Deline › Published

Method

The rows shade the beam as in Appelbaum and Bany's closed form, and the beam and the circumsolar light stay on the plane. Their loss is carried by Deline's DC derate of the array instead (1 to 6). The shadow's height up the table gives the fraction of a string's substrings in shade and the fraction of strings shaded, by the module layout of the rows. The derate is the largest of three limits: few strings shaded, many strings shaded and many substrings shaded. The share of the light that still reaches a shaded cell is the sky and ground light over the plane irradiance. The paper covers the beam alone. The sky light is masked by the sky masking method of the next step, and the ground light by the rows.

The math

1. Light reaching a shaded cell
2. Few strings shaded
3. Many strings shaded
4. Many substrings shaded
5. DC derate of the array

$$E_e = \frac{G_d + G_r}{G_b + G_{cs} + G_d + G_r} \tag{1}$$ $$F_{dc,1} = 1 - \kappa_1 S_{sub}^2 - \kappa_2 S_{sub},\qquad \kappa_1 = (109\,FF_0 - 54.3)\,e^{-4.5X_{str}},\qquad \kappa_2 = \begin{cases} -6X_{str}^2 + 5X_{str} + 0.28 & X_{str} \le 0.65 \\ 1 & X_{str} > 0.65 \end{cases} \tag{2}$$ $$F_{dc,2} = \frac{X_{str} - S_{sub}\,(1 + V_D/V_{mp0})}{X_{str}} \tag{3}$$ $$F_{dc,3} = \kappa_3\,(S_{sub} - 1) + E_e,\qquad \kappa_3 = \max\!\big(\kappa_{3,0},\ E_e - 1\big),\qquad \kappa_{3,0} = (-0.05E_e - 0.01)X_{str} + (0.85\,FF_0 - 0.7)E_e - 0.085\,FF_0 + 0.05 \tag{4}$$ $$F_{dc} = X_{str}\,\max(F_{dc,1}, F_{dc,2}, F_{dc,3}) + (1 - X_{str}) \tag{5}$$

where

Symbol	Meaning	Unit
$E_e$	the share of the plane irradiance that reaches a shaded cell	—
$G_d$	the sky light on the module plane after the sky masking	W/m²
$G_r$	the ground-reflected light on the plane after the rows	W/m²
$G_b$	the direct light on the module plane after the horizon	W/m²
$G_{cs}$	the circumsolar light on the plane	W/m²
$F_{dc,1}$	the shaded string's power over its unshaded power, few strings shaded	—
$S_{sub}$	the fraction of a shaded string's substrings in shade	—
$X_{str}$	the fraction of strings shaded	—
$FF_0$	the module fill factor at standard conditions	—
$\kappa_1, \kappa_2$	the fitted coefficients of the paper	—
$F_{dc,2}$	the shaded string's power over its unshaded power, many strings shaded	—
$V_D$	the bypass diode's forward voltage. 0.5 V	V
$V_{mp0}$	the substring's maximum-power voltage, the module's over its bypass diodes	V
$F_{dc,3}$	the shaded string's power over its unshaded power, many substrings shaded	—
$\kappa_3, \kappa_{3,0}$	the slope of the large-shade limit and its fitted value	—
$F_{dc}$	the array's DC power over its unshaded power	—

References

1. Deline, Dobos, Janzou, Meydbray & Donovan 2013, A simplified model of uniform shading in large photovoltaic arrays, Solar Energy 96 274–282

API

array.shading_model = deline_nonlinear_published

Symbols follow the symbol registry.