Unique solutions to SDE with only Isc, Pmp, and Voc points?

[Mark Campanelli]

Hello world,

  I have reviewed at least one paper that claims to solve for the 5 parameters of single-diode equation using only (0, Isc), (Vmp, Imp), (Voc, 0), and the zero derivative of P w.r.t. V at Vmp. This is only four constraints in a determination of five-parameters. Even though it's a non-linear system of equations in the parameters, it still suggests that the problem may not be well posed due to non-uniqueness (https://en.wikipedia.org/wiki/Well-posed_problem). If not done already, then I think it would be a nice addition to the literature to present a very accessible example of non-uniqueness in this inverse problem. Even better would be to outline "how bad" the situation is in practice for the aforementioned approach.

  Anyone care to collaborate on such a project? (If done already, then could someone share a reference?)

-Mark Campanelli

[Mark Campanelli]

As an update to this post, I have sketched out some preliminaries for the "forward" problem. An important foundation is that the I-V curves (e.g., I as a function of V, for any real V) defined implicitly by the single-diode equation (SDE) exist and are unique for any given choice of the 5 parameters. Does the following sketch of a proof hold water under scrutiny?

1. The SDE is 0 = Iph - Irs * (exp((V + I * Rs)/Vth) - 1) - Gp * (V + I * Rs) - I =: f(V, I; theta), with five parameters in theta, namely, Iph >=0, Irs > 0, Vth > 0, Rs >= 0, and Gp >= 0.
2. Fixing any real-valued voltage V_0, the intermediate value theorem (IVT) can be used to show that there exists at least one current I_0 that satisfies the SDE. A strictly negative partial derivative, ∂f/∂I, ensures that this root is unique.
3. For the solution (V_0, I_0), the implicit function theorem (IFT) can be used to show that there is a local differentiable functional relationship I = g(V) around (V_0, I_0). ∂f/∂I is never zero, so this differentiable function can be continued uniquely for all real V to a "global" differentiable function (which is the unique I-V curve).

(I'm admittedly a bit rusty on the details of how the global continuation in step 3 is guaranteed.)

-Mark

[Anton Driesse]

Hi Mark,

Maybe you can code it up in Lean?  🙂

More down-to-earth, I think there are a number of additional constraints between the parameters if we assume a physical device. Then with the 4 input constraints, the number of solutions may be zero, one or many.  If the 4 input constraints are measurements on a physical device, then it seems like the number of possible solutions should be many. The single solution situation may be possible, but highly improbably.

If I wanted to make an example, I would simply do a fit with a fixed ideality factor (Vth) and then illustrate that I can do a perfect fit using a range of ideality factors.  If you plot all those IV curves you will probably see that they are all very similar.  

Would that be convincing enough?

Anton

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[Mark Campanelli]

Hi Anton,

  With only 4 known points on an I-V curve parameterized with 5 parameters (all nonzero), it is probably as straightforward as knowing that the Jacobian (w.r.t. the five parameters) has a non-trivial null space. (Basically, you just rely on known results from linear theory for a "local" linearization.)

  However, as the problem is stated above, the twist is when one of only three points on an I-V curve is known to be the maximum power point, presumably providing a only a fourth constraint, but likely still "underdetermined".

  All that said, it would be nice to have a more practical demonstration that constructs the multiple solutions. Numerical solutions are one way, but I would find something algebraic more concrete and satisfying. Also, it would be nice to have some sort of "measure" as to how non-unique the solutions can be.

-Mark

[cwh...@sandia.gov]

Mark,

The strategy makes sense to me. I think the parameters bounds will be more than a passing nuisance, as the structure of the solution changes drastically if certain parameters (e.g., IL) are non-positive. And there are parameter combinations where no solution can exist (I'm fairly sure of this, see below). Maybe in some subset of the space where all parameters are positive, one may prove existence and uniqueness.

For an approximate boundary for the non-viable subset, it's easiest (for me) to use one of the exact Lambert W solutions, V=f(I;p) being easier to work with. To first order, this solution reduces to

V = -IRs - nVth log(Io * Rsh / nVth) + smaller posive terms.

The log must be negative, which constrains the quantity Io * Rsh / nVth < 1.

I think it would be useful to provide a formal existence/uniqueness proof, and to provide an example of two parameters sets that satisfy an incomplete set of constraints.

Cliff

Cliff