Single diode models#

Single-diode models are a popular means of simulating the electrical output of a PV module under any given irradiance and temperature conditions. A single-diode model (SDM) pairs the single-diode equation (SDE) with a set of auxiliary equations that predict the SDE parameters at any given irradiance and temperature. All SDMs use the SDE, but their auxiliary equations differ. For more background on SDMs, see the PVPMC website.

Three SDMs are currently available in pvlib: the CEC SDM, the PVsyst SDM, and the De Soto SDM. pvlib splits these models into two steps. The first is to compute the auxiliary equations using one of the following functions:

The second step is to use the output of these functions to compute points on the SDE’s I-V curve. Three points on the SDE I-V curve are typically of special interest for PV modeling: the maximum power (MP), open circuit (OC), and short circuit (SC) points. The most convenient function for computing these points is pvlib.pvsystem.singlediode(). It provides several methods for solving the SDE:

Method	Type	Speed	Guaranteed convergence?
`newton`	iterative	fast	no
`brentq`	iterative	slow	yes
`chandrupatla`	iterative	fast	yes
`lambertw`	explicit	medium	yes

Computing full I-V curves#

Full I-V curves can be computed using pvlib.pvsystem.i_from_v() and pvlib.pvsystem.v_from_i(), which calculate either current or voltage from the other, with the methods listed above. It is often useful to first compute the open-circuit or short-circuit values using pvlib.pvsystem.singlediode() and then compute a range of voltages/currents from zero to those extreme points. This range can then be used with the above functions to compute the I-V curve.

IV curves in reverse bias#

The standard SDE does not account for diode breakdown at reverse bias. The following functions can optionally include an extra term for modeling it: pvlib.pvsystem.max_power_point(), pvlib.singlediode.bishop88_i_from_v(), and pvlib.singlediode.bishop88_v_from_i().

Recombination current for thin film cells#

The PVsyst SDM optionally modifies the SDE to better represent recombination current in CdTe and a-Si modules. The modified SDE requires two additional parameters. pvlib functions can compute the key points or full I-V curves using the modified SDE: pvlib.pvsystem.max_power_point(), pvlib.singlediode.bishop88_i_from_v(), and pvlib.singlediode.bishop88_v_from_i().

Model parameter values#

Despite some models having parameters with similar names, parameter values are specific to each model and thus must be produced with the intended model in mind. For some models, sets of parameter values can be read from external sources, for example:

CEC SDM parameter database can be read using retrieve_sam()
PAN files, which can be read using read_panond()

pvlib also provides a set of functions that can estimate SDM parameter values from various datasources:

Function	SDM	Inputs
`fit_cec_sam()`	CEC	datasheet
`fit_desoto()`	De Soto	datasheet
`fit_desoto_sandia()`	De Soto	I-V curves
`fit_pvsyst_sandia()`	PVsyst	I-V curves
`fit_pvsyst_iec61853_sandia_2025()`	PVsyst	IEC 61853-1 matrix

Single-diode equation#

This section reviews the solutions to the single diode equation used in pvlib-python to generate an IV curve of a PV module.

pvlib-python supports two ways to solve the single diode equation:

Lambert W-Function
Bishop’s Algorithm

The pvlib.pvsystem.singlediode() function allows the user to choose the method using the method keyword.

Lambert W-Function#

When method='lambertw', the Lambert W-function is used as previously shown by Jain, Kapoor [1, 2] and Hansen [3]. The following algorithm can be found on Wikipedia: Theory of Solar Cells, given the basic single diode model equation.

\[I = I_L - I_0 \left(\exp \left(\frac{V + I R_s}{n N_s V_{th}} \right) - 1 \right) - \frac{V + I R_s}{R_{sh}}\]

Lambert W-function is the inverse of the function \(f \left( w \right) = w \exp \left( w \right)\) or \(w = f^{-1} \left( w \exp \left( w \right) \right)\) also given as \(w = W \left( w \exp \left( w \right) \right)\). Defining the following parameter, \(z\), is necessary to transform the single diode equation into a form that can be expressed as a Lambert W-function.

\[z = \frac{R_s I_0}{n N_s V_{th} \left(1 + \frac{R_s}{R_{sh}} \right)} \exp \left( \frac{R_s \left( I_L + I_0 \right) + V}{n N_s V_{th} \left(1 + \frac{R_s}{R_{sh}}\right)} \right)\]

Then the module current can be solved using the Lambert W-function, \(W \left(z \right)\).

\[I = \frac{I_L + I_0 - \frac{V}{R_{sh}}}{1 + \frac{R_s}{R_{sh}}} - \frac{n N_s V_{th}}{R_s} W \left(z \right)\]

Bishop’s Algorithm#

The function pvlib.singlediode.bishop88() uses an explicit solution [4] that finds points on the IV curve by first solving for pairs \((V_d, I)\) where \(V_d\) is the diode voltage \(V_d = V + I*Rs\). Then the voltage is backed out from \(V_d\). Points with specific voltage, such as open circuit, are located using the bisection search method, brentq, bounded by a zero diode voltage and an estimate of open circuit voltage given by

\[V_{oc, est} = n N_s V_{th} \log \left( \frac{I_L}{I_0} + 1 \right)\]

We know that \(V_d = 0\) corresponds to a voltage less than zero, and we can also show that when \(V_d = V_{oc, est}\), the resulting current is also negative, meaning that the corresponding voltage must be in the 4th quadrant and therefore greater than the open circuit voltage (see proof below). Therefore the entire forward-bias 1st quadrant IV-curve is bounded because \(V_{oc} < V_{oc, est}\), and so a bisection search between 0 and \(V_{oc, est}\) will always find any desired condition in the 1st quadrant including \(V_{oc}\).

\[ \begin{align}\begin{aligned}I = I_L - I_0 \left(\exp \left(\frac{V_{oc, est}}{n N_s V_{th}} \right) - 1 \right) - \frac{V_{oc, est}}{R_{sh}} \newline\\I = I_L - I_0 \left(\exp \left(\frac{n N_s V_{th} \log \left(\frac{I_L}{I_0} + 1 \right)}{n N_s V_{th}} \right) - 1 \right) - \frac{n N_s V_{th} \log \left(\frac{I_L}{I_0} + 1 \right)}{R_{sh}} \newline\\I = I_L - I_0 \left(\exp \left(\log \left(\frac{I_L}{I_0} + 1 \right) \right) - 1 \right) - \frac{n N_s V_{th} \log \left(\frac{I_L}{I_0} + 1 \right)}{R_{sh}} \newline\\I = I_L - I_0 \left(\frac{I_L}{I_0} + 1 - 1 \right) - \frac{n N_s V_{th} \log \left(\frac{I_L}{I_0} + 1 \right)}{R_{sh}} \newline\\I = I_L - I_0 \left(\frac{I_L}{I_0} \right) - \frac{n N_s V_{th} \log \left(\frac{I_L}{I_0} + 1 \right)}{R_{sh}} \newline\\I = I_L - I_L - \frac{n N_s V_{th} \log \left( \frac{I_L}{I_0} + 1 \right)}{R_{sh}} \newline\\I = - \frac{n N_s V_{th} \log \left( \frac{I_L}{I_0} + 1 \right)}{R_{sh}}\end{aligned}\end{align} \]

References#

[1] “Exact analytical solutions of the parameters of real solar cells using Lambert W-function,” A. Jain, A. Kapoor, Solar Energy Materials and Solar Cells, 81, (2004) pp 269-277. DOI: 10.1016/j.solmat.2003.11.018

[2] “A new method to determine the diode ideality factor of real solar cell using Lambert W-function,” A. Jain, A. Kapoor, Solar Energy Materials and Solar Cells, 85, (2005) 391-396. DOI: 10.1016/j.solmat.2004.05.022

[3] “Parameter Estimation for Single Diode Models of Photovoltaic Modules,” Clifford W. Hansen, Sandia Report SAND2015-2065, 2015 DOI: 10.13140/RG.2.1.4336.7842

[4] “Computer simulation of the effects of electrical mismatches in photovoltaic cell interconnection circuits” JW Bishop, Solar Cell (1988) DOI: 10.1016/0379-6787(88)90059-2