EELE408 Photovoltaics Apparent motion of the sun Lecture 04: Apparent Motion of the Sum

Dr. Todd J. Kaiser EAST [email protected]

Department of Electrical and Computer Engineering Montana State University - Bozeman SOUTH

WEST NORTH

What time of year is this image true? March 22: Spring Equinox 2 Sept 23: Fall Equinox

Capturing the Light Angle Effects 0⁰

• The power capture by a surface is going to be a function of the angle between the normal to the surface and the light

  0 cos 1

3 All the power is incident on the surface 4

Angle Effects 15⁰ Angle Effects 30⁰

 15 cos  0.9659   30 cos  0.8660

96.6% of the power is incident on the surface 5 86.6% of the power is incident on the surface 6

1 Angle Effects 45⁰ Angle Effects 60⁰

  45 cos  0.7071   60 cos  0.5000

70.7% of the power is incident on the surface 7 50.0% of the power is incident on the surface 8

Angle Effects 75⁰ Angle Effects 90⁰

  75 cos  0.2588   90 cos  0.0000

25.9% of the power is incident on the surface 9 None of the power is incident on the surface 10

Declination Angle of Sun Declination Angle

Summer Solstice d = 23.45º North Mar 23 Pole Declination () = 0º

d

Fall Equinox d = 0º Jun 23 Dec 23 Declination () = -23.45º d Declination () = 23.45º

360    23.45 sin d 80 Winter Solstice 365  Sep 23 d = ‐23.45º Declination () = 0º 11 12 12

2 Elevation Angle () Solar Time

• Local Time (LT): varies with time zones and day light savings • Solar noon Local Solar Time (LST): time when sun is highest in the sky = 0⁰ • Hour Angle (HRA)=15⁰(LST‐12) –Zero at solar noon –Negative in the morning – Positive in the afternoon

The elevation angle is the angular height the sun makes with the horizontal

13 14

Azimuth Angle Elevation and Zenith Angles

Azimuth angle is the angle direction of the sun. f = latitude (Bz@ 45.68⁰) North is at 0 degrees The sun is directly south (180⁰) at solar noon

sin cos  cos sin cosHRA Azimuth  cos1   The elevation angle is zero at sunrise  cos  and sunset Zenith True for mornings Angle The elevation angle is a maximum at use 360⁰‐Azimuth for afternoons N=0⁰ solar noon Elevation or Altitude Angle

W=270⁰ E=90⁰ Elevation  sin 1sin sin  cos cos cosHRA

15 16 S=180⁰

Polar Plot of Sun Position Peak Elevation Angle (Bozeman) for Bozeman

Summer Latitude + Declination Summer   Fall/Spring 45  23.45   Latitude 68.45 Winter 45 Spring/Fall

Latitude - Declination Winter   45  23.45 17 21.55 18

3 Solar Radiation on a Tilted Surface Arbitrary Angles

Smodule  Sincident cos sin  cos    sin cos  

S incident a: sun elevation angle Shorizontal b: module tilt angle y: azimuth angle the panel faces   Q: sun azimuth angle 

Smodule • Most fixed panels are facing toward the equator (south in northern Hemisphere)  = 180⁰   •Tracking Modules   elevation angle t = 90⁰‐ Shorizontal  Sincident sin   tilt angle of panel t Smodule  Sincident sin     declination angle Shorizontal sin     latitude Smodule  sin 19 20   90   at solar noon

Arbitrary Angles Measuring Solar Radiation   N S 

Pyranometers A Acos(

  Smodule  Sincident cos  Sincident S  N  S: Unit vector pointing towards sun  N : Unit vector normal to surface 21 22

Solar Irradiance Data Peak Sun Hours

Area under curves = Solar Insolation

1kW/m2 Sunny Daily Average Data

Daily Solar Daily Data

Insolation n Monthly Average Data Equal Areas Solar Radiatio Solar Cloudy

Day of the Month Time of Day

Peak Sun Hours

23 24

4 MSU Weather Station 1 MSU Weather Station 2

• Sunny • Afternoon Clouds • (7 peak sun hours)

25 26

MSU Weather Station 3 MSU Weather Station 4

• Mostly Cloudy • Overcast: note the scale change

27 28

Isoflux Contour Plots The number of hours the sun is shining

• d = day of the year •  = latitude •  = declination angle

 360    23.45 sin 284  d  365 

1 1  sin sin   sunrise  12  cos   15 cos cos  

1 1  sin sin   sunset  12  cos   15 cos cos   (In solar time)

29 30

5 Hours of Sunlight(Latitude & Day) Daily Solar Irradiance

• HRA = hour angle •  = elevation angle • h = hour of the day

HRA 15 h 12   sin 1sin sin   cos  cos  cos HRA  0.678 H d, 1.3670.7 AM

31 32

Power Density (Latitude & Day) Solar Insolation (latitude & tilt)

Smodule  Sincident cos sin  cos    sin cos  

a: sun elevation angle b: module tilt angle y: azimuth angle the panel faces Q: sun azimuth angle

sunset 0.678 H   1.3670.7 AM hsunrise

33 34

Solar Isolation (latitude & tilt)

Renewable Energy Technology Resource Maps for the United States

• Updated August 2009

35 NREL is a national laboratory of the U.S. Department of Energy Office of Energy Efficiency and Renewable Energy operated by the Alliance for Sustainable Energy, LLC

6 U.S. Renewable Resources Including Transmission Lines

Resource Solar PV/CSP) Wind Geothermal Water Power Biopower Resource Solar PV/CSP) Wind Geothermal Water Power Biopower Theoretical 206,000 GW 8,000 GW 39 GW 140 GW 78 GW Theoretical 206,000 GW 8,000 GW 39 GW 140 GW 78 GW Potential (PV) (onshore) (conventional) Potential (PV) (onshore) (conventional) 11,100GW 2,200 GW 520 GW 11,100GW 2,200 GW 520 GW (CSP) (offshore to (EGS) (CSP) (offshore to (EGS) 50 nm ) 4GW4 GW 50 nm ) 4GW4 GW (co-produced) (co-produced)

U.S. Photovoltaic Solar Resource U.S. Concentrating Solar Resource

U.S. Wind Resource (50m) U.S. Biomass Resource

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