Transposition
Where simultaneous GHI (\(G_{\text{h}}\)) and horizontal Diffuse Irradiance (\(G_{\text{dif}}\)) timeseries are supplied as inputs, a horizontal direct irradiance \(G_{\text{dir}}\) timeseries is calculated as:
$$G_{\text{dir}} \left( t \right) = G_{h} \left( t \right) - G_{\text{dif}} \left( t \right)$$
Direct Irradiance
The direct irradiance on the tilted module surface is calculated from the direct irradiance on the horizontal surface \(G_{\text{dir}}\) as:
$$G_\text{dir,poa} = G_{\text{dir}} \frac{\cos \left( \theta \right)}{\cos \left( \theta_\text{z} \right)} = \mathit{DNI} \cos \left( \theta \right)$$
As described in Sun Position, \(\theta_{\text{z}}\) is the Solar zenith angle (the angle between the vertical and a vector pointing to the sun at the plant location). \(G_{\text{dir,poa}}\) is set to zero for \(\theta_{\text{z}}\) > 88°. This cut-off point for large zenith angles has been found in PVsyst and has been adopted in SolarFarmer.
Ground (Albedo) Irradiance
The reflected irradiance on the tilted module surface is calculated from the Global Horizontal Irradiance \(G_{\text{h}}\) as:
$$G_\text{r,poa} = \frac{1}{2} \rho G_\text{h} \left(1 - \cos \left( \beta \right)\right)$$
Where \(\rho\) is the albedo of the surrounding surface. Some typical albedo values for reference:
Surface | Typical albedo |
---|---|
Fresh asphalt | 0.04 |
Worn asphalt | 0.12 |
Conifer forest | 0.08-0.15 |
Deciduous trees | 0.15-0.18 |
Bare soil | 0.17 |
Green grass | 0.25 |
Desert sand | 0.40 |
New concrete | 0.55 |
Ocean Ice | 0.5-0.7 |
Fresh snow | 0.8-0.9 |
PVsyst emulation: Ground (Albedo) Irradiance
For incidence angles \(\theta > 90^\circ\) (sun behind the module), only the diffuse component of the horizontal irradiance is used, so
$$G_\text{r,poa} = \frac{1}{2} \rho G_{\text{dif}} (1 - \cos{(\beta))}$$
Diffuse Irradiance
Both the Hay and Perez transposition models make use of the Direct Normal Irradiance, \(G_{\text{dni}}\). This is related to direct horizontal irradiance (\(G_{\text{dir}}\)) through:
$$G_{\text{dni}} = \frac{G_{\text{dir}}}{\cos \theta_\text{z}}$$
\(G_{\text{dni}}=0\) for zenith angles \(\theta_{\text{z}} \geq 90^\circ\) (sun below the horizon).
Horizontal Diffuse Irradiance \(G_{\text{dif}}\) is transposed to the plane of each rack (\(G_{\text{dif,poa}}\)) by one of three methods:
Hay Model
The Hay model [4] considers that the diffuse radiation on a horizontal surface is composed of a circum-solar component coming from the direction of the sun and an isotropic diffuse component from the rest of the sky dome. The diffuse radiation on a tilted plane according to the Hay model is expressed as:
$$G_\text{dif,poa} = G_{\text{dif}} \left \lbrack \left( \frac{G_{\text{dni}}}{G_0 \left( n \right)} \right) \frac{a}{\cos \theta_\text{z}} + \left( \frac{1 + \cos\beta}{2} \right) \left( 1 - \frac{G_{\text{dni}}}{G_0 \left( n \right)} \right) \right \rbrack$$
Where
$$a = \max \left(0,\ \cos\theta \right)$$
PVsyst emulation: Hay Model
For zenith angles \(\theta_{\text{z}} \geq 88^\circ\), set \(G_{\text{dni}}=0\) so that:
$$G_{dif,poa} = G_{\text{dif}}\left\lbrack \left( \frac{1 + \cos\beta}{2} \right) \right\rbrack$$
Perez Model
The Perez [5] model considers the circumsolar diffuse, horizon diffuse and isotropic diffuse irradiation. The expression of the diffuse component on tilted surface is shown in the equation below:
$$G_\text{dif,poa} = G_{\text{dif}} \left\lbrack \frac{\left( 1 - F_{1} \right) (1 + \cos\beta)}{2} + F_{1} \frac{a}{b} + F_{2} \sin\beta \right\rbrack$$
Where
$$a = \max \left(0,\ \cos\theta \right)$$
$$b = \max \left(0.087,\cos\theta_\text{z} \right)$$
\(F_1\) and \(F_2\) are empirical coefficients. They express the degree of circumsolar and horizon anisotropy respectively. These coefficients are functions of the sky conditions: the solar zenith angle, the sky clearness (\(\varepsilon\)) and the sky brightness (\(\Delta\)).
$$\varepsilon = \frac{\frac{G_{\text{dif}} + G_{\text{dni}}}{G_{\text{dif}}} + 1.041 \theta_{z}^3} {1 + 1.041 \theta_{z}^3}$$
$$\Delta = \frac{G_{\text{dif}} \mathit{AM}}{G_0 \left( n \right)}$$
\(\mathit{AM}\) is the relative optical air mass as described in Air Mass. The values of \(F_1\) (representing relative circumsolar intensity) and \(F_2\) (representing horizon brightening) are determined from the value of Sky Clearness \(\varepsilon\). Sky Clearness as calculated above is first placed into one of eight bins (inclusive of lower bound, exclusive of upper bound):
\(\varepsilon\) Bin | Lower bound | Upper bound |
---|---|---|
1 Overcast | 1 | 1.065 |
2 | 1.065 | 1.230 |
3 | 1.230 | 1.500 |
4 | 1.500 | 1.950 |
5 | 1.950 | 2.800 |
6 | 2.800 | 4.500 |
7 | 4.500 | 6.200 |
8 Clear | 6.200 | - |
The bin number is then used to determine the value of six further parameters:
\(\varepsilon\) bin | F11 | F12 | F13 | F21 | F22 | F23 |
---|---|---|---|---|---|---|
1 | -0.008 | 0.588 | -0.062 | -0.060 | 0.072 | -0.022 |
2 | 0.130 | 0.683 | -0.151 | -0.019 | 0.066 | -0.029 |
3 | 0.330 | 0.487 | -0.221 | 0.055 | -0.064 | -0.026 |
4 | 0.568 | 0.187 | -0.295 | 0.109 | -0.152 | -0.014 |
5 | 0.873 | -0.392 | -0.362 | 0.226 | -0.462 | 0.001 |
6 | 1.132 | -1.237 | -0.412 | 0.288 | -0.823 | 0.056 |
7 | 1.060 | -1.600 | -0.359 | 0.264 | -1.127 | 0.131 |
8 | 0.678 | -0.327 | -0.250 | 0.156 | -1.377 | 0.251 |
Which finally yield values for \(F_{\text{1}}\) and \(F_{\text{2}}\) as follows:
$$F_{1} = F_{11} + F_{12} \Delta + F_{13} \theta_{z}$$
$$F_{2} = F_{21} + F_{22} \Delta + F_{23} \theta_{z}$$
PVsyst emulation: Perez Model
For zenith angles \(\theta_{\text{z}} \geq 88^\circ\), set \(G_{\text{dif},\text{poa}} = G_{\text{dif}}\)
For incidence angles \(\theta > 90^\circ\) (sun behind the module), set direct normal irradiance \(G_{\text{dni}}\) to zero in the calculation of the sky clearness, so \(\varepsilon = 1 \).