Voltage at the maximum power point - Vmp

To gain the maximum amount of power from the solar cell it should operate at the manximum power voltage. The maximum power voltage is further described by VMP, the maximum power voltage and IMP, the current at the maximum power point.

plot_ideal_cell.png
The maximum power voltage occurs when the differential of the power produced by the cell is zero.

Starting with the IV equation for a solar cell:

I=IL-I0eVVt

Vt=nkTq to simplify the notation in the derivation, where kT/q ~ 0.026 volts and n is the ideality factor. The ideality factor varies with operating point. For these equations the correct value to use is the average from VMP to VOC.

Power  produced by the cell is the product of the voltage and the current, i.e., P = IV.

P=VIL-VI0eVVt

Using differentiation by parts on the second term: u=VI0,  u'=I0, v=eVVt, v'=1VteVVt

The differential of power respect to voltage:

dPdV=IL-VI01VteVVt+I0eVVt

Vmp occurs when dPdV=0

VmpI01VteVmpVt-I0eVmpVt=IL

Detailed steps for rearranging and simplifying:

VmpVteVmpVt-eVmpVt=ILI0

eVmpVtVmpVt-1=ILI0

VmpVt+lnVmpVt-1=lnILI0

Vmp=VtlnILI0-lnVmpVt-1

Using Voc=VtlnILI0

Vmp=Voc-lnVmpVt-1The implicit equation above does not have a simple solution but it converges quickly with iteration. An initial guess of VMP = 0.9 VOC gives an accurate solution in two iterations.

Using Lambert Functions

An exact solution to finding the maximum power voltage is with lambert functions. These are transcendetal functions much like e or trigonometric functions. Lambert is available in most advanced math packages such as Maple, Mathematica and in Python with SciPy, but they are not on most handheld calculators.

As above we need to solve for the voltage when the derivative of the power is zero

P=VIL-VI0eVVt

0=IL-I0expVVt1+VVt

Since V/Vt is much great than 1 we can remove the +1 term: ILI0=expVVt×VVt

The Lambert provides a solution of the form:Y=XeXX=W(Y). Using this relation the previous equatoin becomes:

VVt=WJLJ0

so that we get a simple expression for Vmp.

Vmp=VtWJLJ0

Alternatively, using the full equation above for dP/dV = 0 and not dropping the +1 term the solution becomes:

Vmp=VtWeJLJ0-1

Exact determination of Vmp

Math partly verified using Wolfram Alpha

J=JLJ0[exp(VVt)1]J = J_{L} - J_{0}\left\lbrack \exp\left( \frac{V}{V_{t}} \right) - 1 \right\rbrack

Power = V x J

P=VJLVJ0[exp(VVt)1]

Lose the -1 since we are above 100 mV