PV Partial Shading Calculator: Bypass Diodes and Optimizers

The forward and reverse cell characteristic follows Bishop (1988), as documented by pvlib. With junction voltage $V_d=V+IR_s$:

Here $a_T=nkT/q$, $V_{br}<0$, and the thin-film recombination term is omitted for crystalline silicon. The avalanche term is evaluated on the physical side $V_d>V_{br}$.

Default cell parameters are the illustrative pvlib shading example (attached source s3, page 1): $I_{L,ref}=8.24$ A, $I_{0,ref}=2.36\times10^{-9}$ A, $n=1.3$, $R_s=0.00181$ Ω, $R_{sh,ref}=1000$ Ω, $\alpha_{sc}=0.0042$ A/K, $b=0.002$, $m=3$, $V_{br}=-15$ V. They are not a commercial panel specification. Optional custom values are per full cell, not per module.

The De Soto temperature and irradiance translations are used: $I_L=(G/G_{ref})[I_{L,ref}+\alpha_{sc}(T-T_{ref})]$, $R_{sh}=R_{sh,ref}G_{ref}/G$, and $R_s$ is constant. Saturation current uses $I_0=I_{0,ref}(T/T_{ref})^3\exp[E_{g,ref}/(kT_{ref})-E_g(T)/(kT)]$, with the silicon defaults 1.121 eV and relative bandgap coefficient −0.0002677 K⁻¹. Bandgap divided by elementary charge is stored in volts so the exponent is dimensionless.

The moving band’s overlap with each rectangular cell determines its blocked area fraction $c$. Effective irradiance is $G(c)=G_{POA}[1-c(1-f_{diff})(1-t_{beam})]$. A partially covered cell is represented by its area-average irradiance. Diode and avalanche current terms are cached on a 1041-point junction-voltage grid, with logarithmic spacing near reverse breakdown. Inversion by linear interpolation gives each cell’s terminal voltage at a common string current. The current search combines 801 full-range samples, 200 low-current samples and every third illuminated/shaded cell junction-current knot. These knots resolve the steep voltage change near photocurrent. Samples closer than 10⁻¹⁰ of the largest sampled current are merged before assembling the string curve.

The illuminated-to-shaded cell-current difference is independent of sweep position. At each position only the two boundary-cell current arrays are formed, then reused by substring assembly and every short-circuit root trial. The full-light and full-shadow cell characteristics stay cached throughout the play. Peak lookup uses Boolean index selection to skip flat slopes without repeated filtering of the full sample grid.

Cell voltages add in each two-column substring. The constant-drop diode approximation clamps its terminal voltage to $\max(V_{substring},-V_f)$; three substrings add to one module, and modules add in series. Raw, unclamped trial voltages determine the diode state; they are not the actual cell voltages while the diode carries current. The model does not calculate hot-spot heating.

The short-circuit endpoint is solved with Brent’s method on the assembled cell circuit, using the two samples bracketing zero string voltage. Both charts retain every positive-voltage sample and append $(0,I_{sc})$, so they include the complete generating branch. Global MPPT selects the largest sampled $VI$; the local option selects the highest-voltage sampled local maximum. A peak requires a positive slope followed by a negative slope; flat differences below 10⁻¹⁰ of peak power are ignored. Independently maximizing module power provides the optimizer input-power bound. Optimizer DC output is $\eta_{opt}\sum P_{mpp,module}$, gain is relative to the selected string MPPT, and break-even efficiency is $P_{string}/\sum P_{mpp,module}$.

The Sun Position workflow supplies solar elevation and plane-of-array irradiance. For the optional horizontal-plane construction, elementary ray geometry gives $L_{shadow}=H/\tan\alpha$ and normalized band width $L_{shadow}/L_{axis}$. The sweep position is independent of solar time.

Published references check the unshaded numerical solver, the rounded bottom-edge shading example and the reverse clamp. Physics properties also check bounds, local/global ordering, the independent-module upper bound, series scaling and the effect of diffuse light.

• One series string. A chosen number of modules share exactly the same rectangular shadow; the others are identical and unshaded. Six columns form three full-cell, down-and-back substrings, with one diode per substring.
• Uniform cell temperature, identical cell parameters and constant wiring resistance contained in the cell series resistance. Irradiance affects photocurrent and shunt resistance; diffuse light is not blocked.
• The Bishop and De Soto parameters characterize illustrative crystalline silicon. They are not fitted to a named module; custom values require an appropriate cell characterization.
• Constant bypass-diode forward voltage; reverse leakage, thermal feedback and diode failure are excluded. Avalanche current is an electrical model, not a safe operating limit.
• Module optimizers are fitted to all modules, each reaches its own global MPP, and their constant conversion efficiency is applied to the total power. Voltage/current ratios, string voltage limits, clipping, start-up thresholds and load-dependent efficiency are excluded. This makes the result an idealized DC comparison.
• The local MPPT result is a static example of a basin reached from open circuit. Actual inverter firmware may escape it during a scan.
• Shadow motion is geometric. The horizontal-plane projection uses a two-dimensional ray section with the obstacle oriented across the rays. It supplies an illustrative band width; no daily or annual shadow schedule is inferred.
• Fractions are entered as 0–1 numbers; 0.98 means 98%. Power gains are also shown as percentages.

Screening and teaching for full-cell crystalline-silicon modules with 48–84 series cells in the stated six-column wiring, 1–30 series modules, 100–1400 W/m² plane-of-array irradiance, diffuse fraction 0.01–1 and uniform cell temperature −10 to 80 °C. Positive diffuse light avoids extrapolating the De Soto shunt law to complete darkness. Custom parameter ranges are numerical exploration limits, not guaranteed fits for real cells.

The Sun Position geometry option is restricted to a horizontal receiving plane and solar elevation 5–85°. Keep the projected band no wider than the chosen module dimension. Use a site shade model for tilted panels, trees, multiple objects and annual energy.

MPP and peak locations are sampled; very close or narrow peaks may merge, especially with extreme custom parameters. The standard numerical benchmark agrees within 0.02 W in power and 0.03 V in sampled MPP voltage; this is numerical verification, not a field-accuracy claim. The published “roughly 80%” shading example is deliberately checked with a five percentage-point tolerance.

No prediction of hot-spot temperature, reverse-bias safety, diode rating, insulation, fire protection, equipment compatibility, AC output, annual yield or payback. Half-cut parallel substrings, shingled cells and thin-film modules require their own circuit topology and parameters. Check the module and inverter manufacturers’ requirements. Installation work belongs to a qualified person.