Solar Panel Irradiation Exposure efficiency of solar panels with shadow

Transcription

1 Solar Panel Irradiation Exposure efficiency of solar panels with shadow Frits F.M. de Mul MEDPHYS Software & Services info(at)medphys.nl

2 Solar Panel Irradiation 1. Local Times, corrected for earth s wobbling and axis tilt 2. Irradiation angles and Solar Ray Tracking: zenith (elevation, altitude) and azimuth, as functions of Julian date and time ( Hour Angle ). 3. Exact sunrise and sunset times. 4. Irradiation efficiency compared with normal sun ray incidence. 5. Shadow lines of shading objects of a roof. 6. Clear and cloudy skies (direct vs. Diffuse irradiation). 7. Energy output (kwh/year). Corrections for: Equation of time (eccentricity of orbit and tilt of axis) Atmospheric refraction and extinction (scattering and absorption) Sunrise and sunset shifts due to finite diameter of solar disk 2

3 Solar Panel Irradiation Literature: For solar angles and times: best entrance: see Wikipedia Guttman,A., Extinction coefficient measurements on Clear Atmospheres and Thin Cirrus Clouds", Appl.Opt. 7, 12, 1968, M.L. Roderick, "Methods for Calculating Solar Position and Day Length..", RMTR 137, ISSN , 1992, Western Australian Dept. of Agriculture, Perth, Australia. B.H. Liu, R.C. Jordan, The Interrelationship and Characteristic Distribution of Direct, Diffuse and Total Solar Radiation, Solar Energy, 4, 1960, 1-9. For shadow projection and sun ray tracking: any Analytical Geometry textbook. 3

4 1. Local Times and Hour Angle Definitions: LT : Local Time : actual time at position (h) LST : Local Solar Time : time with the sun s highest altitude at 12:00 h (h). DT : Time difference with GMT : (h) (GMT = Greenwich Mean Time): LSTM : Local Standard Time Meridian: ( 0 ) LSTM = DT * / 24 h 4

5 EoT (min.) 1. Local Times and Hour Angle Equation of time (EoT): (in min.) Empirical relation; corrects for : wobbling of the earth s orbit, the tilt of the earth s axis. EoT 9.87sin(2 ) 360 ( D 81) cos 1.5sin D = number of the day since the start of the year Number of days since start of year 5

6 1. Local Times and Hour Angle Local Solar Time (LST) : (h) Correction of Local Time (LT) for: Equation of Time Position in the time zone LST LT EoT ( 360 LSTM ) ψ = longitude (in 0 ; +/- on eastern/western hemisphere) Hour angle (HRA): ( 0 ) Local Solar Time expressed in degrees Hour angle = 0 at solar 12:00 h, <0, >0 in morning, afternoon 360 HRA ( LST 12) 24 6

7 2. Irradiation angles: Solar Ray Tracking Julian Day Number (JDN) 153( M JDN d 5 2) 365. Y Y 4 M m 12. A 3 ; Y y 4800 y = year (A.D.) ; m = month ; d = day Y 100 A ; Y 32045; m A 12 NB. All these divisions have to be truncated: fractional parts dropped!!! Julian Day Time (JD) JD hour12 JDN 24 min sec Jan AD., 12:00 h => JD = Dec BC., 12:00 h => JD = 0 7

8 2. Irradiation angles: Solar Ray Tracking N Normal on surface Zenith angle Sun ray N S Elevation (Altitude) N Azimuth angle S S 8

9 2. Irradiation angles: Solar Ray Tracking N Elevation (Altitude) and Azimuth angles: θ and φ sin cos( HRA).cos.cos sin.sin cos sin sin.sin cos.cos S HRA = Hour Angle (-/+ in morning / evening) δ = sun declination ξ = local latitude (+/-: on N/S hemisphere) ψ = local longitude (+/-: on E/W hemisphere) φ is measured from North direction (N, E, S, W : => φ = 0, 90, 180, ) φ from formula is to be interpreted as the angle < or > when HRA < or > 0 ζ = zenith angle ( = 90 - θ ) 9

10 2. Irradiation angles: Solar Ray Tracking N Solar Declination δ given by: sin sin. sin ε = obliquity of the ecliptic : approximation: ε = ( ) 0. n S n = Julian day number after 1 Jan = JD λ = solar ecliptic longitude: λ = L sin g sin 2g L = n g = n 10

11 3. Sunrise and Sunset N Sunrise and sunset times: Calculated using linear interpolation of solar zenith angles at subsequent times of the day, S to find times at which solar zenith angle = (or elevation = ). This correction of is applied to account for the finite dimensions of the apparent solar disk. 11

12 4. Irradiation efficiency +Zaxis Normal direction Sun ray Irradiation efficiency: defined as: Wall height -Y-axis 0 +Xaxis Wall at sunside Width of house Shading object Y-value Shading object (e.g. tree) Shading object X-value Roof top height Roof top X- value +Y-axis Irradiation efficiency: Cosine of the angle between solar ray and normal (perpendicular) direction on roof plane. = 1 at perpendicular incidence of sun ray = 0 at parallel incidence 12

13 4. Irradiation efficiency +Zaxis Sun ray Wall height Wall at sunside Shading object (e.g. tree) Roof top height Roof top X-value -Y-axis 0 Width of house +X-axis Shading object Y-value Shading object X- value +Y-axis Z-axis north Sun ray Altitude- or elevation-angle (starting from 0 at ground level) Orientation: +X-axis: perpendicular to sunny wall +Y-axis: along sunny wall, to East +Z-axis: upward +X-axis west south east Azimuth-angle (starting from north = 0) 13

14 4. Irradiation efficiency +X-axis Z-axis τ η n Sun ray θ = altitude (elevation) l r β=ϕ-90 (ϕ = azimuth) m +Y-axis Orientation: +X-axis: perpendicular to sunny wall +Y-axis: along sunny wall, to East +Z-axis: upward Lines l and m in XY-plane Sun ray vector = r Normal vector on roof plane = n. Irradiation efficiency = n n. r r scalar product of vectors n and r E E 1 0 if if n // r n r

15 4. Irradiation efficiency +X-axis Z-axis τ η n Sun ray θ = altitude (elevation) l r β=ϕ-90 (ϕ = azimuth) m +Y-axis Irradiation efficiency E = n r n. r scalar product of vectors n and r Write vectors n and r in X,Y,Zcomponents and calculate their scalar product: cos.sin.sin.sin cos.cos.sin.cos sin.cos E E 1 0 if if n // r n r 15

16 4. Irradiation efficiency Z-axis τ n Correction of E for the transmission through air +X-axis η Sun ray θ = altitude (elevation) l r β=ϕ-90 (ϕ = azimuth) m +Y-axis in atmosphere (due to scattering and absorption) for clear sky: Correction factor CF: CF = exp(- μ.w), μ = attenuation coefficient, in 1/ air-mass ( 0.25) w = amount of air-mass. approx.: w = 1/cos α, with α = ζ if ζ <= 80 α = 80 if ζ > 80 (ζ = zenith angle) 16

17 Correction of zenith angle (in min.) 4. Irradiation efficiency Z-axis τ n Correction for atmospheric refraction: Sun ray r due to changes in refraction coefficient: +X-axis η θ = altitude (elevation) β=ϕ-90 (ϕ = azimuth) m +Y-axis Correction factor l Zenith angle (degr.) 17

18 5. Shadow lines of shading objects Wall height -Y-axis 0 Width of house +X-axis +Zaxis P S Wall at sunside C D E Shading object Y-value B Shading object (e.g. tree) A Orientation: +X-axis: perpendicular to sunny wall +Y-axis: along sunny wall, to East +Z-axis: upward Sun ray Q Shading object X- value R Roof top height Roof top X- value +Y-axis Calculate: 1. Projection of sun ray through top of shading object onto (extended) roof plane. 2. Is crossing point C within roof boundaries? Does roof plane PQRS intersect with shadow plane ABCDE? (Intersection line = CD). 18

19 5. Shadow lines of shading objects +Zaxis P 0 S C D E B Shading object (e.g. tree) Sun ray Q R +Y-axis Question : Is there a shadow line on the roof? Or: Does roof plane PQRS intersect with shadow plane ABCDE? (Intersection line = CD). +X-axis A Answer: Calculate intersection points of (extended) limiting vectors (green) with shadow plane ABCDE, and Check their positions on the roof. 19

20 diffuse fraction D/T diffuse at surface D / extraterrestrial incident 6. Clear and cloudy skies total at surface T / extraterrestrial incident Approximation of diffuse vs. total (= direct + diffuse) irradiation power measurements, for horizontal surface (zenith = 0 ). see: Liu & Jordan. Experimental errors and band widths ± 25 %. NB. For other zenith angles, or other latitudes, minor deviations sky clearness fraction C Approach: 1. Determine D/T from clearness C: D = (1 C) T. 2. Interpolate in D vs. T- curve. 20

21 7. Energy output (in kwh) Energy output (in kwh): 1. Calculate for each hour: Irradiation efficiency, corrected for air transmission and refraction, (%) 2. Multiply with time interval (normally 1 hour) (h) 3. Sum over all hours (per day, month or year) 4. Multiply the result with Solar irradiance power (normally W/m 2 ) Intrinsic (electronic) efficiency of the solar panels (%) Exposed area of the solar panels (m 2 ) Calculation for clear skies and no shadows. 21

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