We're already getting there a bit with smart thermostats for A/C. Refrigerators might be more difficult, although similar temporary demand response schemes could probably be made to work, e.g. every fridge is alerted to shut off in 30 minutes for 15 minutes, they all schedule cooling in advance, and then drop off the charts as a group for a short window of time. Perhaps you saw that a bank of clouds was about to roll over your panels.
Silent loads like your fridge make up a lot of the total power draw, and are also some of the most tied to the concept of permanent base load.
There are already a lot of areas that have peak power pricing and, in fact, a lake I went to as a kid[1] actually took advantage of this by (at a scarily low efficiency) pumping water into the lake during off hours and running the turbines during peak usage - it's basically a giant battery (but, again, super inefficient in terms of the energy required to pump the water into the lake, still efficient enough to be profitable though).
This is the big thing. I think with modern renewables we have to change our behaviour. ie doing washing/drying when its windy, charge cars during the day instead of night.