579cos 120579119911

= 119867119887119877119887 (7)

119877119887= cos120573









119868119889(iso) =

119867119889(1 + cos120573)2

(9)





119868119903=

119867119892120588 (1 minus cos120573)

2

(10)








120597119868119879

120597120573

= 0

120597119868119879

120597120574

= 0

120597119868119879

120597120596

= 0

(11)


120597119868119879

120597120573

= sin120573(119867119900+119867119889

2

minus

119867119892120588

2

)

minus cos120573119867119900


+ cos 120575 sin 120574 sin120596)


120597119868119879

120597120574


minus cos 120574 cos 120575 sin120596 = 0

120597119868119879

120597120596



(12)


120573 = tanminus1 119867119900


+ cos 120575 sin 120574 sin120596)

times [(119867119900+119867119889

2

minus

119867119892120588

2

)


(13)



(14)




)


(15)






119889by means of



119868119889(disk) =

119867119889cos 120579

cos 120579119911

(16)



119868119889= 119865 lowast 119868

119889(iso) + (1 minus 119865) lowast 119868

119889(disk) (17)






119868119889= 119868119889(iso) (1 + 119870sin3 (

120573

2

)) (1 + 119870cos2120579sin3120579119911) (18)








120573 = tanminus1 [ (119867119900+ (1 minus 119865)119867

119889)



times [(119867119900+119865119867119889

2

+ (1 minus 119865)119867119889minus

119867119892120588

2

)


(19)

120574 = tanminus1 [ (119867119900+ (1 minus 119865)119867

119889) cos 120575 sin120596]

times [(119867119900+ (1 minus 119865)119867

119889)


(20)

120596 = sinminus1 [ (119867119900+ (1 minus 119865)119867


plusmn [[(119867119900+ (1 minus 119865)119867

119889) sin 120575]4


minus [(119867119900+ (1 minus 119865)119867

119889) sin 120575 cos 120575]2

timessin2120593cos2120593sin4120574]12

]

times [[(119867119900+ (1 minus 119865)119867

119889) sin 120575]2


(21)




1198891= minus21269 507 1244 553 1305 533 1273 570 1854 117 126

1198892= minus13289 436 1713 475 1796 460 1759 487 2156 095 087

1198893= minus2819 337 2254 370 2361 358 2318 374 2406 072 028

1198894= 9415 219 2708 243 2824 235 2779 238 2601 045 042

1198895= 18792 129 2926 143 3016 138 2983 136 2756 025 050

1198896= 23314 86 3092 95 3153 92 3134 89 2931 017 044

1198897= 21517 106 3192 114 3255 112 3239 109 3000 015 050

1198898= 13784 184 3071 198 3158 194 3135 194 2877 025 055

1198899= 2217 295 2574 319 2677 312 2643 320 2590 050 022

11988910= minus9599 409 2016 440 2099 430 2069 449 2373 075 069

11988911= minus19148 485 1326 532 1394 511 1358 548 1879 118 113

11988912= minus23335 518 1064 572 1121 545 1087 589 1654 135 123



1198891= minus21269 501 658 605 742 535 673 602 989 222 066

1198892= minus13289 423 942 527 1066 458 968 522 1216 219 054

1198893= minus2819 330 1316 424 1494 362 1359 409 1465 187 037

1198894= 9415 247 2047 302 2259 271 2124 288 2013 103 047

1198895= 18792 171 2475 205 2665 187 2556 189 2323 061 062

1198896= 23314 136 2803 158 2958 148 2882 146 2603 039 066

1198897= 21517 157 2927 177 3082 169 3015 168 2697 036 073

1198898= 13784 229 2703 259 2883 247 2803 253 2533 055 065

1198899= 2217 336 2173 378 2349 360 2262 382 2239 090 031

11988910= minus9599 432 1421 495 1564 462 1477 506 1748 145 062

11988911= minus19148 482 738 585 832 516 756 582 1063 219 065

11988912= minus23335 523 592 625 667 556 606 622 935 218 069



1198891= minus21269 575 693 646 793 604 721 658 1147 166 091

1198892= minus13289 495 1056 569 1204 527 1102 581 1438 171 074

1198893= minus2819 401 1605 467 1815 431 1682 474 1812 147 044

1198894= 9415 28 2002 343 2267 307 2095 333 1977 122 056

1198895= 18792 202 2407 246 2666 222 2507 23 2246 079 076

1198896= 23314 168 2714 20 2945 184 2817 185 2490 057 083

1198897= 21517 19 2882 219 3108 207 2998 209 2621 052 090

1198898= 13784 265 2682 301 2919 286 2807 297 2499 070 078

1198899= 2217 365 1981 418 2206 392 2080 423 2075 115 040

11988910= minus9599 49 1529 536 1683 516 1604 553 1935 117 076

11988911= minus19148 557 801 626 913 586 835 639 1256 163 090

11988912= minus23335 615 705 667 794 64 738 684 1276 131 115




1198891= minus21269 567 398 681 494 599 410 665 654 233 051

1198892= minus13289 469 605 604 753 504 625 58 821 274 044

1198893= minus2819 383 1030 502 1266 417 1073 483 1194 241 047

1198894= 9415 287 1574 381 1895 317 1647 357 1592 180 064

1198895= 18792 204 1766 284 2100 228 1836 245 1694 145 076

1198896= 23314 183 2216 239 2546 204 2311 212 2059 100 088

1198897= 21517 211 2483 259 2813 233 2605 242 2276 087 097

1198898= 13784 278 2173 339 2506 305 2292 329 2072 117 080

1198899= 2217 39 1748 457 2024 42 1850 462 1867 146 049

11988910= minus9599 487 1022 571 1214 52 1075 579 1332 188 060

11988911= minus19148 581 593 663 712 611 621 669 967 183 073

11988912= minus23335 626 431 703 522 653 450 706 802 170 074


120597119868119879

120597120573

= minus 119867119900sin120573


minus cos 120575 sin 120574 sin120596)


minus119867119889sin120573

(1 + 119865sin3 (1205732))2

+ 3sin2 (120573

2

)119867119889119865

(1 + cos120573)2

minus119867119889119865 sin120573119902

111990221199023

2

+ 3119865sin2 (120573

2

) 11990211199022119867119889119865

(1 + cos120573)2

+

119867119892120588 sin1205732

+ 119867119889119865 (1 + cos120573) (radic1199021)



(22)

120597119868119879

120597120574


119900cos 120575 sin120593 cos120596 sin 120574

+119867119900cos 120575 cos 120574 sin120596)


+ (1 + 119865sin3 (120573

2

))119867119889119865 (1 + cos120573) 119902

2(radic1199021)



120597119868119879

120597120596

=



+ (1 + 119865sin3 (120573

2

))119867119889119865 (1 + cos120573) 119902

2radic1199021



+ (1 + 119865sin3 (120573

2

))119867119889119865 (1 + cos120573) (3

4

) 1199021


times (sin120596 cos 120575 cos120593) = 0(24)

where





32

1199023= [1 + 119865sin3 (

120573

2

)] (25)







1198891= minus21269 618 224 738 310 647 232 706 400 236 036

1198892= minus13289 521 439 661 595 555 456 63 626 280 041

1198893= minus2819 414 779 559 1046 449 814 526 926 290 052

1198894= 9415 32 1331 44 1736 353 1402 408 1370 233 080

1198895= 18792 258 1888 346 2365 286 1996 317 1786 145 109

1198896= 23314 207 1787 299 2255 231 1871 252 1684 169 107

1198897= 21517 217 1661 317 2120 242 1737 267 1583 184 103

1198898= 13784 282 1454 396 1885 312 1529 358 1444 217 090

1198899= 2217 391 1102 511 1437 426 1163 493 1224 243 113

11988910= minus9599 549 865 629 1084 58 923 638 1186 182 113

11988911= minus19148 631 368 720 484 659 387 713 653 185 106

11988912= minus23335 642 173 758 241 668 178 722 331 226 032


1 11988911 and 119889

12



1198892= minus13289 671 518 732 699 695 565 738 879 137 070

1198893= minus2819 547 1025 631 1377 579 1115 638 1256 188 068

1198894= 9415 431 1864 513 2432 464 2037 518 1832 180 119

1198895= 18792 330 2199 418 2910 362 2388 407 1995 177 170

1198896= 23314 295 2539 373 3292 324 2754 359 2219 152 195

1198897= 21517 298 2184 390 2912 329 2359 370 1968 178 175

1198898= 13784 338 1417 465 2012 372 1515 434 1398 250 125

1198899= 2217 463 1025 580 1445 498 1102 569 1152 244 079

11988910= minus9599 605 565 695 793 635 609 693 825 192 056







119904[37] Daily



119889and 119867










1and 119886



120573 (1198971)

Coefficients120573 (1198972) 120573 (119897

3) 120573 (119897

4) 120573 (119897

5) 120573 (119897

6) 119886

11198862

Correlation 119903





















1and 119886

2of a linear






30 35 40 45 50 55 60

30

35

40

45

50

55O

ptim

um ti

lt an

gle (

deg)

Latitude (deg)

(a)

30 35 40 45 50 55 60

35

40

45

50

55

60

Opt

imum

tilt

angl

e (de

g)

Latitude (deg)

(b)





30 35 40 45 50 55 60

SEM

of t

ilt an

gle v

alue

s

Latitude (deg)

02

04

06

08

1

12

14

16

18

2

22

(a)

30 35 40 45 50 55 60

SEM

of m

onth

ly av

erag

e of d

aily

sola

r rad

iatio

nva

lues

Latitude (deg)

04

06

08

1

12

14

16

(b)










5 Conclusions







References



















































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of



















(14)




)


(15)






119889by means of



119868119889(disk) =

119867119889cos 120579

cos 120579119911

(16)



119868119889= 119865 lowast 119868

119889(iso) + (1 minus 119865) lowast 119868

119889(disk) (17)






119868119889= 119868119889(iso) (1 + 119870sin3 (

120573

2

)) (1 + 119870cos2120579sin3120579119911) (18)








120573 = tanminus1 [ (119867119900+ (1 minus 119865)119867

119889)



times [(119867119900+119865119867119889

2

+ (1 minus 119865)119867119889minus

119867119892120588

2

)


(19)

120574 = tanminus1 [ (119867119900+ (1 minus 119865)119867

119889) cos 120575 sin120596]

times [(119867119900+ (1 minus 119865)119867

119889)


(20)

120596 = sinminus1 [ (119867119900+ (1 minus 119865)119867


plusmn [[(119867119900+ (1 minus 119865)119867

119889) sin 120575]4


minus [(119867119900+ (1 minus 119865)119867

119889) sin 120575 cos 120575]2

timessin2120593cos2120593sin4120574]12

]

times [[(119867119900+ (1 minus 119865)119867

119889) sin 120575]2


(21)




1198891= minus21269 507 1244 553 1305 533 1273 570 1854 117 126

1198892= minus13289 436 1713 475 1796 460 1759 487 2156 095 087

1198893= minus2819 337 2254 370 2361 358 2318 374 2406 072 028

1198894= 9415 219 2708 243 2824 235 2779 238 2601 045 042

1198895= 18792 129 2926 143 3016 138 2983 136 2756 025 050

1198896= 23314 86 3092 95 3153 92 3134 89 2931 017 044

1198897= 21517 106 3192 114 3255 112 3239 109 3000 015 050

1198898= 13784 184 3071 198 3158 194 3135 194 2877 025 055

1198899= 2217 295 2574 319 2677 312 2643 320 2590 050 022

11988910= minus9599 409 2016 440 2099 430 2069 449 2373 075 069

11988911= minus19148 485 1326 532 1394 511 1358 548 1879 118 113

11988912= minus23335 518 1064 572 1121 545 1087 589 1654 135 123



1198891= minus21269 501 658 605 742 535 673 602 989 222 066

1198892= minus13289 423 942 527 1066 458 968 522 1216 219 054

1198893= minus2819 330 1316 424 1494 362 1359 409 1465 187 037

1198894= 9415 247 2047 302 2259 271 2124 288 2013 103 047

1198895= 18792 171 2475 205 2665 187 2556 189 2323 061 062

1198896= 23314 136 2803 158 2958 148 2882 146 2603 039 066

1198897= 21517 157 2927 177 3082 169 3015 168 2697 036 073

1198898= 13784 229 2703 259 2883 247 2803 253 2533 055 065

1198899= 2217 336 2173 378 2349 360 2262 382 2239 090 031

11988910= minus9599 432 1421 495 1564 462 1477 506 1748 145 062

11988911= minus19148 482 738 585 832 516 756 582 1063 219 065

11988912= minus23335 523 592 625 667 556 606 622 935 218 069



1198891= minus21269 575 693 646 793 604 721 658 1147 166 091

1198892= minus13289 495 1056 569 1204 527 1102 581 1438 171 074

1198893= minus2819 401 1605 467 1815 431 1682 474 1812 147 044

1198894= 9415 28 2002 343 2267 307 2095 333 1977 122 056

1198895= 18792 202 2407 246 2666 222 2507 23 2246 079 076

1198896= 23314 168 2714 20 2945 184 2817 185 2490 057 083

1198897= 21517 19 2882 219 3108 207 2998 209 2621 052 090

1198898= 13784 265 2682 301 2919 286 2807 297 2499 070 078

1198899= 2217 365 1981 418 2206 392 2080 423 2075 115 040

11988910= minus9599 49 1529 536 1683 516 1604 553 1935 117 076

11988911= minus19148 557 801 626 913 586 835 639 1256 163 090

11988912= minus23335 615 705 667 794 64 738 684 1276 131 115




1198891= minus21269 567 398 681 494 599 410 665 654 233 051

1198892= minus13289 469 605 604 753 504 625 58 821 274 044

1198893= minus2819 383 1030 502 1266 417 1073 483 1194 241 047

1198894= 9415 287 1574 381 1895 317 1647 357 1592 180 064

1198895= 18792 204 1766 284 2100 228 1836 245 1694 145 076

1198896= 23314 183 2216 239 2546 204 2311 212 2059 100 088

1198897= 21517 211 2483 259 2813 233 2605 242 2276 087 097

1198898= 13784 278 2173 339 2506 305 2292 329 2072 117 080

1198899= 2217 39 1748 457 2024 42 1850 462 1867 146 049

11988910= minus9599 487 1022 571 1214 52 1075 579 1332 188 060

11988911= minus19148 581 593 663 712 611 621 669 967 183 073

11988912= minus23335 626 431 703 522 653 450 706 802 170 074


120597119868119879

120597120573

= minus 119867119900sin120573


minus cos 120575 sin 120574 sin120596)


minus119867119889sin120573

(1 + 119865sin3 (1205732))2

+ 3sin2 (120573

2

)119867119889119865

(1 + cos120573)2

minus119867119889119865 sin120573119902

111990221199023

2

+ 3119865sin2 (120573

2

) 11990211199022119867119889119865

(1 + cos120573)2

+

119867119892120588 sin1205732

+ 119867119889119865 (1 + cos120573) (radic1199021)



(22)

120597119868119879

120597120574


119900cos 120575 sin120593 cos120596 sin 120574

+119867119900cos 120575 cos 120574 sin120596)


+ (1 + 119865sin3 (120573

2

))119867119889119865 (1 + cos120573) 119902

2(radic1199021)



120597119868119879

120597120596

=



+ (1 + 119865sin3 (120573

2

))119867119889119865 (1 + cos120573) 119902

2radic1199021



+ (1 + 119865sin3 (120573

2

))119867119889119865 (1 + cos120573) (3

4

) 1199021


times (sin120596 cos 120575 cos120593) = 0(24)

where





32

1199023= [1 + 119865sin3 (

120573

2

)] (25)







1198891= minus21269 618 224 738 310 647 232 706 400 236 036

1198892= minus13289 521 439 661 595 555 456 63 626 280 041

1198893= minus2819 414 779 559 1046 449 814 526 926 290 052

1198894= 9415 32 1331 44 1736 353 1402 408 1370 233 080

1198895= 18792 258 1888 346 2365 286 1996 317 1786 145 109

1198896= 23314 207 1787 299 2255 231 1871 252 1684 169 107

1198897= 21517 217 1661 317 2120 242 1737 267 1583 184 103

1198898= 13784 282 1454 396 1885 312 1529 358 1444 217 090

1198899= 2217 391 1102 511 1437 426 1163 493 1224 243 113

11988910= minus9599 549 865 629 1084 58 923 638 1186 182 113

11988911= minus19148 631 368 720 484 659 387 713 653 185 106

11988912= minus23335 642 173 758 241 668 178 722 331 226 032


1 11988911 and 119889

12



1198892= minus13289 671 518 732 699 695 565 738 879 137 070

1198893= minus2819 547 1025 631 1377 579 1115 638 1256 188 068

1198894= 9415 431 1864 513 2432 464 2037 518 1832 180 119

1198895= 18792 330 2199 418 2910 362 2388 407 1995 177 170

1198896= 23314 295 2539 373 3292 324 2754 359 2219 152 195

1198897= 21517 298 2184 390 2912 329 2359 370 1968 178 175

1198898= 13784 338 1417 465 2012 372 1515 434 1398 250 125

1198899= 2217 463 1025 580 1445 498 1102 569 1152 244 079

11988910= minus9599 605 565 695 793 635 609 693 825 192 056







119904[37] Daily



119889and 119867










1and 119886



120573 (1198971)

Coefficients120573 (1198972) 120573 (119897

3) 120573 (119897

4) 120573 (119897

5) 120573 (119897

6) 119886

11198862

Correlation 119903





















1and 119886

2of a linear






30 35 40 45 50 55 60

30

35

40

45

50

55O

ptim

um ti

lt an

gle (

deg)

Latitude (deg)

(a)

30 35 40 45 50 55 60

35

40

45

50

55

60

Opt

imum

tilt

angl

e (de

g)

Latitude (deg)

(b)





30 35 40 45 50 55 60

SEM

of t

ilt an

gle v

alue

s

Latitude (deg)

02

04

06

08

1

12

14

16

18

2

22

(a)

30 35 40 45 50 55 60

SEM

of m

onth

ly av

erag

e of d

aily

sola

r rad

iatio

nva

lues

Latitude (deg)

04

06

08

1

12

14

16

(b)










5 Conclusions







References



















































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of




















1198891= minus21269 507 1244 553 1305 533 1273 570 1854 117 126

1198892= minus13289 436 1713 475 1796 460 1759 487 2156 095 087

1198893= minus2819 337 2254 370 2361 358 2318 374 2406 072 028

1198894= 9415 219 2708 243 2824 235 2779 238 2601 045 042

1198895= 18792 129 2926 143 3016 138 2983 136 2756 025 050

1198896= 23314 86 3092 95 3153 92 3134 89 2931 017 044

1198897= 21517 106 3192 114 3255 112 3239 109 3000 015 050

1198898= 13784 184 3071 198 3158 194 3135 194 2877 025 055

1198899= 2217 295 2574 319 2677 312 2643 320 2590 050 022

11988910= minus9599 409 2016 440 2099 430 2069 449 2373 075 069

11988911= minus19148 485 1326 532 1394 511 1358 548 1879 118 113

11988912= minus23335 518 1064 572 1121 545 1087 589 1654 135 123



1198891= minus21269 501 658 605 742 535 673 602 989 222 066

1198892= minus13289 423 942 527 1066 458 968 522 1216 219 054

1198893= minus2819 330 1316 424 1494 362 1359 409 1465 187 037

1198894= 9415 247 2047 302 2259 271 2124 288 2013 103 047

1198895= 18792 171 2475 205 2665 187 2556 189 2323 061 062

1198896= 23314 136 2803 158 2958 148 2882 146 2603 039 066

1198897= 21517 157 2927 177 3082 169 3015 168 2697 036 073

1198898= 13784 229 2703 259 2883 247 2803 253 2533 055 065

1198899= 2217 336 2173 378 2349 360 2262 382 2239 090 031

11988910= minus9599 432 1421 495 1564 462 1477 506 1748 145 062

11988911= minus19148 482 738 585 832 516 756 582 1063 219 065

11988912= minus23335 523 592 625 667 556 606 622 935 218 069



1198891= minus21269 575 693 646 793 604 721 658 1147 166 091

1198892= minus13289 495 1056 569 1204 527 1102 581 1438 171 074

1198893= minus2819 401 1605 467 1815 431 1682 474 1812 147 044

1198894= 9415 28 2002 343 2267 307 2095 333 1977 122 056

1198895= 18792 202 2407 246 2666 222 2507 23 2246 079 076

1198896= 23314 168 2714 20 2945 184 2817 185 2490 057 083

1198897= 21517 19 2882 219 3108 207 2998 209 2621 052 090

1198898= 13784 265 2682 301 2919 286 2807 297 2499 070 078

1198899= 2217 365 1981 418 2206 392 2080 423 2075 115 040

11988910= minus9599 49 1529 536 1683 516 1604 553 1935 117 076

11988911= minus19148 557 801 626 913 586 835 639 1256 163 090

11988912= minus23335 615 705 667 794 64 738 684 1276 131 115




1198891= minus21269 567 398 681 494 599 410 665 654 233 051

1198892= minus13289 469 605 604 753 504 625 58 821 274 044

1198893= minus2819 383 1030 502 1266 417 1073 483 1194 241 047

1198894= 9415 287 1574 381 1895 317 1647 357 1592 180 064

1198895= 18792 204 1766 284 2100 228 1836 245 1694 145 076

1198896= 23314 183 2216 239 2546 204 2311 212 2059 100 088

1198897= 21517 211 2483 259 2813 233 2605 242 2276 087 097

1198898= 13784 278 2173 339 2506 305 2292 329 2072 117 080

1198899= 2217 39 1748 457 2024 42 1850 462 1867 146 049

11988910= minus9599 487 1022 571 1214 52 1075 579 1332 188 060

11988911= minus19148 581 593 663 712 611 621 669 967 183 073

11988912= minus23335 626 431 703 522 653 450 706 802 170 074


120597119868119879

120597120573

= minus 119867119900sin120573


minus cos 120575 sin 120574 sin120596)


minus119867119889sin120573

(1 + 119865sin3 (1205732))2

+ 3sin2 (120573

2

)119867119889119865

(1 + cos120573)2

minus119867119889119865 sin120573119902

111990221199023

2

+ 3119865sin2 (120573

2

) 11990211199022119867119889119865

(1 + cos120573)2

+

119867119892120588 sin1205732

+ 119867119889119865 (1 + cos120573) (radic1199021)



(22)

120597119868119879

120597120574


119900cos 120575 sin120593 cos120596 sin 120574

+119867119900cos 120575 cos 120574 sin120596)


+ (1 + 119865sin3 (120573

2

))119867119889119865 (1 + cos120573) 119902

2(radic1199021)



120597119868119879

120597120596

=



+ (1 + 119865sin3 (120573

2

))119867119889119865 (1 + cos120573) 119902

2radic1199021



+ (1 + 119865sin3 (120573

2

))119867119889119865 (1 + cos120573) (3

4

) 1199021


times (sin120596 cos 120575 cos120593) = 0(24)

where





32

1199023= [1 + 119865sin3 (

120573

2

)] (25)







1198891= minus21269 618 224 738 310 647 232 706 400 236 036

1198892= minus13289 521 439 661 595 555 456 63 626 280 041

1198893= minus2819 414 779 559 1046 449 814 526 926 290 052

1198894= 9415 32 1331 44 1736 353 1402 408 1370 233 080

1198895= 18792 258 1888 346 2365 286 1996 317 1786 145 109

1198896= 23314 207 1787 299 2255 231 1871 252 1684 169 107

1198897= 21517 217 1661 317 2120 242 1737 267 1583 184 103

1198898= 13784 282 1454 396 1885 312 1529 358 1444 217 090

1198899= 2217 391 1102 511 1437 426 1163 493 1224 243 113

11988910= minus9599 549 865 629 1084 58 923 638 1186 182 113

11988911= minus19148 631 368 720 484 659 387 713 653 185 106

11988912= minus23335 642 173 758 241 668 178 722 331 226 032


1 11988911 and 119889

12



1198892= minus13289 671 518 732 699 695 565 738 879 137 070

1198893= minus2819 547 1025 631 1377 579 1115 638 1256 188 068

1198894= 9415 431 1864 513 2432 464 2037 518 1832 180 119

1198895= 18792 330 2199 418 2910 362 2388 407 1995 177 170

1198896= 23314 295 2539 373 3292 324 2754 359 2219 152 195

1198897= 21517 298 2184 390 2912 329 2359 370 1968 178 175

1198898= 13784 338 1417 465 2012 372 1515 434 1398 250 125

1198899= 2217 463 1025 580 1445 498 1102 569 1152 244 079

11988910= minus9599 605 565 695 793 635 609 693 825 192 056







119904[37] Daily



119889and 119867










1and 119886



120573 (1198971)

Coefficients120573 (1198972) 120573 (119897

3) 120573 (119897

4) 120573 (119897

5) 120573 (119897

6) 119886

11198862

Correlation 119903





















1and 119886

2of a linear






30 35 40 45 50 55 60

30

35

40

45

50

55O

ptim

um ti

lt an

gle (

deg)

Latitude (deg)

(a)

30 35 40 45 50 55 60

35

40

45

50

55

60

Opt

imum

tilt

angl

e (de

g)

Latitude (deg)

(b)





30 35 40 45 50 55 60

SEM

of t

ilt an

gle v

alue

s

Latitude (deg)

02

04

06

08

1

12

14

16

18

2

22

(a)

30 35 40 45 50 55 60

SEM

of m

onth

ly av

erag

e of d

aily

sola

r rad

iatio

nva

lues

Latitude (deg)

04

06

08

1

12

14

16

(b)










5 Conclusions







References



















































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of




















1198891= minus21269 567 398 681 494 599 410 665 654 233 051

1198892= minus13289 469 605 604 753 504 625 58 821 274 044

1198893= minus2819 383 1030 502 1266 417 1073 483 1194 241 047

1198894= 9415 287 1574 381 1895 317 1647 357 1592 180 064

1198895= 18792 204 1766 284 2100 228 1836 245 1694 145 076

1198896= 23314 183 2216 239 2546 204 2311 212 2059 100 088

1198897= 21517 211 2483 259 2813 233 2605 242 2276 087 097

1198898= 13784 278 2173 339 2506 305 2292 329 2072 117 080

1198899= 2217 39 1748 457 2024 42 1850 462 1867 146 049

11988910= minus9599 487 1022 571 1214 52 1075 579 1332 188 060

11988911= minus19148 581 593 663 712 611 621 669 967 183 073

11988912= minus23335 626 431 703 522 653 450 706 802 170 074


120597119868119879

120597120573

= minus 119867119900sin120573


minus cos 120575 sin 120574 sin120596)


minus119867119889sin120573

(1 + 119865sin3 (1205732))2

+ 3sin2 (120573

2

)119867119889119865

(1 + cos120573)2

minus119867119889119865 sin120573119902

111990221199023

2

+ 3119865sin2 (120573

2

) 11990211199022119867119889119865

(1 + cos120573)2

+

119867119892120588 sin1205732

+ 119867119889119865 (1 + cos120573) (radic1199021)



(22)

120597119868119879

120597120574


119900cos 120575 sin120593 cos120596 sin 120574

+119867119900cos 120575 cos 120574 sin120596)


+ (1 + 119865sin3 (120573

2

))119867119889119865 (1 + cos120573) 119902

2(radic1199021)



120597119868119879

120597120596

=



+ (1 + 119865sin3 (120573

2

))119867119889119865 (1 + cos120573) 119902

2radic1199021



+ (1 + 119865sin3 (120573

2

))119867119889119865 (1 + cos120573) (3

4

) 1199021


times (sin120596 cos 120575 cos120593) = 0(24)

where





32

1199023= [1 + 119865sin3 (

120573

2

)] (25)







1198891= minus21269 618 224 738 310 647 232 706 400 236 036

1198892= minus13289 521 439 661 595 555 456 63 626 280 041

1198893= minus2819 414 779 559 1046 449 814 526 926 290 052

1198894= 9415 32 1331 44 1736 353 1402 408 1370 233 080

1198895= 18792 258 1888 346 2365 286 1996 317 1786 145 109

1198896= 23314 207 1787 299 2255 231 1871 252 1684 169 107

1198897= 21517 217 1661 317 2120 242 1737 267 1583 184 103

1198898= 13784 282 1454 396 1885 312 1529 358 1444 217 090

1198899= 2217 391 1102 511 1437 426 1163 493 1224 243 113

11988910= minus9599 549 865 629 1084 58 923 638 1186 182 113

11988911= minus19148 631 368 720 484 659 387 713 653 185 106

11988912= minus23335 642 173 758 241 668 178 722 331 226 032


1 11988911 and 119889

12



1198892= minus13289 671 518 732 699 695 565 738 879 137 070

1198893= minus2819 547 1025 631 1377 579 1115 638 1256 188 068

1198894= 9415 431 1864 513 2432 464 2037 518 1832 180 119

1198895= 18792 330 2199 418 2910 362 2388 407 1995 177 170

1198896= 23314 295 2539 373 3292 324 2754 359 2219 152 195

1198897= 21517 298 2184 390 2912 329 2359 370 1968 178 175

1198898= 13784 338 1417 465 2012 372 1515 434 1398 250 125

1198899= 2217 463 1025 580 1445 498 1102 569 1152 244 079

11988910= minus9599 605 565 695 793 635 609 693 825 192 056







119904[37] Daily



119889and 119867










1and 119886



120573 (1198971)

Coefficients120573 (1198972) 120573 (119897

3) 120573 (119897

4) 120573 (119897

5) 120573 (119897

6) 119886

11198862

Correlation 119903





















1and 119886

2of a linear






30 35 40 45 50 55 60

30

35

40

45

50

55O

ptim

um ti

lt an

gle (

deg)

Latitude (deg)

(a)

30 35 40 45 50 55 60

35

40

45

50

55

60

Opt

imum

tilt

angl

e (de

g)

Latitude (deg)

(b)





30 35 40 45 50 55 60

SEM

of t

ilt an

gle v

alue

s

Latitude (deg)

02

04

06

08

1

12

14

16

18

2

22

(a)

30 35 40 45 50 55 60

SEM

of m

onth

ly av

erag

e of d

aily

sola

r rad

iatio

nva

lues

Latitude (deg)

04

06

08

1

12

14

16

(b)










5 Conclusions







References



















































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of




















1198891= minus21269 618 224 738 310 647 232 706 400 236 036

1198892= minus13289 521 439 661 595 555 456 63 626 280 041

1198893= minus2819 414 779 559 1046 449 814 526 926 290 052

1198894= 9415 32 1331 44 1736 353 1402 408 1370 233 080

1198895= 18792 258 1888 346 2365 286 1996 317 1786 145 109

1198896= 23314 207 1787 299 2255 231 1871 252 1684 169 107

1198897= 21517 217 1661 317 2120 242 1737 267 1583 184 103

1198898= 13784 282 1454 396 1885 312 1529 358 1444 217 090

1198899= 2217 391 1102 511 1437 426 1163 493 1224 243 113

11988910= minus9599 549 865 629 1084 58 923 638 1186 182 113

11988911= minus19148 631 368 720 484 659 387 713 653 185 106

11988912= minus23335 642 173 758 241 668 178 722 331 226 032


1 11988911 and 119889

12



1198892= minus13289 671 518 732 699 695 565 738 879 137 070

1198893= minus2819 547 1025 631 1377 579 1115 638 1256 188 068

1198894= 9415 431 1864 513 2432 464 2037 518 1832 180 119

1198895= 18792 330 2199 418 2910 362 2388 407 1995 177 170

1198896= 23314 295 2539 373 3292 324 2754 359 2219 152 195

1198897= 21517 298 2184 390 2912 329 2359 370 1968 178 175

1198898= 13784 338 1417 465 2012 372 1515 434 1398 250 125

1198899= 2217 463 1025 580 1445 498 1102 569 1152 244 079

11988910= minus9599 605 565 695 793 635 609 693 825 192 056







119904[37] Daily



119889and 119867










1and 119886



120573 (1198971)

Coefficients120573 (1198972) 120573 (119897

3) 120573 (119897

4) 120573 (119897

5) 120573 (119897

6) 119886

11198862

Correlation 119903





















1and 119886

2of a linear






30 35 40 45 50 55 60

30

35

40

45

50

55O

ptim

um ti

lt an

gle (

deg)

Latitude (deg)

(a)

30 35 40 45 50 55 60

35

40

45

50

55

60

Opt

imum

tilt

angl

e (de

g)

Latitude (deg)

(b)





30 35 40 45 50 55 60

SEM

of t

ilt an

gle v

alue

s

Latitude (deg)

02

04

06

08

1

12

14

16

18

2

22

(a)

30 35 40 45 50 55 60

SEM

of m

onth

ly av

erag

e of d

aily

sola

r rad

iatio

nva

lues

Latitude (deg)

04

06

08

1

12

14

16

(b)










5 Conclusions







References



















































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of



















1and 119886



120573 (1198971)

Coefficients120573 (1198972) 120573 (119897

3) 120573 (119897

4) 120573 (119897

5) 120573 (119897

6) 119886

11198862

Correlation 119903





















1and 119886

2of a linear






30 35 40 45 50 55 60

30

35

40

45

50

55O

ptim

um ti

lt an

gle (

deg)

Latitude (deg)

(a)

30 35 40 45 50 55 60

35

40

45

50

55

60

Opt

imum

tilt

angl

e (de

g)

Latitude (deg)

(b)





30 35 40 45 50 55 60

SEM

of t

ilt an

gle v

alue

s

Latitude (deg)

02

04

06

08

1

12

14

16

18

2

22

(a)

30 35 40 45 50 55 60

SEM

of m

onth

ly av

erag

e of d

aily

sola

r rad

iatio

nva

lues

Latitude (deg)

04

06

08

1

12

14

16

(b)










5 Conclusions







References



















































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of


















30 35 40 45 50 55 60

30

35

40

45

50

55O

ptim

um ti

lt an

gle (

deg)

Latitude (deg)

(a)

30 35 40 45 50 55 60

35

40

45

50

55

60

Opt

imum

tilt

angl

e (de

g)

Latitude (deg)

(b)





30 35 40 45 50 55 60

SEM

of t

ilt an

gle v

alue

s

Latitude (deg)

02

04

06

08

1

12

14

16

18

2

22

(a)

30 35 40 45 50 55 60

SEM

of m

onth

ly av

erag

e of d

aily

sola

r rad

iatio

nva

lues

Latitude (deg)

04

06

08

1

12

14

16

(b)










5 Conclusions







References



















































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of


















5 Conclusions







References



















































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of









































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of

















research article an algorithm to determine the optimum tilt...

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