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 only when measured power falls by more than the deadband; it keeps the previous direction inside that band. This is a voltage-reference, deadband variant, not the exact duty-cycle sign rule of Femia (2005), Eq. (1), PDF p. 2. 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.

Sampling interval. Four time constants leave $e^{-4}=1.83\%$ of a first-order voltage transient. The 4τ card is a first-order estimate, not a universal minimum. Femia (2005), PDF p. 5 (printed p. 967), Eq. (14), instead estimates power settling for a lightly damped second-order converter using its damping and natural frequency. Such a plant can settle more slowly than this assumed voltage servo. Select the hardware period beyond measured converter and filter settling; a 63.2% response time alone does not characterize ringing.

Dynamic voltage-step threshold. Femia (2005), PDF pp. 7–8 (printed pp. 969–970), Eqs. (15)–(23), compares perturbation-induced power change with irradiance-induced power change. Adapt the criterion to a settled voltage-reference perturbation: define $H=-\tfrac12\partial^2 I/\partial V^2$, $K_G=\partial I/\partial G$ at fixed voltage, and $C=HV_{mp}+I_{mp}/V_{mp}=-\tfrac12\partial^2P/\partial V^2$, all at each module MPP. The local estimates are $|\Delta P_V|\approx C(\Delta V)^2$ and $|\Delta P_G|\approx V_{mp}K_G|\dot G|T_s$. Hence

Here the plant has unity steady voltage-reference gain; no duty-cycle gain is inferred. The fitted single-diode model supplies the local fixed-voltage sensitivity $K_G$, rather than assuming a universal material constant. With $D=I_0\exp[(V+IR_s)/a]/a+1/R_p$, implicit differentiation gives $K_G=(I_{ph}/G)/(1+R_sD)$ and $H=I_0\exp[(V+IR_s)/a]/[2a^2(1+R_sD)^3]$.

The cloud changes by $G-G_{min}$ in one second, so the maximum ramp rate is $(G-G_{min})/(1\,s)$. Compute the threshold at 129 irradiance levels across that range at the selected cell temperature and report the largest assessed value and its irradiance. The array threshold is the module threshold times the series count; parallel count cancels. The threshold grows with $\sqrt{|\dot G|T_s}$ and is zero for constant irradiance. The warning compares the chosen step against this near-MPP estimate. It assumes settled samples, small perturbations and a locally smooth curve; it is not a guarantee during acquisition or large excursions, and it omits deadband margin. The separate one-step/deadband warning remains relevant.

Loss and efficiency definitions. The ±one-step loss card uses the exact static curve. Femia's Eqs. (24)–(25), PDF p. 9 (printed p. 971), instead estimate a worst instantaneous loss for a four-point swing reaching two steps from the MPP. The captured-energy percentages here are $100\int P(t)\,dt/\int P_{MPP}(t)\,dt$, including acquisition and cloud ramps; they are different from that worst-loss efficiency and do not represent converter efficiency.

Animated operating point. At playback time, select the latest controller sample and its held voltage command. Evolve that sample's voltage exponentially to the current time and bound it between zero and the current open-circuit voltage, using the same trajectory as the energy integral below. Both the moving marker and its voltage/power caption use this continuous operating point; history charts retain the sampled data. At a controller sample time the marker coincides with its sampled voltage.

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 evaluates the held command's exponential voltage response at the current time and bounds it by instantaneous Voc; time-history charts retain controller-sampled values.
• Captured energy and the animated operating point use the same first-order trajectory between updates, continuous irradiance and instantaneous Voc bound.
• Dynamic P&O guidance uses local derivatives at each MPP and the maximum rate of the specified uniform cloud ramps. It assesses 129 equally spaced irradiance levels at the selected temperature, not every possible curve or converter. The threshold excludes a safety margin for the comparison deadband and is advisory for the P&O variant simulated here.

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 first-order time constant. The 4τ estimate does not replace the slower settling bound of a lightly damped second-order converter (Femia 2005, PDF p. 5, Eq. 14). The dynamic voltage-step threshold adapts Femia's near-MPP criterion (PDF p. 8, Eq. 23) to the fitted module and this cloud profile. It is an approximate lower threshold; exceeding it does not guarantee correct tracking with deadbands, unsettled samples, quantization, off-MPP operation or a different irradiance history. 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.