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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 :py:func:`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

If lower accuracy (within ~1%) is allowable, the special points can be estimated much more quickly using :py:func:`pvlib.singlediode.batzelis_keypoints`.

Computing full I-V curves

Full I-V curves can be computed using :py:func:`pvlib.pvsystem.i_from_v` and :py:func:`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 :py:func:`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: :py:func:`pvlib.pvsystem.max_power_point`, :py:func:`pvlib.singlediode.bishop88_i_from_v`, and :py:func:`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: :py:func:`pvlib.pvsystem.max_power_point`, :py:func:`pvlib.singlediode.bishop88_i_from_v`, and :py:func:`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:

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

Function SDM Inputs
:py:func:`~pvlib.ivtools.sdm.fit_cec_sam` CEC datasheet
:py:func:`~pvlib.ivtools.sdm.fit_desoto` De Soto datasheet
:py:func:`~pvlib.ivtools.sdm.fit_desoto_sandia` De Soto I-V curves
:py:func:`~pvlib.ivtools.sdm.fit_pvsyst_sandia` PVsyst I-V curves
:py:func:`~pvlib.ivtools.sdm.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:

  1. Lambert W-Function
  2. Bishop's Algorithm

The :func:`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 :func:`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}.

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}}

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`