I'm sure that 20W the whole panel generates every few days for an hour or two will offset the cost
You can—and it's usually safe to, in the translation from actual researchers to PR departments—ceertainly doubt the viability of this claim, but the whole article is devoted to at least explicitly claiming that they're aware of the obvious problems, and can overcome them:
> … the efforts to collect energy from falling raindrops have faced a technical hurdle that has made the concept inefficient and impractical. … as one might expect, the amount of power per raindrop is incredibly small. …
> Now, a team of researchers says they have found a design and configuration that greatly reduces the coupling capacitance issue and one they claim could make energy-harvesting rain panels a practical reality.
One key seems to be that, although you're obviously meant to think of rooftop panels, it seems more to be about large-scale installations:
> “The peak power output of the bridge array generators is nearly 5 times higher than that of the conventional large-area raindrop energy with the same size, reaching 200 watts per square meter,” Li explained, “which fully shows its advantages in large-area raindrop energy harvesting.”
(I do notice on re-reading that it says "… may lead to the development of rooftop, power-generating rain panels", but (1) one can freely claim that anything may lead to something else, and (2) one can claim that anything may lead to anything else, so I prefer to go by quotes from the researchers themselves.)
So your hypothetical 200 watts per square meter at 100% efficiency assuming spherical cow types of ideal conditions could possibly provide that power for 50 minutes per year under ideal circumstances or average 0.02w/m2 over a year. And average roughly 0.002w/m2 in extremely rainy though not world record setting locations.
Now, you ignore the kinetic energy in rainfall and try and harvest the potential energy when it lands at a high altitude, allowing a large collection area and large differences in altitude. But we call those things dams.
And if it wasn't free or near-free it's competing with putting one more panel or adding one more battery to the system. Or... just big, cheap funnel that is feeding water generator Sure you lose some of the velocity but it is so much simpler system
Like, generating energy from tiny movements have its niches (like sensors powered by piezoelectrics from vibrations of machine they monitor), but using it for raindrops just feels like a waste of time, most areas don't even get enough rain for that to matter.
Now if they figured out to do it to airflow with few moving parts, that would be more interesting.
"Technology improvements" can't get around the fundamental limit in the amount of energy available to harvest.
Edit: Oh, that's horribly misleading. They seem to be reporting the theoretical maximum output of this harvesting technology - e.g., how much energy could be recovered if they started blasting the panel with a firehose. The amount available in rain is on the order of .2-.4 W/m^2. At that rate you'd never recover the amount of energy as was used to produce the panel.
This webpage has already burned more energy than that panel will ever produce.
For comparison, even on the rainiest day I've measured in the last year from my 6.6 kW photovoltaic array on my house, I made 8 kWh of energy (on a good summer's day I get around 40 kWh), with the lowest output across most of the day being 450W (less at sunrise and sunset obviously).
Other have posted calculations of the maximum energy that can be expected to be collected from raindrops, and from those calculations, I doubt that with a raindrop capture device as large as my PV array you could make 8 kWh in a month.