The billing cycle/hours actually make a bit of sense when you look into it. I say this with a caveat that I don't know the specifics of your electricity system/grids.
I work for a power generation company over a bunch of different sites, but, I'm primarily based at a 320MW gas turbine peaking station for the South Western Australia isolated smart grid. (By the way, count yourself lucky, we don't have any on-peak off-peak pricing here. its all a fixed 22.6 cents per kWh) We don't always run, we just come on when demand is high, maybe 2-3 times a week. Our primary power production here for base load is coal fired power stations with Gas Turbines doing the rest, and wind/PV panels trying to keep up, there are some hydro plants, but these seldom run for the fact that they're based on damming smaller rivers, and we'd run out of water if we kept the gates open all the time. Hydro is usually reserved for black starting, which is where the grid goes into frequency distress and shuts down. The hydro stations have big diesel generators which can start their machinery, and they wind up, providing enough power for gas turbines to start up, which then provides enough power for coal plants to turn on. We have no nuclear power in Australia.
The hours that are displayed on that website are definitely the peak hours for us. If were going to turn on, 80% of the time, its during those hours. This is when people wake up, its cold, they turn on their heater, make coffee, cook breakfast, etc... They then go to work, during the day, they aren't using power, things quiet down. They come home, turn on their TV, air conditioner, cook dinner, etc... Then they settle down, turn things off, watch TV, and then go to sleep.
I don't know how the base power is provided where you are, but, I'm not sure if they can simply vary the inlet valves for hydro production. I would guess that what happens is they keep the inlet fairly constant for base load power, and then turn on gas turbines, or some other stable, non-steam based unit. Gas is pretty expensive, and turbines use a fair bit of it. We store 1.6 million litres of Diesel on site in case something happens to the natural gas supply. 1.6 million litres will run the unit for about 12 hours. So, peak periods are expensive.
Also, are you sure that the electricity prices have risen because the power usage has gone down? Just because two things coincide doesn't make them causal. I'd bet that fuel prices, grocery prices, car prices, etc... have all gone up since 2005 as well, it doesn't necessarily follow a direct relationship. The most common causes of increased power prices are infrastructure costs, fuel prices, and government rebates/regulations