applied sciences
Article Shading by Overhang PV Collectors
Joseph Appelbaum * , Avi Aronescu and Tamir Maor
Tel Aviv University School of Electrical Engineering, Tel Aviv 69978, Israel; [email protected] (A.A.); [email protected] (T.M.) * Correspondence: [email protected]
Received: 1 August 2019; Accepted: 30 September 2019; Published: 12 October 2019
Abstract: Photovoltaic modules integrated into buildings may provide shading to windows, doors and walls to protect against sun rays and at the same time generate ancillary electrical energy. The study develops the methodology for calculating the shadow variation cast by overhangs on doors, windows, carports, and calculates the annual incident energy (beam, diffuse and global energy) on overhangs made up of conventional and bifacial PV modules. The methodology of the present study is different from published articles including software programs. The study starts with shadows on walls cast by a horizontal pole and follows by shadows on walls cast by horizontal plates, inclined pole, inclined plate, and shaded area. The study deals also with overhangs placed one above the other. The calculation of the diffuse radiation involves the calculation of view factors to sky, to ground and between overhangs. In addition, the present study suggests using bifacial PV modules for overhangs and calculates the contribution of the reflective energy (5% and more) from walls and ground to the rear side of the bifacial PV module.
Keywords: BIPV; window shadowing; overhang PV collectors; bifacial PV collectors
1. Introduction Photovoltaic modules are often integrated into the building envelope and may provide shading, and at the same time perform as an ancillary source of electrical power. Building-integrated photovoltaics (BIPV) are photovoltaic materials used for many years as replacements for conventional building material in some parts of the building such as roofs, canopies, facades, solar carports and others. Overhangs on south-facing walls in northern latitudes are protruding structures and are important in shading windows and doors from undesired solar heat by blocking the sunlight during the summer months. References that illustrate the concept of BIPV are in mentioned in [1–9] to name only a few. There are many tools available for shading analysis and designing overhang shading structures (Solar Pathfinder, SunEye, Pilkington, Autodesk ECOTECT, METEONORM, Shadow Analyser, Sombrero, EnergyPlus, ESP-r, and many more). EnergyPlus is a building energy simulation program to model energy consumption and water use in buildings. ESP-r is a program that simulates the energy and environmental performance of buildings. Commercial ray tracing software programs are available and may be used for shading analysis (3Delight, Anim8or, ASAP). Not many publications deal with analytical expressions of the shadow variation on windows, walls and on grounds cast by overhang structures [10–15]. The methodology of the present study is different to in the above-mentioned articles including software programs. The study develops basic mathematical expressions, step by step, for calculating the shadow variation cast by overhangs on windows, carports, and for calculating the annual incident energy on overhangs made up of conventional and bifacial PV modules. The calculation of the diffuse radiation involves the calculation of view factors to sky, to ground and between overhangs. The study starts with the shadow on a wall cast by a horizontal pole and follows by shadows on a wall cast by a horizontal plate, inclined pole, inclined plate, shaded areas and overhangs placed one
Appl. Sci. 2019, 9, 4280; doi:10.3390/app9204280 www.mdpi.com/journal/applsci Appl. Sci. 2019, 9, 4280 2 of 14 Appl. Sci. 2019, 9, x FOR PEER REVIEW 2 of 14 Appl. Sci. 2019, 9, x FOR PEER REVIEW 2 of 14 otherabove (see the Figotherure (see 1). Figure The present1). The present study study provides provides an insight an insight and and methodology methodology for for calculating calculating other (see Figure 1). The present study provides an insight and methodology for calculating shadowsshadows and and incident incident radiation radiation on on PV PV modules modules deployed deployed on on overhangs overhangs in in BIPV. BIPV. shadows and incident radiation on PV modules deployed on overhangs in BIPV.
FigureFigure 1. 1.OverhangOverhang photovoltaic photovoltaic (PV (PV)) collector collector (blue) (blue) on on building building façade façade.. Figure 1. Overhang photovoltaic (PV) collector (blue) on building façade. 2.2. Materials Materials and and Methods Methods 2. Materials and Methods TheThe variation variation of of shadows shadows on on vertical vertical objects objects (walls, (walls, windows, windows, doors, etc.) doors, is important etc.) is importantinformation informationfor designingThe variation for shadingdesigning of elements shadows shading to elements on block vertical solar to block objects heat insolar buildings. (walls, heat in windows, buildings. The following doors, The following analysis etc.) is starts importantanalysis with startsshadowsinformation with castshadows for by designing poles cast andby polesshading extends and toelements extends plates. to blockplates. solar heat in buildings. The following analysis starts with shadows cast by poles and extends to plates. 2.1.2.1. Shadow Shadow on on a S aouth South Facing Facing W Wallsalls Cast Cast by by a H a Horizontalorizontal Pole Pole 2.1. Shadow on a South Facing Walls Cast by a Horizontal Pole AA horizontal horizontal pole pole bc bc of of length HH attachedattached perpendicularperpendicular to to a wall,a wall, is shownis shown in Figurein Fig2ure. The 2. The right rightedge edgeA “c” horizontal "c" casts casts a shadow apole shadow bc on of theon length the wall wall byH attached theby the solar solar raysperpendicular rays at point at point “a”. to "a". a wall, is shown in Figure 2. The right edge "c" casts a shadow on the wall by the solar rays at point "a".
FigureFigure 2. 2.HorizontalHorizontal pole pole attached attached perpendicular perpendicular to toa wall a wall.. Figure 2. Horizontal pole attached perpendicular to a wall. FromFrom Fig Figureure 22 it it follows: follows: From Figure 2 it follows: (1) Fx = K sin γ, Fy = J sin α (1) FKFJxysin , sin FKFJsin , sin (1) JKHKxy/ cos , cos (2) J = K/ cos α, H = K cos γ (2) JKHK/ cos , cos By substituting Equation (2) into(2) Equation (1), we get the coordinates of the shadow point "a", By substituting Equation (2) into Equation (1), we get the coordinates of the shadow point “a”, i.e., i.e., theBy shadow substituting variation Equation on the (2)wall into during Equation the day (1), maybe we get calculated: the coordinates of the shadow point "a", the shadow variation on the wall during the day maybe calculated: i.e., the shadow variation(3) on the wall during the day maybe calculated: FHx tan FH tan (3) x (4) Fx = H tan γ (3) FHy tan / cos FH tan / cos (4) wherey Fy = H tan α/cosγ (4) wheresin sin sin cos cos cos (5) sin sin sin cos cos cos wherecot [sin cos cos tan ] / sin (6)(5) cot [sin cos cos tan ] / sin and sin α = sin(6)φ sin δ + cos φ cos δ cos ω (5) and 15T 180 (7) cot γ = [sin φ cos ω cos φ tan δ]/ sin ω (6) 15is Tthe 180sun elevation(7) angle, is the sun azimuth− angle with respect to south, is the solar angleand ( is 0the noon, sun elevation 0 morning angle, and is the sun 0 azimuthafternoon) angle and with T respectis the to solar south, time. Theis the solar solar angle ( 0 noon, 0 morning and ω=150 Tafternoon)180 and T is the solar time. The solar (7) inclination angle is given by: − inclination angle(284 nis ) given by: 23.45sin (284 n ) (8) 23.45sin 365 (8) 365 Appl. Sci. 2019, 9, 4280 3 of 14
α is the sun elevation angle, γ is the sun azimuth angle with respect to south, ω is the solar angle (ω = 0 noon, ω < 0 morning and ω > 0 afternoon) and T is the solar time. The solar inclination angle δ is given by: Appl. Sci. 2019, 9, x FOR PEER REVIEW (284 + n) 3 of 14 δ = 23.45 sin (8) 365 Figure 3 shows the shadow component , F and F , of a pole attached prependicular to a wall Figure3 shows the shadow component, Fx and Fy,y of a pole attached prependicular to a wall at o atthe the origin origin 0,0,0 0,0,0 calculated calculated according according Equations Equation (3)–(8)s (3)– for(8) Telfor Aviv Tel Aviv (latitude (latitudeφ = 32 ◦N)32 forN the) for months the monthsDecember December to April to on April 21st day.on 21 Thest day. arrows The indicate arrows the indicate shadow the length shadow at given length hours at given beforenoon. hours 0 On vernal equinox (δ = 0 ) the Fy component0 is constant during the day. In afternoon hours, the beforenoon. On vernal equinox ( 0 ) the F component is constant during the day. In afternoon shadow is symmetrical with respect to the Y axis.y hours, the shadow is symmetrical with respect to the Y axis.
Figure 3. Shadow trajectory by horizontal pole on a wall, December to April on 21st day. Figure 3. Shadow trajectory by horizontal pole on a wall, December to April on 21st day. 2.2. Shadow on Walls Cast by Horizontal Plate 2.2. Shadow on Walls Cast by Horizontal Plate A horizontal plate bcde of length L and width H, attached perpendicular to a wall on side bd locatedA horizontal above a windowplate bcde (not of length shown), L and is shown width in H, Figure attached4. Theperpendicular trajectory ofto thea wall shadow on side on bd the locatedwindow above is shown a window in Figure (not5 on shown), 21st January is shown before in noon Figure (blue 4. The lines) trajectory and at 12:00 of the noon shadow (dashed on lines). the windowA plate mayis shown be considered in Figure as5 on composed 21st January of an before infinite noon number (blue of lines) poles; and therefore at 12:00 Equations noon (dashed (3)–(8) lines).are applied A plate to may obtain be theconsidered shadow as trajectory composed on theof an window. infinite At number 08:00 theof poles shadow; therefore takes the Eq formuation ofsa (3)parallelogram–(8) are applied (blue) to andobtain at noonthe shadow time the trajectory shadow ison rectangular the window. (yellow). At 08:00 the shadow takes the form ofNow a parallelogram we extend the (blue) calculation and at of noon a shadow time the on shadow a wall for is therectangular case where (yellow). the plate is inclined with an angle ε downwards, see Figure6. From Figure6 we write: H = A sin ε, Py = Fy + A cos ε and by substituting in Equation (4) we calculate the coordinates of point “a”, i.e., Px(see Equation (3)) and Py: Px = A sin ε tan γ (9)
Py = (A sin ε tan α/ cos γ) + Acosε (10) The shadow variation (Equations (9), (10)) on the wall during the day may now be obtained similarly to the procedure for the horizontal plate.
Figure 4. Horizontal plate attached perpendicular to a wall.
Appl. Sci. 2019, 9, x FOR PEER REVIEW 3 of 14
Figure 3 shows the shadow component , Fx and Fy , of a pole attached prependicular to a wall at the origin 0,0,0 calculated according Equations (3)–(8) for Tel Aviv (latitude 32o N ) for the months December to April on 21st day. The arrows indicate the shadow length at given hours 00 beforenoon. On vernal equinox ( ) the Fy component is constant during the day. In afternoon hours, the shadow is symmetrical with respect to the Y axis.
Figure 3. Shadow trajectory by horizontal pole on a wall, December to April on 21st day.
2.2. Shadow on Walls Cast by Horizontal Plate
A horizontal plate bcde of length L and width H, attached perpendicular to a wall on side bd located above a window (not shown), is shown in Figure 4. The trajectory of the shadow on the window is shown in Figure 5 on 21st January before noon (blue lines) and at 12:00 noon (dashed lines). A plate may be considered as composed of an infinite number of poles; therefore Equations (3)Appl.–(8) Sci.are2019 applied, 9, 4280 to obtain the shadow trajectory on the window. At 08:00 the shadow takes 4the of 14 form of a parallelogram (blue) and at noon time the shadow is rectangular (yellow).
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Appl. Sci. 2019, 9, x FOR PEER REVIEW 4 of 14 Figure 4. Horizontal plate attached perpendicular to a wall. Figure 4. Horizontal plate attached perpendicular to a wall.
Figure 5. Shadow variation on a window on 21st January (blue lines) and at 12:00 noon (dashed lines).
Now we extend the calculation of a shadow on a wall for the case where the plate is inclined with an angle downwards, see Figure 6. Figure 5. Shadow variation on a window on 21st January (blue lines) and at 12:00 noon (dashed lines). Figure 5. Shadow variation on a window on 21st January (blue lines) and at 12:00 noon (dashed lines).
Now we extend the calculation of a shadow on a wall for the case where the plate is inclined with an angle downwards, see Figure 6.
Figure 6. Inclined plate attached to a wall. Figure 6. Inclined plate attached to a wall. 2.3. Shadow of a Horizontal Pole on a Horizontal Plate From Figure 6 we write: A horizontal pole bc of length H attached perpendicular to a wall at point b (origin 0,0,0), is shown HAPFAsin ,yy cos in Figure7. The plate is located at a distance R under the pole. The right edge "c" of the pole casts a
and by substituting in Equation (4) we calculate the coordinates of point "a", i.e., Px (see Equation shadow at point F on the plate,Figure and point 6. Inclined “d” cast plate a shadowattached to at a point wall. E, i.e., the length dc of the pole
cast(3)) theand shadow Py : EF on the plate at a right angle to the wall. The length bd of the pole cast a shadow bE on theFrom wall. Fig Theure shadow 6 we write: length EF is given(9) by (see Appendix A in [16]): PAx sin tan HAPFAsin , cos yy (10) PAy ( sin tan / cos ) Acos EF = ZF = H(1 R/Fy) (11) and by substituting in Equation (4) we calculate the −coordinates of point "a", i.e., P (see Equation The shadow variation (Equations (9), (10)) on the wall during the day may nowx be obtained and the distance OE is: (3))similarly and P toy : the procedure for the horizontal plate. OE = XF = RFx/Fy (12) PA sin tan (9) x where2.3. ShadowFx and ofF ay Hisorizontal given in P Equationsole on a Horizontal (3) and (4),Plate respectively. PA( sin tan / cos ) Acos (10) y A horizontal pole bc of length H attached perpendicular to a wall at point b (origin 0,0,0), is shownThe in shadow Figure 7.variation The plate (Eq isuation locateds (9), at a(10)) distance on the R wall under during the pole.the day The may right now edge be " obtainedc " of the similarlypole casts to a the shadow procedure at point for Ftheon horizontal the plate, plate.and point " d " cast a shadow at point E , i.e., the length dc of the pole cast the shadow EF on the plate at a right angle to the wall. The length bd of the 2.3. Shadow of a Horizontal Pole on a Horizontal Plate pole cast a shadow bE on the wall. The shadow length EF is given by (see Appendix A in [16]): A horizontal pole bc of length(11) H attached perpendicular to a wall at point b (origin 0,0,0), is EF ZFy H(1 R / F ) shown in Figure 7. The plate is located at a distance R under the pole. The right edge " c " of the and the distance OE is: pole casts a shadow at point F on the plate, and point " d " cast a shadow at point E , i.e., the length OE X RF /F (12) dc of the poleF cast x the y shadow EF on the plate at a right angle to the wall. The length bd of the
polewhere cast F ax shadowand Fy bEis given on the in wall.Equation The sshadow (3) and length(4), respectively. EF is given by (see Appendix A in [16]): (11) EF ZFy H(1 R / F ) and the distance OE is: (12) OE XF RF x/F y
where Fx and Fy is given in Equations (3) and (4), respectively. Appl. Sci. 2019, 9, 4280 5 of 14 Appl.Appl. Sci. Sci. 2019 2019, 9,, 9x, FORx FOR PEER PEER REVIEW REVIEW 5 5of of 14 14
FigureFigureFigure 7. 7. 7. ShadowShadow Shadow of ofof a aahorizontal horizontalhorizontal pole polepole on onon a aahorizontal horizontalhorizontal plate plate. plate. .
FigureFigureFigure 8 8 shows 8 shows shows the the shadow the shadow shadow componenet, componenet, componenet,EF andEFEFOE and ,and of a OE poleOE, H, of of= a 0.941 a pole polem attached HmHm0.9410.941 prependicular attached attached prependiculartoprependicular a wall at a distance to to a a wall wallR = at at 3.0 a am distance distanceabove aRm horizontalRm3.03.0 aboveabove plate, a a calculated horizontal horizontal according plate, plate, calculated calculated to Equations according according (11)–(12) to to for Tel Aviv (latitude φ = 32 N) for the months May,o o June and July on the 21st day. The arrows indicate EqEquationuations s(11) (11)–(12)–(12) for for Tel Tel Aviv ◦Aviv (latitude (latitude 3232NN) for) for the the months months May, May, June June and and July July on on the the 21 21stst the shadow length at given hours before noon. day.day. The The arrows arrows indicate indicate the the shadow shadow length length at at given given hours hours before before noon. noon.
Figure 8. Shadow trajectory of a horizontal pole on a horizontal plate. FigureFigure 8. 8. Shadow Shadow trajectory trajectory of of a ahorizontal horizontal pole pole on on a ahorizontal horizontal plate plate. . 2.4. Shadow of an Inclined Pole on a Horizontal Plate 2.4.2.4. Shadow Shadow of of a na nInclined Inclined P olePole on on a aH Horizontalorizontal P latePlate Proceeding with our analysis, we develop now the expression for a shadow cast on a horizontal plate by an inclined pole of length A attached downwards to a wall with an angle ε with respect to the wall, see Figure9. Based on Appendix A in [ 16] and after some mathematical manipulation one obtains the coordinates of points F and E, respectively:
Px(R A cos ε) XF = − (13) Py A cos ε −
YF = R (14)
A sin ε(Py R) ZF = − (15) Py A cos ε − XE = RPx/Py (16) Y = R (17) E ZE = 0 (18) FigureFigure 9. 9. Shadow Shadow of of an an inclined inclined pole pole on on a ahorizontal horizontal plate plate. .
ProceedingProceeding with with our our analysis, analysis, we we develop develop now now the the expression expression for for a ashadow shadow cast cast on on a ahorizontal horizontal plateplate by by an an inclined inclined pole pole of of length length AA attached attached downwards downwards to to a awall wall with with an an angle angle with with respec respect t toto the the wall, wall, see see Fig Figureure 9 .9 Based. Based on on Appendix Appendix A A in in [16] [16] and and after after some some mathematical mathematical manipulation manipulation oneone obtains obtains the the coordinates coordinates of of points points F F and and EE, respectively:, respectively: Appl. Sci. 2019, 9, x FOR PEER REVIEW 5 of 14
Figure 7. Shadow of a horizontal pole on a horizontal plate.
Figure 8 shows the shadow componenet, EF and OE , of a pole Hm 0.941 attached prependicular to a wall at a distance Rm 3.0 above a horizontal plate, calculated according to Equations (11)–(12) for Tel Aviv (latitude 32o N ) for the months May, June and July on the 21st day. The arrows indicate the shadow length at given hours before noon.
Appl. Sci. 2019, 9, x FORFigure PEER REVIEW 8. Shadow trajectory of a horizontal pole on a horizontal plate. 6 of 14
Appl. Sci. 2019, 9, 4280 6 of 14 2.4. ShadowPRA( of an I cosnclined ) Pole on a Horizontal Plate X x F (13) PAy cos (14) YRF
Asin (PRy ) Z F (15) PAy cos (16) XE RP x/ P y (17) YRE (18) Z E 0
2.5. Shadow by Inclined Plates We now analyze the mutual shading by two parallel plates, one above the other, attached to a wall and inclined with an angle downwards. The length and width of the plates are L and A , Figure 9. Shadow of an inclined pole on a horizontal plate. respectively, as shown inFigure Figure 9. Shadow 10. of an inclined pole on a horizontal plate. 2.5.Based Shadow on by Appendix Inclined Plates A in [16], the length of segments g and EF are given by: Proceeding with our analysis, we develop now the expression for a shadow cast on a horizontal g()/ R P P (19) plate byWe an now inclinedxy analyze pole the of mutual length shading A attached by two downwards parallel plates, to a one wall above with thean other,angle attached with torespec a wallt and inclined with an angle(20) ε downwards. The length and width of the plates are L and A, respectively, to theEF wall, A(1 see R Fig / Pyure ) 9. Based on Appendix A in [16] and after some mathematical manipulation as shown in Figure 10. one obtainsRADcos the coordinates (21) of points F and E , respectively:
Figure 10. Inclined plates one above another. Figure 10. Inclined plates one above another. Based on Appendix A in [16], the length of segments g and EF are given by: The shaded area is given by: (22) g = (R Px)/Py (19) EF() L g × and the relative shaded area S sh is given by: EF = A(1 R/Py) (20) − sh EF() L g R Px R S (1 ) (1 ) R =(23)A cos ε + D (21) A L Py Py L
The shaded area is given by: o For the case of horizontal plates ( 90 ) we have (see Equation (21) EF (L g) (22) × − R D, g ( D Px )/P, y EF A(1D/P) y and by substituting in Equations (9) and (10) we and the relative shaded area Ssh is given by: obtain: EF(H tan R cos ) / tan (24)EF (L g) R Px R Ssh = − = (1 ) (1 · ) (23) (25)A · L − Py · − Py L gR sin / tan · and theFor shaded the case area of is horizontal plates (ε = 90◦) we have (see Equation (21) R = D, g = (D sh × PxS)/Py, EFEF () L= A g(1 (26)D/P y) and by substituting in Equations (9) and (10) we obtain: − 0 EF A 0 gL where and EF = (H tan α R cos γ)/tanα (24) − Appl. Sci. 2019, 9, x FOR PEER REVIEW 7 of 14
Appl. Sci. 2019, 9, 4280 7 of 14 EF 0 and g 0 ; otherwise EF 0 and g 0
sh sh and S 0 for non-negative values for bothg = REFsin andγ/ tan g α otherwise S 0 . (25) For long collectors, LH , the shaded area is given by L() H x (see Figure 11) where and the shaded area is xkcos( ) (27) sh sc S = EF (L g) (26) and × − where 0 EF A and 0 g L; EF 0 and g 0; otherwise EF = 0 and g = 0 and Ssh 0 for kR /≤ tan ≤ (28) ≤ ≤ ≥ ≥ ≥ EF g Ssh = non-negativeSubstituting values Equation for both (28) inand Equationotherwise (27) we get: 0. L H L(H x) For long collectors, >>(29), the shaded area is given by (see Figure 11) where xRcos(sc ) / tan − and the relative shaded area is given by: x = k cos(γs γc) (27) L( H x ) R cos( ) − S sh 1 sc (30) and LHH tan k = R/ tan α sh (28) The exposed width x is bounded by the width H of the collector; see Figure 11, i.e., S 0 .
FigureFigure 11. 11. ShadedShaded area area on on bottom bottom collector collector..
3. ResultsSubstituting Equation (28) in Equation (27) we get:
x = R cos(γs γc)/ tan α (29) 3.1. Shaded Area − andThe the relativerelative shadedshaded areaarea, is in given percentage, by: on a bottom collector cast by an overhang top collector, horizontally installed, is depicted in Figure 12. The collector is of a long edge L and a width Hm 0.941 L(H x) R cos(γs γc) . The figure shows theSsh daily= variation− = of1 the relative− shaded area on April 21st (dotted(30) lines) and on June 21st (solid lines), for threeL H distances− DmHtan1.0,α 2.0, 3.0 . The starting and leaving × T T sh times Theof the exposed solar rays width onx theis bounded bottom collector by the width are denotedH of the by collector; cr and see Figurecs , respectively. 11, i.e., S On0. April ≥ 21st, no shading occurs between about 09:30 and 14:30 on the bottom collector for Dm 3.0 . The shaded3. Results area is smaller in winter months. The relative shaded area, in percentage, on the ground cast by an overhang roof of a carport is 3.1. Shaded Area depicted in Figure 13. The roof is of a long edge and the overhang width is Hm 5.0 . The figure showsThe the relative daily variation shaded area,of the in relative percentage, shaded on aarea bottom on April collector 21st cast (dotted by an lines) overhang and on top June collector, 21st (solidhorizontally lines), for installed, two distances is depictedDm in3.0,5.0 Figure 12.. At The noon collector hours is ofin asummer long edge months,L anda the width solarH =rays0.941 stillm . The figure shows the daily variation of the relative shaded area on April 21st (dotted lines) and on penetrate the parking area, and in winter months the parking area enjoys the sun. June 21st (solid lines), for three distances D = 1.0, 2.0, 3.0 m. The starting and leaving times of the solar rays on the bottom collector are denoted by Tcr and Tcs , respectively. On April 21st, no shading occurs between about 09:30 and 14:30 on the bottom collector for D = 3.0m. The shaded area is smaller in winter months. Appl.Appl. Sci. Sci.2019 2019, 9,, 4280 9, x FOR PEER REVIEW 8 of8 14of 14
Appl. Sci. 2019, 9, x FOR PEER REVIEW 8 of 14
Figure 12. Daily variation of percentage shaded area on the bottom collector on April ( ... ) and June (___)Figure 21st, 12.H = Daily0.941 variationm, D = 1.0, of percentage 2.0, 3.0 m. shaded area on the bottom collector on April (…) and June (___) 21st, Hm 0.941 , Dm 1.0, 2.0, 3.0 . The relative shaded area, in percentage, on the ground cast by an overhang roof of a carport is depicted in Figure 13. The roof is of a long edge and the overhang width is H = 5.0 m. The figure shows the daily variation of the relative shaded area on April 21st (dotted lines) and on June 21st (solid lines), forFigure two distances12. Daily variationD = 3.0, of 5.0 percentagem. At noon shaded hours area in summeron the bottom months, collector the solar on April rays (…) still and penetrate June (___) 21st, Hm 0.941 , Dm 1.0, 2.0, 3.0 . the parking area, and in winter months the parking area enjoys the sun.
Figure 13. Daily variation of percentage shaded area on the ground on April (…) and June (___) 21st, Hm 5.0 , Dm 3.0, 5.0 .
3.2. Incident Radiation on Overhang PV Collectors
FigureInFigure this 13. 13.Daily section Daily variation variation we develop of percentageof percentage the equations shaded shaded area forarea on the theon groundthe incident ground on solar April on April (radiation... (…)) and and June on June a (___) single(___ 21st,) 21 andst, on multipleH =Hm5.0 overhangsm5.0, D =, 3.0,Dm attached 5.03.0,m. 5.0 to. south-facing vertical walls.
3.2. Incident Radiation on Overhang PV Collectors 3.2.1.3.2. Incident Single ORadiationverhang on C ollectorOverhang PV Collectors In this section we develop the equations for the2 incident solar radiation on a single and on multiple The global solar irradiance G , in (/)Wm, incident on an inclined overhang PV collector with overhangsIn this attached section to south-facing we develop vertical the equations walls. for the incident solar radiation on a single and on multiple overhangs attached to south-facing vertical walls. an angle consists of the direct beam Gb , diffuse irradiance on a horizontal plane Gdh , reflected 3.2.1.irradiance SingleOverhang from the wall Collector above the overhang, and the sky and wall view factors: 3.2.1. Single Overhang Collector 2 (31) TheG global Gb cos solar irradianceF H sky G dh G refε, in wall (W F H/m wall) , G incident g on an inclined overhang PV collector with an 2 angle ε consistsThe global of thesolar direct irradiance beam G bG, di, inffuse (/)Wm irradiance, incident on a horizontal on an inclined plane overhangGdh, reflected PV collector irradiance with fromwhere the wall is above the angle the overhang, between the and solar the skybeam and and wall the view normal factors: to the collector given by [16]: an angle consists of the direct beam Gb , diffuse irradiance on a horizontal plane Gdh , reflected cos cos sin sin cos cos( ) (32) irradiance from the wall above the overhang,sc and the sky and wall view factors: Gε = Gb cos θ + FH sky Gdh + re fwall FH wall Gg (31) The angles involved in solar· calculation→ are· shown(31) in Fig· ure→ 14. · G Gb cos F H sky G dh ref wall F H wall G g where θis theis the angle sun between elevation the angle; solar beam is andthe thecollector normal inclination to the collector angle givenwith respect by [16]: to horizontal where is the angle between the solar beam and the normal to the collector given by [16]: plane, 90 ( is the collector angle with respect to the wall) and s c is the = + ( ) difference between the suncos andθ collectorcos β sin azimuthsα (32)sin β withcos α respectcos γs toγ csouth. G is the global incident(32) cos cos sin sin cos cos( sc ) − g The angles involved in solar calculation are shown in Figure 14. irradianceThe angles on involvedthe wall; inFH solar sky is calculation the view factor are shown of the in surface Figure 14H. to sky; FH wall is the view factor α is the is sun the elevation sun elevation angle; angle;wallβ is the collector is the collector inclination inclination angle with angle respect with to respect horizontal to horizontal plane, of the surface to wall; ref is the reflectance of the wall above the collector. β =plane,90 ε (ε is90 the collector ( is angle the with collector respect angle to the with wall) respect and γ = to γ thes γ wall)c is the and diff erence s between c is the −The collector is open to sky therefore the view factor to sky of a− single collector is [17]: thedifference sun and collector between azimuths the sun and with collector respect toazimuths south. G withg is the respect global to incident south. G irradiance is the global on the incident wall; (33) g FH sky (1 cos ) / 2 (1 sin ) / 2 irradiance on the wall; F is the view factor of the surface H to sky; F is the view factor and view factor of the collectorH sky to the wall is given by: H wall of the surface to wall; ref wall is the reflectance of the wall above the collector. The collector is open to sky therefore the view factor to sky of a single collector is [17]: (33) FH sky (1 cos ) / 2 (1 sin ) / 2 and view factor of the collector to the wall is given by: Appl. Sci. 2019, 9, x FOR PEER REVIEW 9 of 14
(34) FH wall (1 cos ) / 2 (1 sin ) / 2 refwall F G Appl. Sci.The2019 reflected, 9, 4280 irradiance, H wall g , of the wall on the collector is small with respect9 of 14to 60o f the beam and diffuse irradiance. For , for example, FH wall 0.067 and multiplying by a high
FreflectanceH sky is the coefficient view factor of the of the wall surface wouldH resultto sky; onlyFH inwall aboutis the 2% view of the factor global of irradiance the surface GH .to wall; → → re f wall is the reflectance of the wall above the collector.
FigureFigure 14. 14.Angles Angles involvedinvolved inin solarsolar radiationradiation calculations. calculations.
The collector is open to sky therefore the view factor to sky of a singletop collector is [17]: The yearly direct beam incident irradiation on the top collector, qb , per unit area, is given in
2 Wh/ m by: FH sky = (1 + cosβ)/2 = (1 + sin ε)/2 (33) → n365 TS top andq viewbb factor of G thecos collector T (35) to the wall is given by: nT1 R top 2 and the diffuse incident radiationFH qwall ,= per (1 unitcos area,β)/2 in= Wh (1 / sin m εis)/ given2 (see Equation (33)) by: (34) → d − − n365 T SS top 1 sin wall qThe reflected() irradiance, Gre T f FH wall Gg, of the wall on the collector is small with respect to d dh (36)· → · 2 nT1 f SR = = the beam and diffuse irradiance. For ε 60◦, for example,FH wall 0.067 and multiplying by a high → reflectancewhere ΔT is coe theffi cientsummation of the walltime would interval result (for onlysolar indata about sampled 2% of every the global hour irradiance T=1) fromG εsun. rise TR top The yearly direct beam incident irradiation on the top collector, q , per unit area, is given in to sunset TS on the collector for the beam irradiance, and from sun riseb TSR to sun set TSS for the Wh/m2 by: diffuse irradiance. The other summation is nfrom=365 TJanuaryS 1 (n = 1) to December 31 (n = 365). top X X q = Gb cos θ∆T (35) b 3.2.2. Multiple Overhang Collectors n=1 TR top and theIn dimultipleffuse incident overhang radiation collectorsq ,the per top unit collector area, in casts Wh/ ma shadow2 is given on (see the Equation bottom collector (33)) by: forming a shaded area as shown in the Figured 11. b The yearly direct beam incident irradiation on nthe=365 bottomTSS collector, q , per unit area, is given top 1 + sin ε X X b 2 q = ( ) Gdh∆T (36) in Wh/ m by: d 2 × n=1 TSR n365 T S qbot. Gcos (1 S sh ) T wherebb∆T is the summation time interval (37) (for solar data sampled every hour ∆T = 1) from sun rise
nT1 R TR to sunset TS on the collector for the beam irradiance, and from sun rise TSR to sun set TSS for the diwhereffuse irradiance.S sh is given The by other Equation summation (29). is from January 1 (n = 1) to December 31 (n = 365).
bot. 3.2.2.The Multiple yearly Overhang diffuse incident Collectors irradiation on the bottom collector, qd , per unit area, is given in 2 WhIn/ m multiple by: overhang collectors the top collector casts a shadow on the bottom collector forming a n365 T shaded area as shownSS in the Figure 11. qbot.. F bot G T Thed yearly d direct dh beam incident(38) irradiation on the bottom collector, qb , per unit area, is given in nT1 b 2 SR Wh/m by: bot. where the sky view factor F of the bottomT collector is calculated based on the "Crossed-String d nX=365 XS qbot. = G cos θ(1 Ssh)∆T (37) Method" [18], see Figure 15: b b − n=1 TR where Ssh is given by Equation (29). Appl. Sci. 2019, 9, 4280 10 of 14
bot. The yearly diffuse incident irradiation on the bottom collector, qd , per unit area, is given in Wh/m2 by: nX=365 XTSS bot. = bot. qd Fd Gdh∆T (38) n=1 TSR bot. where the sky view factor Fd of the bottom collector is calculated based on the “Crossed-String Appl.Method” Sci. 2019 [18, 9],, x see FOR Figure PEER REVIEW 15: 10 of 14 AC + BC AB b = Fd − (39) b AC BC AB 2AC Fd (39) resulting in: 2AC resulting in: 2 2 1/2 bot. H + D + H cos ε [D + (H sin ε) ] 2= 2 1/2 Fd − (40) bot. HDHDH cos [ ( sin ) ] 2H F (40) d 2H
Figure 15. Calculation of sky view factor. Figure 15. Calculation of sky view factor. The yearly global incident irradiation on the top and bottom collectors, respectively, in Wh/m2, are, The yearly global incident irradiation on the top and bottom collectors, respectively, in Wh/ m 2 top = top + top , are, qy qb qd (41) top top top q q q (41) y b d bot. bot. bot. qy = q + q (42) bot... bot bot (42) b d qy q b q d Multiplying Equations (41) and (42) by the collector area H L results in the collectors’ yearly HL× incidentMultiplying energies Eq inuationWh. Tables (41)1 and shows (42) theby the results collector of yearly area incident results energies, in the in collectorskWh, on’ the yearly top incident energies in Wh . Table 1 shows the results of yearly incident energies, in kWh , on the top and bottom PV overhang collectors for three distances R = 1.0, 2.0, 3.0 m, H = 0.941m, L = 30m, ε = and bottom PV overhang collectors for three distances Rm 1.0,2.0,3.0 , 70◦(β = 20◦), (see Figure 11 and Equations (35)–(38), (40)–(42)) for Tel Aviv (latitude φ = 32◦N). oo EHb top0.941, Ed mtop ,, LE g 30top mare , the 70 beam, ( 20 di )ff,use (see and Figure global 11 and energy Equation on thes (35 top)–( collector,38), (40)–(42) respectively) for Tel Aviv and − − − Eb bot., Ed bot.,32Ego Nbot. areEEE the,, beam, diffuse and global energy on the bottom collector, respectively. The (latitude− − − ). b top d top g top are the beam, diffuse and global energy on the top collector, percent difference in global energy between the top and bottom collectors are 48.87, 32.95, 24.48%, for EEE,, respectivelyR = 1.0, 2.0, 3.0 andm , respectively.b bot... d bot This g bot di ffareerence the stems beam, from diffuse shading and on global the bottom energy collector on the and bottom from collector,its lower skyrespectively. view factor. The percent difference in global energy between the top and bottom collectors are 48.87, 32.95, 24.48%, for Rm 1.0,2.0,3.0 , respectively. This difference stems from shading on the bottomTable collector 1. Annual and incident from its energies lower on sky top view and bottom factor. PV overhang collectors, H = 0.941m, L = 30m, ε = 70◦; see Figure 11.
Table 1. AnnualEb top(kWh incident) Ed top energies(kWh) onEg top( topkWh ) and Eb bottombot.(kWh ) PV Ed overhangbot.(kWh) collectors,Eg bot.(kWh ) R(m) Distance − − o − − − − H0.941 m , LBeam-top 30 m , 70Di;ff seeuse-top Figure 11Global-top. Beam-bottom Diffuse-bottom Global-bottom 1.0 38,644 15,297 53,941 20,654 6,928 27,582 2.0 38,644 15,297 53,941 27,418 8,750 36,168 Rm()3.0 38,644 15,297 53,941 30,805 9,392 40,197 Eb top () kWh Ed top () kWh Eg top () kWh Eb bot. () kWh Ed bot. () kWh Eg bot. () kWh Distance 3.3. Bifacial PV ModulesBeam-top Diffuse-top Global-top Beam-bottom Diffuse-bottom Global-bottom Bifacial1.0 PV modules38,644 have15,297 been developed53,941 with the purpose20,654 of enhancing6,928 the power27,582 output over conventional2.0 (mono-facial)38,644 PV15,297 modules because53,941 bifacial modules27,418 can absorb8,750 solar radiation36,168 from both the front3.0 and the3 rear8,644 side. Their15,297 applications 53,941 are useful when30,805the nearby9,392 ground or other40,197 surfaces are
3.3. Bifacial PV Modules Bifacial PV modules have been developed with the purpose of enhancing the power output over conventional (mono-facial) PV modules because bifacial modules can absorb solar radiation from both the front and the rear side. Their applications are useful when the nearby ground or other surfaces are highly reflective [19–22]. The application and mainly the physics of bifacial cells are presented in the dissertation [19]. A study by simulation and field testing of vertical installed bifacial modules appear in [20]. Reference [21] compares the performance of vertical and south facing bifacial PV collector fields. A recent article [22] on simulating the energy yield of a bifacial Appl. Sci. 2019, 9, 4280 11 of 14 Appl. Sci. 2019, 9, x FOR PEER REVIEW 11 of 14 photovoltaichighly reflective plant [19 investigates–22]. The application the influence and mainly of ground the physics size, cast of bifacial ground cells shadow are presented and ground in the reflectivitydissertation on [19 the]. A energy study by yield. simulation The only and reference, field testing as of far vertical as we installed know, on bifacial bifacial modules PV modules appear suggestedin [20]. Reference for window [21] compares shading theby overhangsperformance is of mentioned vertical and in south [23]. The facing overhang bifacial PV use collectors white semitransparentfields. A recent articlereflector [22 sheet] on simulating placed behind the energy the bifacial yield modules of a bifacial to enhance photovoltaic the reflected plant investigates radiation onthe the influence rear side of ground of the size, PV module. cast ground The shadow present and study ground suggests reflectivity using on bifacial the energy PV modules yield. The in only its naturalreference, manner as far withoutas we know, providing on bifacial additional PV modules external suggested reflective for material, window i.e., shading using by the overhangs natural reflectanceis mentioned from in walls [23]. Theand overhanggrounds of uses the whitebuilding semitransparent to assess the additional reflector sheet energy placed contribution behind theof thebifacial rear side modules of the to PV enhance bifacial the module. reflected radiation on the rear side of the PV module. The present study suggests using bifacial PV modules in its natural2 manner without providing additional external The global incident irradiance G , in (/)Wm, on the front side of the PV collector modules reflective material, i.e., using the natural reflectance from walls and grounds of the building to assess inclined by with respect to a wall is given by [24] (see Figure 16): the additional energyf contribution f of the f rear side of the PV bifacial module. G G cos F G ref F G 2 (43) The globalb incident H sky irradiance dh wallGε, H in wall (W/m g ), on the front side of the PV collector modules inclined by εThewith global respect incident to a wall irradiance is given by on [24 the] (see rear Figure side 16 of): the PV collector modules consists of the irradiance from ground, from shaded wall S and from unshaded wall: f f f rG = G rcos θ + F G r + re f F G (43) G al F G refε Fb G refH sky Fdh wall G H(44)wall g H grd. dh wall . s H · wall . s dh→ wall . ush· H wall . u sh g· → · f f whereThe FH global sky isincident the view irradiance factor of on front the rearsurface side ofH the to PV sky collector; FH wall modules is the view consists factor of the of irradiancethe front
from ground, from shadedr wall S and from unshaded wall: r surface H to wall; FH grd . is the view factor of the rear surface to ground; FH wall. s is the view = r + r r + r factor of rear G sideε al surfaceFH grd to. shadedGdh re fwallwall.s; FFH wall.sG isdh there fviewwall.ush factorFH wall of .ush rearG g surface H (44) to · → · · H→ wall. ush · → · f unshadedf wall; al is the ground albedo; refwall is the reflectancef of the wall above the collector; where FH sky is the view factor of front surface H to sky; FH wall is the view factor of the front surface ref → r ref → r H wall. s is theF reflectance of the shaded wall; wall . ush His the reflectanceF of the unshaded wall below to wall; H grd. is the view factor of the rear surface to ground; H wall.s is the view factor of rear the collector. → Fr →H al side surface to shaded wall; H wall.ush is the view factor of rear surface to unshaded wall; is the The different viewf factor expressions→ were developed in [24] and adaptations were made to the ground albedo; re f is the reflectance of the wall above the collector; re fwall.s is the reflectance of the configuration in Figwallure 16. In multiple overhangs, the bottom of the top collector receives also shaded wall; re fwall.ush is the reflectance of the unshaded wall below the collector. reflected radiation from the top of the bottom collector.
FigureFigure 16. 16. ViewView factor factor to to sky, sky, wall wall and and ground ground..
The different view factor expressions were developed in [24] and adaptations were made to the The view factor of the rear-side of the collector to shaded wall is developed in [24] and is: configuration in Figure 16. In multiple overhangs, the bottom of the top collector receives also reflected H S [ H2 S 2 2 HS cos ] 1/2 radiationr from the top of the bottom collector. FH wall. s (45) The view factor of the2H rear-side of the collector to shaded wall is developed in [24] and is: Note that the shaded wall S varies with time during the day with the position of the sun, see 1/2 Figure 5. H + S [H2 + S2 2HS cos ε] r = FH wall.s − − (45) The view factor of the rear→-side of the collector 2 toH unshaded wall is given in [24] and after manipulation we get: Note that the shaded wall S varies with time during the day with the position of the sun, see W[ H2 S 2 2 HS cos ] 1/2 FigureF r 5. wall. ush H wall. ush 2H (46) 2 2 1/2 [(SHWH cos wall. ush ) ( sin ) ] 2H The view factor between the rear-side of collector to the ground is based on [24] and after manipulation the view factor is: Appl. Sci. 2019, 9, 4280 12 of 14
The view factor of the rear-side of the collector to unshaded wall is given in [24] and after manipulation we get: 1/2 W +[H2+S2 2HS cos ε] Fr = wall.ush − H wall.ush 2H → 2 2 1/2 (46) [(S H cos ε+Wwall.ush) +(H sin ε) ] − − 2H The view factor between the rear-side of collector to the ground is based on [24] and after manipulation the view factor is:
1/2 H S W + [(H sin ε)2 + (S H cos ε + W )2] r = wall.ush wall.ush FH grd. − − − (47) → 2H Bifacial PV Overhang The contribution of the annual energy on the rear side of the bifacial overhang collector is demonstrated by an example for the following overhang given data:
0 H = 0.941m, L = 30m, S + Wwall.ush = 3.0m, ε = 70 , al = 0.1, re fwall.s = 0.1, re fwall.ush = 0.4
The rear side annual energy components are (using Equations (44)–(47)): ground reflected energy: 1055 kWh ; shaded wall reflected energy: 684 kWh ; unshaded wall reflected energy: 1281 kWh . The total annual reflected incident energy on the rear side of the overhang collector amounts to 3020 kWh, comprising 5.6% of the front side overhang energy. Based on the above-mentioned energies one may calculated the energy contribution for different reflectance coefficients of the ground, shaded wall and unshaded wall.
4. Discussion Not much analytical work has been published on the shadow variation on windows, walls and grounds including shaded areas cast by protruding structures such as overhangs. There are many tools available for shading analysis and designing overhang shading structures; however, they do not provide the insight and the mathematical methodology how these tools were derived, what are the assumptions, what was considered and was neglected. The general analytical expressions developed in the present study specify explicitly the various parameters of shading by overhangs and permit the flexibility in parameter variation. Based on the methodology and the expressions, one may extend the study on shadows for different overhang configurations. The novel contribution of the present study includes also the incident radiation on PV overhang collectors, installed one above the other, in the presence of shading. The effect of the sky view factor affecting the diffuse radiation on PV collectors was recently introduced and implemented in the calculation of incident radiation on PV modules on overhangs. A novel application of bifacial PV modules was introduced in the present study including the computation of the energy gain of the rear side of the bifacial PV modules.
5. Conclusions Building integrated photovoltaic structures are playing nowadays an important role in solar electric energy generation. Photovoltaic modules integrated into buildings may provide shading to windows, doors and carports, and, at the same time, generate electrical energy. The study develops basic mathematical expressions, step by step, for shadows cast by overhangs (single and multiple overhangs placed one above the other) and calculates the annual incident energy (beam, diffuse and global energy) on overhangs made up of PV modules. The calculation of the diffuse incident radiation involves the view factors to sky, to ground and between overhangs. The present study suggests using bifacial PV modules for overhangs to increase the PV output power contributed by the reflective energy from walls and ground to the rear side of the bifacial PV module. A power gain of more than 5% was demonstrated. The new approach for analyzing shadows cast by overhangs and the determination Appl. Sci. 2019, 9, 4280 13 of 14 of the incident radiation on PV overhang collectors provides an inside and methodology of shadow analysis on protruding structures in BIPV.
Author Contributions: J.A.—conceptualization, writing—original draft preparation; A.A.—software; T.M.—methodology. Funding: This research received no external funding and performed for a degree in engineering. Conflicts of Interest: The authors declare no conflict of interest.
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