PV Single-Diode Model: Datasheet I-V Curves and MPPT

The module equation is $I=I_{ph}-I_0[\exp((V+IR_s)/a)-1]-(V+IR_s)/R_p$, with $a=nN_{cell}kT/q$. Power is $VI$. Villalva et al., open companion, pp. 2–5, motivates coupled adjustment of the resistances through the datasheet MPP.

Exact endpoint refinement. For each trial $R_s$, solve the linear equations in $I_0$ and $g_p=1/R_p$ obtained by subtracting the short-circuit equation from the open-circuit and MPP equations. Recover $I_{ph}$ from the short-circuit equation. Bracket and bisect $R_s$ until $I_{mp}-V_{mp}H/(1+R_sH)=0$, where $H=I_0\exp((V_{mp}+I_{mp}R_s)/a)/a+g_p$. This is the stationary-power form of a coupled Rs/Rp iteration, derived here; it avoids a grid-dependent peak search. The fit flag rejects solutions without positive physical parameters.

Temperature and irradiance. Following the coefficient-based approach of Villalva eqs. (3) and (6), $I_{ph}(G,T)=[I_{ph,ref}+\alpha_{sc}\Delta T]G/G_{ref}$. An endpoint correction gives $I_0(T)=[I_{ph,ref}+\alpha_{sc}\Delta T-(V_{oc,ref}+\beta_{oc}\Delta T)/R_p]/[\exp((V_{oc,ref}+\beta_{oc}\Delta T)/a(T))-1]$. Thus Voc follows the supplied coefficient at reference irradiance. Rs, Rp and n remain constant. Newton iteration solves the implicit current; a bracketed diode-voltage solve locates the MPP. Array voltage and current scale with series and parallel module counts.

Tracker experiment. Both algorithms step the array voltage reference. Between samples the assumed first-order voltage loop follows $V_k=V_{cmd}+(V_{k-1}-V_{cmd})\exp(-T_s/\tau)$. P&O reverses its last perturbation when measured power falls by more than the deadband. INC uses $I+V\Delta I/\Delta V$ with a deadband and handles negligible voltage change separately. Irradiance changes may confuse either finite-difference method. Curves and samples are deterministic; measurement noise is not added.

Four time constants leave $e^{-4}=1.83\%$ of a first-order voltage transient. The minimum-period card uses this derived settling estimate. Femia (2005) and the open Energies (2021) study explain why actual sampling must follow converter/filter dynamics. The ±step loss card evaluates the static curve exactly; it is not an average oscillation-loss prediction.

Energy between controller samples. Each sampled voltage command is held until the next update. Power is integrated along the continuous first-order trajectory $V(t_k+u)=V_{cmd,k}+[V(t_k)-V_{cmd,k}]\exp(-u/\tau)$, bounded by zero and the instantaneous open-circuit voltage. The energy grid is separate from the MPPT sample grid and independent of playback time. Integration segments split at the 2, 3, 5 and 6 s irradiance corners and at 2, 6 and 16 time constants after each update, resolving fast settling even when the sample period is long.

Gauss–Legendre quadrature compares four and eight points per segment, refining to 16 and 32 if needed. The largest change in either captured-energy percentage, or in available energy as a percentage of its refined value, must be at most 0.001 percentage point. The available-energy integral includes four seconds at starting irradiance, two seconds at minimum irradiance and the two one-second ramps. Captured energy is not estimated by joining controller power samples with straight lines. The reported refinement change is a numerical convergence estimate, not a model-accuracy guarantee.

• Identical silicon modules, uniform irradiance and cell temperature, constant series and shunt resistance, no interconnect loss. Cell temperature is supplied directly.
• STC data are nominal values. n is an explicit assumption, not an independently extracted material constant. Several parameter sets may fit a datasheet.
• The KC200GT preset uses the manufacturer's 0.00318 A/K coefficient; Villalva rounds this to 0.0032 A/K. Its n=1.3 is from the paper.
• The CEC preset is a numerical benchmark. Its n is derived from the published modified ideality voltage; the environmental law here is coefficient-based and is not the CEC/De Soto environmental model.
• The MPPT plant is an assumed first-order voltage servo with no overshoot or switching model. The power deadband represents a comparison threshold, not a stochastic noise model. Both algorithms begin at the same fraction of Voc.
• The simulated cloud affects every cell equally. Playback displays the most recent sampled voltage on the continuously changing static curve; time-history values are sampled values.
• Captured energy uses the exponential voltage response between updates, with continuous irradiance and the instantaneous Voc bound. The animation retains the most recent sampled voltage as its marker; its curve uses the exact irradiance at playback time.

Use consistent STC points with 0 < Imp < Isc and 0 < Vmp < Voc. Custom modules must have a positive integer series-cell count, 0.5 ≤ n ≤ 2, positive Isc coefficient and negative Voc coefficient. A failed fit means the supplied points and assumed n do not admit this positive-resistance model; check the points and adjust n using measured curves.

The selected cell temperature range is −20 to 70 °C; the temperature-family curves extend another 15 K in either direction. Starting irradiance and the entire MPPT experiment must stay within 50–1200 W/m². The minimum experiment irradiance is G × cloud ratio; settings below 50 W/m² are blocked. A low-light uncertainty warning appears whenever this minimum is below 200 W/m². Low-light and extreme-temperature predictions are extrapolations, not a calibrated accuracy guarantee. Validate against additional manufacturer curves, especially below 200 W/m².

Villalva Table II lists Rs=0.221 Ω and Rp=415.405 Ω for the KC200GT. These rounded approximate parameters do not exactly meet the endpoint and stationary-MPP equations used here. The exact refinement yields about 0.23077 Ω and 597.37 Ω while retaining the published Isc, Voc, Imp and Vmp. The reference case therefore checks the independent datasheet points, not those approximate resistances. The CEC benchmark independently checks fitted resistances.

Sampling guidance is conditional on the supplied time constant. Real converters may have resonances, delays, saturation and interacting control loops. Hardware PI tuning needs a plant model or measured response. This tool does not size protection, cables or legal string-voltage limits. Partial shading and bypass diodes require a multi-substring model (catalog subtask #841); double-diode physics is outside scope. Installation work belongs to a qualified person.