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POA Ground Reflected

You are here:
  1. Home
  2. Modeling Steps
  3. 1. Weather and Design
  4. Plane of Array (POA) Irradiance
  5. Calculating POA Irradiance
  6. POA Ground Reflected

Irradiance on a tilted surface that is reflected off the ground, E_{g} , is calculated as a function of the irradiance on the ground, usually assumed to be GHI, the reflectivity of the ground surface, known as albedo, and the tilt angle of the surface, \theta_{T,surf} :

E_{g}=GHI\times albedo\times \frac{\left ( 1-\cos \left ( \theta_{T,surf} \right ) \right )}{2}

 

 

The model for ground reflected irradiance E_{g} develops from the following assumptions:

  1. The array is infinitely long.
  2. Irradiance on the ground is uniform and equal to GHI, i.e., horizon blocking and near-field shading by the array is ignored.
  3. Irradiance reflects from the ground equally in all directions, i.e., the ground is a diffuse or Lambertian reflector.
  4. The ground is visible to the array from the point of intersection of the array’s slope projected to the ground, to the infinite horizon.

With these assumptions the model for E_{g} can be derived using a view or configuration factor. The view factor quantifies the fraction of diffuse reflected irradiance from one surface A_{1} that impinges on a second surface A_{2}. In the equation above for E_g, the term \frac{1-\cos{\theta_{T,surf}}}{2}=F_{A_1 \rightarrow A_2} is the view factor from surface A_1, the ground, to surface A_2, the module.

To derive this model, consider an infinite strip dA_{1} on the ground of differential width  dx that is parallel to the array. Denote the array’s face as A_{2}, and the view factor from dA_{1} to A_{2}  as F_{dA_{1} \rightarrow A_{2}}. The irradiance per unit length (W / m) reflecting from the strip dA_{1} is GHI \times albedo \times dx.  The contribution to irradiance per unit length (W / m) on A_{2} from reflected irradiance from dA_{1} is

dE_{g}=GHI \times albedo \times F_{dA_{1} \rightarrow A_{2}} \times dx

Let x be a coordinate axis on the ground with origin under the lower, forward edge of the array, parallel to the rear-facing normal vector to the array, with negative direction toward the horizon behind the array. Consider A_{1} to be the half-plane on the ground defined by x\leq x_{0},  and A_{2}  to be the rear-facing side of an infinitely long strip of width L above the half plane with edges parallel to the ground. The total irradiance on A_{2} per unit area (W/m2) from irradiance reflected from A_{1} is

\begin{align*} E_g & = \frac{1}{A_2}\int dE_g = \frac{1}{L_2}\int_{-\infty}^{x_0}GHI\times albedo \times F_{dA_1\rightarrow A_2}dx \\ & = GHI \times albedo \times \int_{-\infty}^{x_0}\frac{F_{dA_1\rightarrow A_2}}{L_2}dx \\ & = GHI \times albedo \times F_{A_1 \rightarrow A_2} \end{align}

 

The view factor F_{dx \rightarrow A_{2}} is given by the formula

F_{dA_{1} \rightarrow A_{2}} = \frac{1}{2} \left( \sin \left( \phi_{2} \right) - \sin \left( \phi_{1} \right) \right)

 ( see case B-71 at http://www.thermalradiation.net/tablecon.html)

Rewriting the terms in the equation for F_{dA_{1} \rightarrow A_{2}}  as

\sin \phi_{2} = \cos \left(90 - \phi_{2} \right ) = \frac{x - h \cot \theta_{T,surf}}{\sqrt{\left(x - h \cot \theta_{T,surf} \right )^2 + h^2}}

\sin \phi_{1} = \cos \left(90 - \phi_{1} \right ) = \frac{x - h \cot \theta_{T,surf} - L_{2} \cos \theta_{T,surf}}{\sqrt{\left(x - h \cot \theta_{T,surf} - L_{2} \cos \theta_{T,surf} \right )^2 + \left(L_{2} \sin \theta_{T,surf} + h \right )^2}}

Substituting and evaluating the integral F_{A_1 \rightarrow A_2} = \int_{-\infty}^{x_0}\frac{F_{dA_1 \rightarrow A_2}}{L_2} dx obtains

F_{A_{1} \rightarrow A_{2}} = \frac{1}{2} \left [L_{2} \cos \theta_{T,surf} - \sqrt{\left( x_{0} - h \cot \theta_{T,surf} \right)^2 + h^2} + \sqrt{ \left( x_{0} - h \cot \theta_{T,surf} - L_{2} \cos \theta_{T,surf} \right)^2 + \left(L_{2} \sin \theta_{T,surf}+ h \right)^2} \right]

When the full half plane is visible to the rear facing side of the array, x_{0} = h \cot \theta_{T,surf} which reduces the above expression to

F_{A_{1} \rightarrow A_{2}} = \frac{1}{2} \left(1 + \cos \theta_{L,surf} \right )

When A_{2} is taken to be the front-facing surface and A_{1} is the half-plane x\geq h \cot \theta_{T,surf}, the view factor F_{A_{1}\rightarrow A_{2}} is calculated in a similar manner to obtain

F_{A_{1} \rightarrow A_{2}} = \frac{1}{2}\left( 1 - \cos \theta_{T,surf} \right )

Content contributed by Sandia National Laboratories

Modeling Steps
331. Weather and Design
3Sun Position
Solar Position Algorithm (SPA)
Basic Solar Position Models
Sandia’s Ephemeris Model
9Irradiance & Insolation
Extraterrestrial radiation
Air Mass
2Direct Normal Irradiance
Piecewise Decomposition Models
DIRINT Model
Global Horizontal Irradiance
Diffuse Horizontal Irradiance
1Spectral Content
AM 1.5 Standard Spectrum
2Weather Data Sources for Performance Modeling
National Solar Radiation Database
Spectral irradiance dataset from Albuquerque
4Weather Observations
Air Temperature
Wind Speed and Direction
Precipitation
Air Pressure
5Array Orientation
Fixed tilt
Single Axis Tracking
Two-Axis Tracking
2Array Orientation Errors
Effect of Array Tilt Errors
Effect of Array Azimuth Errors
8Plane of Array (POA) Irradiance
8Calculating POA Irradiance
POA Beam
Angle of Incidence
1POA Ground Reflected
Albedo
5POA Sky Diffuse
Isotropic Sky Diffuse Model
Simple Sandia Sky Diffuse Model
Hay and Davies Sky Diffuse Model
Reindl Sky Diffuse Model
Perez Sky Diffuse Model
4Shading, Soiling, and Reflection Losses
4Incident Angle Reflection Losses
Physical IAM Model
ASHRAE IAM Model
Martin and Ruiz IAM Model
Sandia IAM Model
112. DC Module IV Characteristics
2Module Temperature
Sandia Module Temperature Model
Faiman Module Temperature Model
2Cell Temperature
Sandia Cell Temperature Model
PVsyst Cell Temperature Model
2Effective Irradiance
Spectral Response
Spectral Mismatch Models
2Single Diode Equivalent Circuit Models
De Soto “Five-Parameter” Module Model
PVsyst Module Model
3Point-value models
Sandia PV Array Performance Model
Loss Factor Model
1PVWatts
Improvements to PVWatts
43. DC Array IV
Mismatch Losses
DC Component Health
DC Wiring Losses
Array Utilization
74. DC to AC Conversion
CEC Inverter Test Protocol
Operating Temperature
Sandia Inverter Model
Driesse Inverter Model
Inverter Saturation or “Clipping”
Loss of Grid
Advanced Inverter Features
55. AC System Output
AC Wiring Losses
4PV Performance Metrics
Performance Ratio
Normalized Efficiency
Performance Index
Annual Yield
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