Skip to main content

​
Summary

The 5-parameter single-diode model, following the De Soto et al. (2006) formulation, is an equivalent circuit that represents a PV module as a light-generated current source, a diode, a , and a . Given the scaled module parameters from Parameter Translation, the model solves the circuit equation at the module level to determine the (VmpV_{mp}, ImpI_{mp}, PmpP_{mp}) and (VocV_{oc}). All modules within a DC field are assumed to operate under identical conditions, so the equation is solved once per DC field and the results are scaled: voltage is multiplied by the number of modules in series, and current by the number of parallel strings.

​
Inputs

NameSymbolUnitsDescription
PhotocurrentIphI_{ph}ALight-generated current
Saturation CurrentI0I_0ADiode reverse saturation current
Series ResistanceRsR_sΩSeries resistance (includes module internal resistance and DC wiring resistance)
Shunt ResistanceRshR_{sh}ΩShunt resistance of module
Diode Ideality Factorγ\gamma—Diode ideality factor
Number of CellsNsN_s—Cells in series within module
Cell TemperatureTcT_c°COperating cell temperature

​
Outputs

NameSymbolUnitsDescription
Max Power VoltageVmpV_{mp}VVoltage at maximum power point
Max Power CurrentImpI_{mp}ACurrent at maximum power point
Max PowerPmpP_{mp}WVmp×ImpV_{mp} \times I_{mp}
Open-Circuit VoltageVocV_{oc}VVoltage at open-circuit (I=0I = 0)

​
Detailed Description

​
Circuit Equation

De Soto’s single-diode equivalent circuit models four physical mechanisms: IphI_{ph} is the driven by absorbed irradiance, I0I_0 and γ\gamma govern within the cells (exponential diode term), RshR_{sh} accounts for leakage current paths through the cells, and RsR_s captures ohmic losses due to series resistance within the module and DC wiring. All five parameters are scaled from reference to operating conditions by Parameter Translation. The current–voltage relationship that defines the is: I=Iph−I0(exp⁡ ⁣(q(V+IRs)NskTcγ)−1)−V+IRsRshI = I_{ph} - I_0 \left(\exp\!\left(\frac{q(V + IR_s)}{N_s k T_c \gamma}\right) - 1\right) - \frac{V + IR_s}{R_{sh}} where q=1.602×10−19q = 1.602 \times 10^{-19} C is the elementary charge, k=1.381×10−23k = 1.381 \times 10^{-23} J/K is the Boltzmann constant, and TcT_c is the cell temperature converted from °C to Kelvin. For convenience, PlantPredict defines the modified thermal voltage Vth=NsγkTc/qV_{th} = N_s \gamma k T_c / q. The circuit equation is implicit in both II and VV because of the IRsIR_s coupling. PlantPredict uses different numerical strategies depending on which quantity needs to be calculated.

​
Maximum Power Point

PlantPredict introduces the internal voltage Vint=V+IRsV_{int} = V + IR_s, which absorbs the voltage drop due to the series resistance. In this form, both II and VV can be explicitly solved from VintV_{int}: I(Vint)=Iph−I0(eVint/Vth−1)−VintRshI(V_{int}) = I_{ph} - I_0 (e^{V_{int}/V_{th}} - 1) - \frac{V_{int}}{R_{sh}} V(Vint)=Vint−I(Vint) RsV(V_{int}) = V_{int} - I(V_{int}) \, R_s Power is therefore a single-variable function P(Vint)=V(Vint)×I(Vint)P(V_{int}) = V(V_{int}) \times I(V_{int}). PlantPredict finds the maximum by solving dP/dVint=0dP/dV_{int} = 0 using Newton-Raphson iteration on the analytically derived first and second derivatives. Once the optimal internal voltage Vint,mpV_{int,mp} has converged: Imp=I(Vint,mp)I_{mp} = I(V_{int,mp}) Vmp=Vint,mp−ImpRsV_{mp} = V_{int,mp} - I_{mp} R_s Pmp=Vmp×ImpP_{mp} = V_{mp} \times I_{mp}

​
VV Given II

Because the I-V relationship is implicit, solving the circuit equation for VV at a given current is transcendental—no closed-form elementary solution exists. However, the equation can be reformulated into the Lambert W canonical form eaX+b=cX+de^{aX+b} = cX + d, which is well-behaved and guarantees convergence. The resulting Lambert W expression is evaluated numerically via Newton-Raphson iteration. The open-circuit voltage VocV_{oc} is obtained as the special case with I=0I = 0.

​
II Given VV

When the voltage is fixed—for instance, when set by the inverter at an operating point away from MPP (e.g., clipping)—the IRsIR_s term makes the equation implicit in II. PlantPredict uses the same Lambert W reformulation and Newton-Raphson evaluation to solve for II at the imposed voltage.

​
References

  • De Soto, W., Klein, S. A., & Beckman, W. A. (2006). Improvement and validation of a model for photovoltaic array performance. Solar Energy, 80(1), 78–88.