Sub-module I-V curves, PVsyst application

Method

Each table of the scene carries its modules on an even grid that fills it edge to edge. The grid's rows across the width and its module orientation are the pair whose width-to-length ratio is nearest the table's, from the module's long-to-short side ratio, portrait first on a tie. A module holds 3 sub-modules side by side across its short side, each running its long side and protected by one bypass diode. A module of twin half cells holds two halves along its long side. The two sub-modules of one diode work in parallel. Six cells cross a module's short side. Its long side holds the cells in series times the cells in parallel over 6. Each sub-module is tested at its four corners, set half a cell in from its lower and upper edges. A corner counts as shaded under the shadow of a neighbouring table or of an object, or under a tree line that blocks at least half the direct light. A string is a run of a table's modules, taken column by column for strings side by side along the length and row by row for strings stacked across the width. The placements that name one string form that string. At each step with at least 5 W/m² of direct light on the plane, the field's strings are grouped by the shade on their sub-modules. The step counts when the shaded corners cover from 1 % to 98 % of the strings' modules, a quarter of a sub-module for each corner. A sub-module with 0, 1, 2, 3 or 4 shaded corners has none, 10 %, a third, two thirds or all of its cells at the shaded irradiance. A lit cell receives the direct light, the ring around the sun, the diffuse light and the ground reflection on the plane after their shading, the incidence-angle loss and soiling. A shaded cell receives the diffuse light and the ground reflection. Each cell follows the module's one-diode curve at the step's cell temperature, scaled to one cell of one branch. Under reverse voltage it follows the quadratic reverse characteristic. The cells of a sub-module add their voltages. The sub-modules of one diode share its current. The diode keeps the group's voltage above the negative of its forward voltage. The modules of a string add their voltages. The strings of one MPPT input add their currents. The input works at its highest power at or above the inverter's MPPT minimum voltage. A string on an input of its own finds its own highest power. The shaded power over the power of the same strings unshaded, less the beam shaded fraction, gives the electrical loss as a share of the direct light and the ring around the sun. The electrical shading row applies it to the module power before the electrical row. The row is computed only on steps where a sub-module corner is in shade. If not given: every module has 3 bypass diodes. A module with twin half cells has 2 cells in parallel.

The math

1. Module grid of table $t$
2. Corner test points of a sub-module
3. Cell voltage at a branch current
4. Sub-module voltage
5. Diode group and string
6. Power of one MPPT input
7. Linear share of the shaded corners
8. Electrical loss on the direct light

$$(o_t, n_u, n_a) = \arg\min_{n_u n_a = M_t} \left| \ln \frac{n_u\,a_u(o)}{n_a\,a_a(o)\,W_t/L_t} \right| \tag{1}$$ $$v \in \left\{ v_0 + \tfrac{1}{2} c,\ v_1 - \tfrac{1}{2} c \right\},\qquad c = \frac{W_t}{n_u\,r},\qquad r = \begin{cases} N_s N_p / 6 & \text{portrait} \\ 6 & \text{landscape} \end{cases} \tag{2}$$ $$N_p i = \begin{cases} I_{ph} - I_0\left(e^{V_d/a} - 1\right) - \dfrac{V_d}{R_{sh}} & N_p i < I_{ph} \\ I_{ph} + \dfrac{N_p B_{rev}}{N_s^2}\,V_d^2 & N_p i \ge I_{ph} \end{cases},\qquad v(i) = \frac{V_d - N_p i\,R_s}{N_s} \tag{3}$$ $$V_k(i) = \frac{N_s}{d}\left[(1 - s_k)\,v_u(i) + s_k\,v_s(i)\right],\qquad s_k = \left(0,\ 0.1,\ \tfrac{1}{3},\ \tfrac{2}{3},\ 1\right) \tag{4}$$ $$V_g(I) = \max\!\left(V_\parallel(I),\ -V_D - R_D I\right),\qquad \sum_j i_j\!\left(V_\parallel\right) = I,\qquad V_{str}(I) = \sum_{m} \sum_{g} V_{g,m}(I) \tag{5}$$ $$P_{sh} = \max_{V \ge V_{min}} V \sum_c w_c\,I_c(V),\qquad P_{mpp} = \max_{V} V\,I_0(V) \tag{6}$$ $$L = \frac{\sum_c n_c \sum_{m \in c} \sum_{j \in m} k_j / (4 J)}{\sum_c n_c M_c},\qquad f_{elec} = 0 \text{ unless } 0.01 \le L \le 0.98 \tag{7}$$ $$F_{el} = \frac{1 - P_{sh}/P_{mpp}}{(G_b + G_c)/G_u} - F_b,\qquad G_u = G_b + G_c + G_d,\qquad f_{elec} = \max(0, F_{el})\,\frac{G_b + G_c}{G_b} \tag{8}$$

where

Symbol	Meaning	Unit
$o_t$	the module orientation on table $t$: portrait (long side up the slope) or landscape	—
$n_u, n_a$	the module rows across the table's width and the modules along its length	—
$M_t$	the modules of table $t$	—
$a_u(o), a_a(o)$	the module's side up the slope and along the table over its short side: the long-to-short side ratio and 1 in portrait, 1 and that ratio in landscape	—
$W_t, L_t$	the width and the length of table $t$	the scene's unit
$v$	the up-slope coordinate of a test point	the scene's unit
$v_0, v_1$	the lower and upper edges of the sub-module	the scene's unit
$c$	the height of one cell up the slope	the scene's unit
$r$	the cells of one module up the slope	—
$N_s$	the cells in series of the module (module.n_series)	—
$N_p$	the cells in parallel of the module (module.n_parallel, 2 for twin half cells)	—
$i$	the current of one branch of cells	A
$I_{ph}, I_0, a, R_s, R_{sh}$	the module's one-diode parameters at the cell's irradiance and the step's cell temperature	A, A, V, Ω and Ω
$V_d$	the module's diode voltage	V
$B_{rev}$	the reverse characteristic of one cell, 0.02222 A/V²	—
$v(i)$	the voltage of one cell at branch current $i$	V
$V_k(i)$	the voltage of a sub-module with $k$ shaded corners at branch current $i$	V
$d$	the bypass diodes of a module, 3	—
$s_k$	the share of a sub-module's cells shaded with $k$ shaded corners	—
$v_u, v_s$	the voltage of a lit and of a shaded cell	V
$V_g(I)$	the voltage of one diode group at string current $I$	V
$V_\parallel(I)$	the common voltage of the group's sub-modules carrying $I$ together	V
$i_j(V)$	the current of sub-module $j$ of the group at voltage $V$	A
$V_D, R_D$	the bypass diode's forward voltage and resistance, 0.7 V and 0.01 Ω	—
$V_{str}(I)$	the voltage of a string at current $I$	V
$P_{sh}$	the power of the input with the step's shade, per string	W
$V_{min}$	the inverter's MPPT minimum voltage (inverter.v_mppt_min_v)	V
$w_c$	the share of the field's strings in shading state $c$	—
$I_c(V)$	the current of a string in state $c$ at voltage $V$	A
$P_{mpp}$	the power of the same strings unshaded at their maximum, per string	W
$I_0(V)$	the current of an unshaded string at voltage $V$	A
$L$	the linear share of the shaded corners over the strings' modules	—
$n_c$	the strings in shading state $c$	—
$k_j$	the shaded corners of sub-module $j$	—
$J$	the sub-modules of one module	—
$M_c$	the modules of a string in state $c$	—
$F_{el}$	the electrical loss as a share of the direct light and the ring around the sun	—
$G_b, G_c$	the direct light and the ring around the sun on the plane after the horizon, the incidence-angle loss and soiling, before shading	W/m²
$G_d$	the diffuse light and the ground reflection on the plane after their shading, the incidence-angle loss and soiling	W/m²
$G_u$	the irradiance of a lit cell	W/m²
$F_b$	the beam shaded fraction of the step	—
$f_{elec}$	the share of the direct light lost to the electrical effect	—

Note

Compared with PVsyst 8.1.5's detailed electrical shading on a field of 80 fixed racks of 3 strings of 30 twin half-cell modules in portrait on one MPPT input, read through its exact shading projection, the annual electrical shading loss agrees within 1 % and the annual energy within 0.03 %.

References

1. PVsyst 8 help, Electrical shadings: Module Layout
2. PVsyst 8 help, Reverse characteristic of a cell
3. Bishop 1988, Computer simulation of the effects of electrical mismatches in photovoltaic cell interconnection circuits, Solar Cells 25, 73-89

API

array.shade_mode = module_layout

Symbols follow the symbol registry.