# 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 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.

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

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.

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

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

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

## 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

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

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 Ns V_{th}} \right) - 1 \right) - \frac{V_{oc, est}}{R_{sh}} \newline\\I = I_L - I_0 \left(\exp \left(\frac{n Ns V_{th} \log \left(\frac{I_L}{I_0} + 1 \right)}{n Ns V_{th}} \right) - 1 \right) - \frac{n Ns 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 Ns 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 Ns 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 Ns V_{th} \log \left(\frac{I_L}{I_0} + 1 \right)}{R_{sh}} \newline\\I = I_L - I_L - \frac{n Ns V_{th} \log \left( \frac{I_L}{I_0} + 1 \right)}{R_{sh}} \newline\\I = - \frac{n Ns 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