Your chart significantly overstates the degree to which energy is getting cheaper, by focusing on a particular component which has gotten cheaper faster than the overall system. After a PV panel is manufactured, it still needs to be transported somewhere, mounted, and attached to inverters and other supporting equipment; the price of these things is going down, but not nearly as quickly, and not in an exponential-to-zero way.
In general it's true that the energy cost doesn't fall as fast as the chart the amount of installed solar power seems to rise exponentially at 28% per year. The effect I'm talking about will happen as a larger share of the power gets produced by solar.
As far as driving the other costs to zero, one idea is to put the solar cells into surfaces that get installed anyway. That's what Solar City is doing with their new rooftop tiles. Those also have the advantage that they have lower transport costs than traditional rooftop tiles and outperform them in durability.
In related news China build it's first solar cell expressway. It's way to expensive at current prices but as the technology advances it might get cheap enough to be practical and maybe as they thinking about building up the technology they also manage to produce tiles that are less prone to potholes then the asphalt roads we have currently similarly to how Solar Cities rooftop tiles outperform conventional tiles in many regards.
1. Something that is already expected to be on a finite time horizon might be a good candidate. Say we wanted to clean up a landfill, and built a recycling facility next to it. We already know that the facility has a limited life, because the landfill will eventually be emptied - excess energy could just be used to reduce the time, or it could only operate with the surplus energy in exchange for taking more time. The point is, use the energy to manipulate the time domain of the problem.
2. What about projects that are too energy intensive to have been implemented previously? There's a slew of ideas that were never implemented because the energy simply wasn't there - different manufacturing processes, waste disposal, what have you - which means the economics of their operation wasn't really considered. If the surplus energy makes them technically feasible, it may also be economically feasible to operate them intermittently.
3. Make the energy more mobile, and take it somewhere else where energy conditions are different in the near term. Ship containers of super-heated salt. Extremely long-distance transmission lines. A fleet of airships to bounce microwaves around. The efficiency losses are now a feature instead of a bug.
There are energy storage mechanisms that last longer than batteries. I don't know the exact economics or mechanics, but excess energy could be used to pump water to a higher elevation to store the potential energy.
When the energy is needed, at night or in the winter, the water would be allowed to flow back down through turbines to reclaim some of that energy.
Excess energy could also be used for hydrolysis; the hydrogen could be stored for later use.
I'm not 100% sure what you're asking, but from Wikipedia:
...current best processes for water electrolysis have an effective electrical efficiency of 70-80%,[38][39][40] so that producing 1 kg of hydrogen (which has a specific energy of 143 MJ/kg or about 40 kWh/kg) requires 50–55 kWh of electricity. At an electricity cost of $0.06/kWh, as set out in the Department of Energy hydrogen production targets for 2015[41], the hydrogen cost is $3/kg. https://en.m.wikipedia.org/wiki/Hydrogen_economy
Some quick googling indicates a kilogram of hydrogen seller for around $14, give or take.
There is inefficiency in hydrogen storage, but it should keep longer than a lithium ion battery.
The negative prices are a failure of the market / regulation, they don't actually mean that you have free energy.
That being said, the question for the most economical opportunistic use of intermittent energy makes sense.
Hardware produces a certain amount of energy and when too much energy gets produced that has the potential to damage the grid. It's no market failure if there are people willing to pay to have that energy used up to prevent the grid from being damaged.
The normal energy market needs roughly the same amount of energy every day. It's currently not setup to use three times as much energy at days where three times the amount of energy gets produced.
What's valuable to buy for current energy consumers is the guarantee that they can get a certain amount of energy at the time they want to consume that energy.
Epistemological status: Superficial personal speculation .
This posts follows What could be done with RNA and DNA sequencing that's 1000x cheaper than it's now? in thinking about technical challenges we will face in the next decades. It's an attempt by my to play a bit with futurism.
Both solar and wind energy have exponential efficiency improvements:
A majority of Indian new coal power now costs more then new wind and solar power. This means that we can expect a lot of new wind and solar power installations to be build.
Last year, in 2017, in Germany there were 146 hours in which energy prices were negative. This means that during those hours wholesale customers got payed to consume energy.
Unfortunately, batteries are expensive and while it might be economical to load a battery in the day and release the energy at night it won't be economical to load the battery in summer and release it in winter.
As a result batteries don't work as energy sink for summer days with too much energy production.
Maybe, we can construct our wind turbines in a way that we can dumb excess energy into them when there isn't wind? It would however be more desirable to find a way to actually spend that free energy productively.
Most machines that we currently build are optimized to run as much as possible and thus don't provide good energy sinks. A machine that produces spoons is most economical when it runs 24/7.
This means that there's an economic opportunity to build machines that can work as energy sinks for days with negative energy costs.
It's likely a good idea to build data-centers in a way that they are not only able to run in a low energy mode but also in a mode where they burn as much energy as possible. As a result it might b worthwhile for some cloud provider to have in a year specific times where the compute is radically cheaper than on other days. Maybe that compute will go into crypto-mining but it we can also think about other uses for it.
The data-centers likely won't be enough of a data sink and we need new ideas about what to do with other excess energy.
Do you have any energy about how to build good energy sinks that might use energy usefully and that are economical if they are only used for a few weeks/days a year?