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:
$$E_{d}=DHI\times \left[ \left( 1-F_{1} \right) \left( \frac{1+\cos \left ( \theta_{T} \right)}{2} \right) + F_1\left( \frac{a}{b} \right)+F_2 \sin \left (\theta_T \right) \right]$$,
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])
| 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])
| 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