Perez Sky Diffuse Model

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While the sky diffuse model presented up to this point separated the isotropic, circumsolar, and horizon components explicitly, Perez developed a more complex model that relies on a set of empirical coefficients for each term.

The basic form of the model is:

$E_{d}=DHI\times \left [ \left ( \left ( 1-F_{1} \right )\left ( \frac{1+\cos \left ( \theta_{T} \right )}{2} \right )+F_{1}\left ( \frac{a}{b} \right )+F_{2} \sin \left ( \theta_{T} \right ) \right ]$ ,

where $F_{1}$ and $F_{2}$ are complex empirically fitted functions that describe circumsolar and horizon brightness, respectively.

$a = \max \left ( 0,\cos \left ( AOI \right ) \right )$ , and $b=\max \left ( \cos \left ( 85^{\circ} \right ),\cos \left ( \theta_{Z} \right ) \right )$ .

$DHI$ is diffuse horizontal irradiance,
$AOI$ is the angle of incidence between the sun and the plane of the array.
$\theta_{Z}$ is the solar zenith angle.
$\theta_{T}$ is the array tilt angle from horizontal.

$F_{1}=\max \left [ 0,\left ( f_{11}+f_{12}\Delta +\frac{\pi \theta_{Z}}{180^{\circ}}f_{13} \right) \right ]$ ,

$F_{2}= f_{21}+f_{22}\Delta +\frac{\pi \theta_{Z}}{180^{\circ}}f_{23}$

The $f$ coefficients are defined for specific bins of clearness ( $\varepsilon$ ), which is defined as:

$\varepsilon =\frac{(DHI+DNI)/DHI+\kappa \theta_{Z}^{3} }{1+\kappa \theta_{Z}^{3} }$ ,

where $DNI$ is direct normal irradiance and $\kappa$ is a constant equal to $1.041$ for angles are in radians, or $5.535\times 10^{-6}$ for angles in degrees.

$\Delta =\frac{DHI\times AM_{a}}{E_{a}}$

where $AM_{a}$ is the absolute air mass, and $E_{a}$ is extraterrestrial radiation.

Perez has published a number of different versions of the $f$ coefficients fitted to various data sets [2, 3 , 4]. Table 1 shows the $f$ coefficient values published in [3] for irradiance. The $\varepsilon$ bin refers to bins of clearness, $\varepsilon$ , defined in Table 2.

Table 1. Perez model coefficients for irradiance (from Table 6 in [3])

$\varepsilon$ bin	f11	f12	f13	f21	f22	f23
1	-0.008	0.588	-0.062	-0.06	0.072	-0.022
2	0.13	0.683	-0.151	-0.019	0.066	-0.029
3	0.33	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	1.06	-1.6	-0.359	0.264	-1.127	0.131
8	0.678	-0.327	-0.25	0.156	-1.377	0.251

Table 2. Sky clearness bins (from Table 1 in [3])

$\varepsilon$ 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	—

References

[1] Loutzenhiser P.G. et. al. “Empirical validation of models to compute solar irradiance on inclined surfaces for building energy simulation” 2007, Solar Energy vol. 81. pp. 254-267
[2] Perez, R., Seals, R., Ineichen, P., Stewart, R., Menicucci, D., 1987. A new simplified version of the Perez diffuse irradiance model for tilted surfaces. Solar Energy 39 (3), 221–232.
[3] Perez, R., Ineichen, P., Seals, R., Michalsky, J., Stewart, R., 1990. Modeling daylight availability and irradiance components from direct and global irradiance. Solar Energy 44 (5), 271–289.
[4] Perez, R. et. al 1988. “The Development and Verification of the Perez Diffuse Radiation Model”. SAND88-7030

Content contributed by Sandia National Laboratories