There's a lot of demand that can be shifted to high-power periods, basically anything that requires heat/cold. Modern insulation is fairly good and we do have a lot of liquids with a high thermal capacity. Generating warm water for buildings and cold coolant for aircons during peak power production would be technologically feasible, it just doesn't pay off currently since the power production curve is essentially flat (lot of base power with some peak added on top).
In the future it might also be interesting to move industries with high power demand to places with more solar power - electric arc furnaces, aluminum production, ... Some of those require a base load, but a lot of them can survive a few hours with limited power.