Add "logwright" SDE solver functions

[kandersolar]

• Closes Add an SDE solver using the "LogWright" function  #1856
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• Tests added
• Updates entries in docs/sphinx/source/reference for API changes.
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This PR implements two new functions in pvlib.singlediode: _logwright_v_from_i and _logwright_i_from_v. The reference from #1856 (link) provides the v_from_i equation. I used the same technique to derive the i_from_v equation. I plan to document the equations in our singlediode.rst page.

The new base functions are then integrated into the rest of the SDE solving machinery. Here are some runtimes for the pvsystem.i_from_v, v_from_i, and singlediode functions comparing method='lambertw' with method='logwright':

[kandersolar]

@cwhanse I wonder if you can provide some insight on the calculation result differences I am seeing, relative to the values from method='lambertw'. In particular the identified vmp and its downstream values are much less accurate than the other values, so I suspected something with the golden section search may be the cause. However, setting a stricter tolerance in the golden section search doesn't make much difference. I also looked at the differences from logwright to newton and lambertw to newton, and they're all about the same as this:

[cwhanse]

@kandersolar you are plotting Vmp (logWright) - Vmp (lambertw)?

What happens if you increase the iterations in tools._logwrightomega?

[cwhanse]

The curves in the precise_iv_curve*.json files are precise (to 15 significant figures), maybe we can benchmark the three solvers against those (lambertw, log Wright, netwon). Sorry if the format is inconvenient.

[kandersolar]

@kandersolar you are plotting Vmp (logWright) - Vmp (lambertw)?

Yep.

What happens if you increase the iterations in tools._logwrightomega?

No meaningful change. I also tried switching to scipy's implementation of the Wright Omega function, and no change there either.

The curves in the precise_iv_curve*.json files are precise (to 15 significant figures), maybe we can benchmark the three solvers against those (lambertw, log Wright, netwon).

Good idea! I will try that.

[kandersolar]

Here is a comparison of the various methods, calculating error relative to the "Case 1" known values in https://github.com/cwhanse/ivcurves/tree/main/ivcurves/test_sets:

So newton and brentq do quite well, but the two lambertw-style approaches have some trouble finding the MPP.

Putting aside logwright for the moment, I think I have narrowed the MPP issue down to the golden section search function performing poorly at the peak of the P-V curve, which is relatively flat and broad. Specifically, P varies by only an ULP or two (~10^-15) over a voltage range of order 10^-7, so unsurprisingly the golden section search has trouble navigating that region. Here is a plot showing the P-V curve and comparing identified MPPs (horizontal lines show a 1-ULP step, since matplotlib refuses to show tick labels at such a small scale):

Just to see if another search method might work better, I swapped scipy's powell in place of our golden section search and got somewhat better convergence in this single example, but still not as good as the bishop/brentq method. I am curious how it is that brentq seems to handle this problem so reliably.

Anyway, this is starting to sound like a bug report for _golden_sect_DataFrame. Perhaps this should be a new issue?

[cwhanse]

Nicely laid out. Your findings are consistent with my own assessments.

The lambert W methods find the maximum of the (V, P) curve. The bishop88 methods find the zero of the dP/dV curve. I'm not sure the difference is a bug in the usual sense. When that approach was taken, I'm sure we thought 8 digits of accuracy sufficed, and may not have realized the shortcoming in that approach.

[adriesse]

Interesting observations and discussions! If the blue line above from the precise IV curve, would it be possible to make it even more precise, or perhaps more ideal, by removing the dithering so it steps up on the left and down on the right of Pmp?

[adriesse]

I seem to recall another discussion about (vectorized?) root finding recently, but I can't remember where. Could someone point me to it?

[cwhanse]

#1661

[kandersolar]

Or #974.

If the blue line above from the precise IV curve, would it be possible to make it even more precise, or perhaps more ideal, by removing the dithering so it steps up on the left and down on the right of Pmp?

Shame on me for not labeling it! The blue line in that plot was from lambertw_i_from_v. The precise curves would presumably not suffer from such numerical artifacts, but they also doesn't have nearly the voltage resolution needed to show them anyway.

[adriesse]

Or #974.

That's the one, thanks!

[mikofski]

there was a similar issue in pvmismatch which was using argmax to find the max power point (MPP), but in issue 65 there were discontinuities/jumps similar to the issue described for LambertW approach. A simple solution was to calculate the derivative around MPP using central difference and interpolate to find a better zero.

[adriesse]

If the blue line above from the precise IV curve, would it be possible to make it even more precise, or perhaps more ideal, by removing the dithering so it steps up on the left and down on the right of Pmp?

Shame on me for not labeling it! The blue line in that plot was from lambertw_i_from_v. The precise curves would presumably not suffer from such numerical artifacts, but they also doesn't have nearly the voltage resolution needed to show them anyway.

Does the dithering also appear in V and I, or only their product? If only the latter, perhaps there is a way to make a cleaner product?

[echedey-ls]

Mmmm, can I be of help with this PR? I see the discussion revolves around the precision of each algorithm, but that doesn't invalidate the fact that logwright is still faster (if I understood correctly).

[kandersolar]

The convergence error discussed above is large enough to make me uncomfortable, but probably not too large to keep the method from being useful. The other thing is that only the v_from_i equation has a reference; I derived the i_from_v equation myself.

This PR currently shows that the logwright method has promise, but IMHO a short paper defining the complete method and comparing speed+accuracy with other methods needs to be written before this PR gets picked up again.