Global tilted irradiance

The Global Tilted Irradiance (GTI), also known as the Plane-Of-Array (POA) irradiance, is the total hemispherical solar irradiance incident on a titled plane. The orientation of a tilted plane is normally defined by a tilt angle and an azimuth angle. When the condition allows, a plane is usually tilted equatorward. In broadband values, i.e. integrated over the whole Electromagnetic spectrum, it normally has SI units of W m⁻². In a sense, single-axis trackers and dual-axis trackers are tilted planes, only that their tilt angles and azimuth angles change with time.

The GTI is the sum of the beam (or direct), diffuse and ground reflected irradiances incident on that same tilted plane ^[1].

Information about time series of the GTI resource is useful for applications such as flat-plate collectors, which vary from photovoltaic (PV) modules, to passive collectors used as solar heaters for water or air applications. The emerging bifacial solar photovoltaics (BPV) have the ability to exploit the GTI incident on the front and rear sides of the module, and hence have an improved solar-cell efficiency per unit area as opposed to the more common monofacial PVs ^[2][3][4][5].

Pyranometers, which have a 180° field of view, may measure the GTI when fixed on a tilted plane. If the tilt angle is 0°, this implies a flat plane, and hence the GTI is reduced to the Global Horizontal Irradiance (GHI). In the special scenario when a pyranometer is placed on a plane that is tracking the sun, then the Global Normal Irradiance (GNI) is the measured variable. Similarly, if a pyranometer is attached to a single-axis tracker, the measured variable is the GTI on the tracker orientation.

In the absence of such measurements, which are quite scarse ^[6][7], the GTI may be deduced from transposition models which require measurements or modelled values of the Direct (or Beam) Normal Irradiance (BNI or DNI), the Diffuse Horizontal Irradiance (DHI), the GHI, and the ground albedo ^{[8][9][10][11][12][13][14]}. Sky models are an alternative, which utilize similar inputs, to provide the angular diffuse radiance distribution of the whole sky, from which the GTI on any tilted plane may be computed ^[15][16][17]. In such models the beam component on the tilted plane is computed from the BNI. The GTI may also be computed by means of physical models solving the radiative transfer equation in the atmosphere ^[18][19], such models require very specific information on the composition of the atmosphere for accurate simulations.

An example of where to retrieve GTI data is NASA POWER, providing climatological monthly mean GTI derived from satellite-based hourly data for equatorward tilted planes with tilt angles equal to 0°, the latitude minus 15°, the latitude, the latitude plus 15°, and 90°.^[20]

Formulae

In the absence of measurements, the GTI may be calculated.

On either side of a tilted plane, the GTI is the sum of the Beam Tilted Irradiance (BTI), the Downward Diffuse Tilted Irradiance (DDTI), and the upward Reflected Tilted Irradiance (RTI). As both the DDTI and RTI are diffuse components, then the total Diffuse Tilted Irradiance (DTI) incident on the tilted plant is their sum. The following equations present these relations:

${\displaystyle GTI=BTI+DDTI+RTI}$

${\displaystyle DTI=DDTI+RTI}$

The BTI is computed from the BNI and the angle of incidence θ_i as:

${\displaystyle BTI=BNI*\max(\cos(\theta _{i}),0)}$

where the max function is placed to avoid negative values. It means that the no beam irradiance is incident within the field of view of the tilted plane in question.

With knowledge of the angular distribution of the downward diffuse radiance across the whole dome of the sky, and the upward ground reflected diffuse radiance, the DDTI and RTI on either side of a plane are computed as:

${\displaystyle DDTI^{+}=\int _{\phi =0}^{2\pi }\int _{\theta =0}^{\pi /2}{L^{\downarrow }(\theta ,\varphi )\max {\left(\cos {(\theta )}\cos {(\beta )}+\sin {(\theta )}\sin {(\beta )}\cos {(\varphi -\alpha )},0\right)}\sin {(\theta )}d\theta d\varphi }}$

${\displaystyle RTI^{+}=\int _{\phi =0}^{2\pi }\int _{\theta =pi/2}^{\pi }{L^{\uparrow }(\theta ,\varphi )\max {\left(\cos {(\theta )}\cos {(\beta )}+\sin {(\theta )}\sin {(\beta )}\cos {(\varphi -\alpha )},0\right)}\sin {(\theta )}d\theta d\varphi }}$

${\displaystyle DDTI^{-}=\int _{\phi =0}^{2\pi }\int _{\theta =0}^{\pi /2}{L^{\downarrow }(\theta ,\varphi )\max {\left(\cos {(\theta )}\cos {(\beta +\pi )}+\sin {(\theta )}\sin {(\beta +\pi )}\cos {(\varphi -\alpha )},0\right)}\sin {(\theta )}d\theta d\varphi }}$

${\displaystyle RTI^{-}=\int _{\phi =0}^{2\pi }\int _{\theta =pi/2}^{\pi }{L^{\uparrow }(\theta ,\varphi )\max {\left(\cos {(\theta )}\cos {(\beta +\pi )}+\sin {(\theta )}\sin {(\beta +\pi )}\cos {(\varphi -\alpha )},0\right)}\sin {(\theta )}d\theta d\varphi }}$

where + denotes the front side of the plane, − the rear side of the plane, L^↓ the downwelling diffuse sky radiance, L^↑ the upwelling reflected radiance, θ is element zenith angle, φ the element azimuth angle, β the tilt angle of the plane measured from the horizon, and α the azimuth angle of the plane.

Using these relations, then the GTI on either side of the plane is computed as:

${\displaystyle GTI^{+}=BTI+DDTI^{+}+RTI^{+}}$

${\displaystyle GTI^{-}=BTI+DDTI^{-}+RTI^{-}}$

Applications

• Photovoltaic systems
• BPVs
• Solar thermal collectors:
  • Solar water heating
  • Solar air heat
• Agrivoltaics
• Daylighting (architecture)

See also

• Solar irradiance
• Sunlight
• Diffuse sky radiation
• Direct normal irradiance