Incidence Angle Modifier
Five reflection models are available in SolarFarmer: ASHRAE, CIEMAT and three pointwise-based models (Fresnel normal glass, anti-reflective glass and custom profiles). CIEMAT is only applicable if the model has the three components of the incident irradiation available.
ASHRAE
The ASHRAE incidence angle modifier's behaviour is defined by a single parameter, \(b_0\), with a typical value of 0.05, and the incidence angle \(\theta \left(j, t \right)\) for submodule \(j\) at orientation \(o_j\).
The beam irradiance is modified like this:
$$G_\text{dir,poa,iam} \left( o_j, t \right) = G_\text{dir,poa,soil} \left(o_j, t \right) \left( 1 - b_0 \left( \frac{1}{\cos \theta \left( j, t \right) } - 1 \right) \right)$$
SolarFarmer currently does not account for reflection and absorption of incident diffuse irradiance, because there is no single incidence angle associated with the diffuse or reflected irradiances. A recommendation by PVsyst is to take weighted averages of the modifiers calculated at discreet intervals of zenith and incidence angle over the sky and ground visible to the active surface of a module positioned at the nominal tilt and azimuth angle for the Layout Region. Where \(\theta\) denotes incidence angle and \(A\) is azimuth from the module, diffuse and reflected incidence angle modifiers would be calculated as follows over the sky and ground as seen from the module:
$$\IAM_{dif/r} = \frac{ \sum_{\theta} { \sum_{A = 0}^{360} { \left( 1 - b_0 \left( \frac{1}{\cos \theta} - 1 \right) \right) \cos \theta \sin \theta }}} { \sum_{\theta} { \sum_{A = 0}^{360} { \cos \theta \sin \theta }}}$$
Then the diffuse components modified for incident angle would be given by the following expressions:
$$G_\text{dif,poa,iam} \left(o_j, t \right) = G_\text{dif,poa,soil} \left(o_j, t \right) \IAM_{\text{dif}}$$
$$G_\text{r,poa,iam} \left(o_j, t \right) = G_\text{r,poa,soil} \left(o_j, t \right) \IAM_\text{r}$$
Note that this is an approximation as it doesn't consider shading obstacles of the specific orientation of modules in the array which may be affected by the underlying terrain. This method may be implemented in a future version of SolarFarmer
The ASHRAE model was not specifically obtained for PV modules and presents some problems, especially with high angles of incidence. The ASHRAE implementation for PV modules is described here:
CIEMAT
The CIEMAT model was proposed by N. Martin and J.M. Ruiz in [6] and is intended specifically for the analysis of PV modules of different technologies.
This model proposes three angular losses factor, one for each component of the incident radiation on tilted surfaces: \(F_\text{dir}\), \(F_\text{dif}\), and \(F_\text{r}\) for the direct (beam), diffuse and reflected components respectively. These loss factors are calculated as follows:
$$F_\text{dir} \left( \theta \right) = \frac{ \exp \left( -\cos \theta \ / {a_\text{r}} \right) - \exp \left( -1 / {a_\text{r}} \right) } {1 - \exp \left( -1 / {a_{r}} \right) }$$
$$F_\text{dif} \left( \beta \right) \cong \exp \left [ - \frac{1}{a_r} \left( c_1 \left( \sin \beta + \frac{\pi - \beta - \sin \beta }{1 + \cos \beta } \right) + c_2 \left( \sin \beta + \frac{\pi - \beta - \sin \beta }{1 + \cos \beta } \right)^2 \right) \right ]$$
$$F_\text{r} \left( \beta \right) \cong \exp \left [- \frac{1}{a_r} \left( c_1 \left( \sin \beta + \frac{\beta - \sin \beta }{1 - \cos \beta } \right) + c_2 \left( \sin \beta + \frac{\beta - \sin \beta }{1 - \cos \beta} \right)^2 \right) \right ]$$
Note that only \(F_\text{dir}\) has a time dependency for fixed-tilt arrays.
Where \(a_\text{r}\) is the angular losses coefficient, obtained empirically to fit in each case. Typical values of \(a_\text{r}\) are:
| Module conditions | \(a_\text{r}\) |
|---|---|
| Commercial clean modules | 0.16 - 0.17 |
| Modules with moderate quantity of dust | 0.20 |
| Significant dust | 0.27 |
The approximate solutions for \(F_\text{dif}\) and \(F_\text{r}\) presented above include two fitting parameters, \(c_1\) and \(c_~2\).
$$c_{1} = \frac{4}{3 \pi}$$
The value of \(c_2\) depends on the value of \(a_\text{r}\); typical values are shown below.
| \(a_\text{r}\) | c2 |
|---|---|
| 0.16 | -0.074 |
| 0.17 | -0.069 |
| 0.18 | -0.064 |
Finally, the effective irradiances reaching the cells in a module are calculated from the irradiances incident on the module (normally after soiling, but not shown here) as:
$$G_\text{dir,poa,iam} \left(o_j, t \right) = \left( 1 - F_\text{dir} \right) G_\text{dir,poa,soil} \left(o_j, t \right)$$
$$G_\text{dif,poa,iam} \left(o_j, t \right) = \left( 1 - F_\text{dif} \right) G_\text{dif,poa,soil} \left(o_j, t \right)$$
$$G_\text{r,poa,iam} \left(o_j, t \right) = \left( 1 - F_\text{r} \right) G_\text{r,poa,soil} \left(o_j, t \right)$$
Pointwise
SolarFarmer also allows the user to choose between two pre-defined pointwise models for \(\IAM\) and to define any custom pointwise profile. The pointwise model simply interpolates between measured incidence angle - transmission data point pairs to supply an instantaneous value for transmission into a module through its encapsulation at a specific incidence angle. The available pointwise models are based on Fresnel equations for normal glass and anti-reflective (AR) coated glass. The \(\IAM\) coefficients used for these two IAM models are shown below as a function of the angle of incidence (AOI):
| AOI (degrees) | 0 | 30 | 50 | 60 | 70 | 75 | 80 | 85 | 90 |
|---|---|---|---|---|---|---|---|---|---|
| Normal glass | 1 | 0.998 | 0.981 | 0.948 | 0.862 | 0.776 | 0.636 | 0.403 | 0 |
| AR-coated glass | 1 | 0.999 | 0.987 | 0.962 | 0.892 | 0.816 | 0.681 | 0.440 | 0 |
For a custom pointwise model to be applicable, the user must supply a series of (usually measured) data points for transmission of the module surface vs. incidence angle. The data point pairs supplied are incidence angle (in degrees) and transmission (as a fraction between 0 and 1, where 1 is perfect transmission). There is no limit to the number of data point pairs that can be supplied.
In SolarFarmer, the direct, diffuse and reflected components of light are treated separately:
Direct \(G_\text{dir,iam}\)
As the incidence angle of this component is unique and known, the corresponding \(\IAM\) modifier is simply interpolated from the supplied measured. Linear interpolation is used.
Diffuse \(G_\text{dif,iam}\)
Diffuse irradiance is assumed to arrive equally from all parts of the sky visible to the module. Diffuse irradiance multiplied by the appropriate IAM is integrated over this portion of the sky.
Diffuse \(G_\text{r,iam}\)
Integrate over area below horizon, multiplying reflected irradiance by appropriate \(\IAM\) for incidence angle.
A comparison of the supported \(\IAM\) is graphically illustrated below:
