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DOC: move proof that estimated Voc is in q4
* closes pvlib#518 * add singlediode.rst and add to index
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‎docs/sphinx/source/index.rst

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api
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comparison_pvlib_matlab
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variables_style_rules
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singlediode
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Indices and tables

‎docs/sphinx/source/singlediode.rst

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.. _singlediode:
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Single Diode Model
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==================
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This section reviews the solutions to the single diode model used in
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pvlib-python to generate an IV curve of a PV module.
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pvlib-python supports two ways to solve the single diode model, by passing the
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a ``method`` keyword to the :func:`pvlib.pvsystem.singlediode` function:
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1. Lambert W-Function
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2. Bishop's Algorithm
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Lambert W-Function
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------------------
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When ``method='lambertw'``, the Lambert W-Function is used as previously shown
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by Jain and Kapoor [1]. The following algorithm can be found on
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`Wikipedia: Theory of Solar Cells
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<https://en.wikipedia.org/wiki/Theory_of_solar_cells>`_ given the basic single
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diode model equation.
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.. math::
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I = I_L - I_0 \left(\exp \left(\frac{\left(V + I R_s \right)}{n Ns V_{th}} \right) - 1 \right)
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- \frac{\left(V + I R_s \right)}{R_{sh}}
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Lambert W-function is the inverse of the function
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:math:`f \left( w \right) = w \exp \left( w \right)` or
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:math:`w = f^{-1} \left( w \exp \left( w \right) \right)` also given as
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:math:`w = W \left( w \exp \left( w \right) \right)`. Rearranging the single
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diode equation above with a Lambert W-function yields the following.
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.. math::
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z = \frac{R_s I_0}{n Ns V_{th} \left(1 + \frac{R_s}{R_{sh}} \right)} \exp \left(
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\frac{R_s \left( I_L + I_0 \right) + V}{n Ns V_{th} \left(1 + \frac{R_s}{R_{sh}}\right)}
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\right)
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The the module current can be solved using the Lambert W-function.
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.. math::
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I = \frac{I_L + I_0 - \frac{V}{R_{sh}}}{1 + \frac{R_s}{R_{sh}}}
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- \frac{n Ns V_{th}}{R_s} W(z)
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Bishop's Algorithm
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------------------
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The function :func:`pvlib.singlediode.bishop88` uses an explicit solution [2]
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that finds points on the IV curve by first solving for pairs :math:`(V_d, I)`
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where :math:`V_d` is the diode voltage :math:`V_d = V + I*Rs`. Then the voltage
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is backed out from :math:`V_d`. Points with specific voltage, such as open
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circuit, are located using the bisection search method, ``brentq``, bounded
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by a zero diode voltage and an estimate of open circuit voltage given by
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.. math::
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V_{oc, est} = n Ns V_{th} \log \left( \frac{I_L}{I_0} + 1 \right)
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We know that :math:`V_d = 0` corresponds to a voltage less than zero, and
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we can also show that when :math:`V_d = V_{oc, est}`, the resulting
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current is also negative, meaning that the corresponding voltage must be
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in the 4th quadrant and therefore greater than the open circuit voltage
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(see proof below). Therefore the entire forward-bias 1st quadrant IV-curve
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is bounded, and a bisection search within these points will always find
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desired condition.
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.. math::
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I = I_L - I_0 \left(\exp \left(\frac{V_{oc, est}}{n Ns V_{th}} \right) - 1 \right)
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- \frac{V_{oc, est}}{R_{sh}} \newline
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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)
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- \frac{n Ns V_{th} \log \left(\frac{I_L}{I_0} + 1 \right)}{R_{sh}} \newline
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I = I_L - I_0 \left(\exp \left(\log \left(\frac{I_L}{I_0} + 1 \right) \right) - 1 \right)
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- \frac{n Ns V_{th} \log \left(\frac{I_L}{I_0} + 1 \right)}{R_{sh}} \newline
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I = I_L - I_0 \left(\frac{I_L}{I_0} + 1 - 1 \right)
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- \frac{n Ns V_{th} \log \left(\frac{I_L}{I_0} + 1 \right)}{R_{sh}} \newline
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I = I_L - I_0 \left(\frac{I_L}{I_0} \right)
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- \frac{n Ns V_{th} \log \left(\frac{I_L}{I_0} + 1 \right)}{R_{sh}} \newline
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I = I_L - I_L - \frac{n Ns V_{th} \log \left( \frac{I_L}{I_0} + 1 \right)}{R_{sh}} \newline
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I = - \frac{n Ns V_{th} \log \left( \frac{I_L}{I_0} + 1 \right)}{R_{sh}}
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References
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----------
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[1] A. Jain, A. Kapoor, "Exact analytical solutions of the
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parameters of real solar cells using Lambert W-function", Solar
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Energy Materials and Solar Cells, 81 (2004) 269-277.
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[2] "Computer simulation of the effects of electrical mismatches in
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photovoltaic cell interconnection circuits" JW Bishop, Solar Cell (1988)
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https://doi.org/10.1016/0379-6787(88)90059-2

‎pvlib/pvsystem.py

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@@ -1963,52 +1963,12 @@ def singlediode(photocurrent, saturation_current, resistance_series,
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open-circuit.
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If the method is either ``'newton'`` or ``'brentq'`` and ``ivcurve_pnts``
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are indicated, then :func:`pvlib.singlediode.bishop88` is used to
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are indicated, then :func:`pvlib.singlediode.bishop88` [4] is used to
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calculate the points on the IV curve points at diode voltages from zero to
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open-circuit voltage with a log spacing that gets closer as voltage
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increases. If the method is ``'lambertw'`` then the calculated points on
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the IV curve are linearly spaced.
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The ``bishop88`` method uses an explicit solution from [4] that finds
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points on the IV curve by first solving for pairs :math:`(V_d, I)` where
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:math:`V_d` is the diode voltage :math:`V_d = V + I*Rs`. Then the voltage
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is backed out from :math:`V_d`. Points with specific voltage, such as open
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circuit, are located using the bisection search method, ``brentq``, bounded
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by a zero diode voltage and an estimate of open circuit voltage given by
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.. math::
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V_{oc, est} = n Ns V_{th} \\log \\left( \\frac{I_L}{I_0} + 1 \\right)
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We know that :math:`V_d = 0` corresponds to a voltage less than zero, and
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we can also show that when :math:`V_d = V_{oc, est}`, the resulting
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current is also negative, meaning that the corresponding voltage must be
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in the 4th quadrant and therefore greater than the open circuit voltage
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(see proof below). Therefore the entire forward-bias 1st quadrant IV-curve
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is bounded, and a bisection search within these points will always find
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desired condition.
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.. math::
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I = I_L - I_0 \\left(\\exp \\left(\\frac{V_{oc, est}}{n Ns V_{th}} \\right) - 1 \\right)
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- \\frac{V_{oc, est}}{R_{sh}} \\newline
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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)
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- \\frac{n Ns V_{th} \\log \\left(\\frac{I_L}{I_0} + 1 \\right)}{R_{sh}} \\newline
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I = I_L - I_0 \\left(\\exp \\left(\\log \\left(\\frac{I_L}{I_0} + 1 \\right) \\right) - 1 \\right)
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- \\frac{n Ns V_{th} \\log \\left(\\frac{I_L}{I_0} + 1 \\right)}{R_{sh}} \\newline
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I = I_L - I_0 \\left(\\frac{I_L}{I_0} + 1 - 1 \\right)
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- \\frac{n Ns V_{th} \\log \\left(\\frac{I_L}{I_0} + 1 \\right)}{R_{sh}} \\newline
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I = I_L - I_0 \\left(\\frac{I_L}{I_0} \\right)
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- \\frac{n Ns V_{th} \\log \\left(\\frac{I_L}{I_0} + 1 \\right)}{R_{sh}} \\newline
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I = I_L - I_L - \\frac{n Ns V_{th} \log \\left( \\frac{I_L}{I_0} + 1 \\right)}{R_{sh}} \\newline
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I = - \\frac{n Ns V_{th} \\log \\left( \\frac{I_L}{I_0} + 1 \\right)}{R_{sh}}
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References
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-----------
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[1] S.R. Wenham, M.A. Green, M.E. Watt, "Applied Photovoltaics" ISBN

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