Solar Module Partial Shading: Bypass Diodes, Multi-Peak MPPT and Hot Spots (2D Cell Map)

Presets

Live readouts

Graphs

Module current against voltage (shaded and unshaded)

Module power against voltage (local maxima in orange)

Tracked power and global maximum against time

Global maximum power against shadow position

Numerical checks

running...

How to use

Choose a preset, then move the shadow with the sliders (in 2D you can drag it on the module). The green curve is the shaded module, the dashed grey one the same module in full sun; orange dots are the local power maxima and the red dot is where the tracker is operating. Try: (1) Pole across one substring and watch a second P-V hill appear; (2) Hill-climbing trap then Same, global scan and compare tracked and available power; (3) Bird dropping, no diode and read the worst-cell temperature; (4) switch the bypass diodes off and on with the same shade; (5) enable shadow drift to see the peaks move. Drag on the module to move the shadow. Space pauses, R resets.

The science

Cell model. Every cell is a single-diode circuit with series resistance R_s, shunt resistance R_sh and a reverse-breakdown term (Bishop). With x = V + I R_s:

I = I_L - I₀ [ e^{x/(n V_t)} - 1 ] - (x / R_sh) [ 1 + a (1 - x/V_br)^-m (x < 0) ], V_t = kT/q, I_L = (G/1000) I_sc [1 + α(T - 25 C)]

The page solves this for the cell voltage at a given current by bisection. I₀ is fixed by the open-circuit voltage at standard conditions and scales with temperature as T³ exp[-E_g/(nk)(1/T - 1/T_ref)]. Cell temperature follows the NOCT rule T = T_amb + (NOCT - 20) G/800.

Series strings and bypass diodes. The 60 cells are wired in series as three substrings of 20 cells (two columns of ten). In a series string all cells carry the same current, so the voltage at current I is the sum V_sub(I) = Σ V_cell(I). A cell with less light has a lower I_L; when the string current exceeds it, that cell is driven to negative voltage. A bypass diode across the substring conducts once V_sub falls below -V_f, clamping the substring at about -0.45 V and letting the current of the lit substrings pass:

V_module(I) = Σ_k=1..3 max( V_sub,k(I), -V_f )

The module curve is therefore a staircase: each shaded substring adds a step and a local maximum on the P-V curve. The shaded cell geometry is exact: every cell takes the fraction of its area under the rectangular shadow.

MPPT. Perturb-and-observe changes the operating voltage by a fixed step and keeps the direction while power rises. On a multi-peak curve it can climb the nearest hill and stay there although a higher one exists; the global-scan mode sweeps the whole curve periodically. The tracking efficiency is the tracked power over the global maximum.

Hot spot. At the operating current the worst shaded cell sits at negative voltage V_c and dissipates P = |V_c| I; its temperature is estimated as T_cell + P R_th. With working bypass diodes the voltage across the cell is limited to roughly -(V_f + the voltage of the other cells in the substring), so the dissipation stays small; without a diode the whole module current flows through the shaded cell.

Checks (run in the page). (1) the solver against the closed form V = nV_t ln((I_L - I)/I₀ + 1) for R_s = 0 and R_sh infinite; (2) the unshaded module maximum power against 60 times a separately scanned single-cell maximum; (3) one fully dark, bypassed substring against (2/3) P₀ - I V_f; (4) the number of P-V maxima for three different shade levels equals three.

Simplifications. A uniform cell temperature (no thermal coupling between cells), hard rectangular shadow with one depth, a constant diode drop, an ideal DC-side tracker with no converter loss, no spectral or angle-of-incidence effects, and a one-dimensional thermal model for the hot spot. Real modules differ in cell layout (half-cut cells, landscape orientation) and breakdown behaviour. The time axis of energy is a time-lapse of 600 s per real second.