The current system allows rich customers to negligently waste electricity that poor customers need to stay warm. If the system prioritized usage and charged the corporations $5/kWh while charging poor customers the standard rate, people wouldn't have frozen to death.
GP's point still remains. Even if you do charge corporations $5 kWh, that's likely not enough to solve the power deficit. After all, corporate offices aren't exactly power suckers (a 1500W space heater is equal to at least a few dozen LED office lights), and the heavy users (factories) are usually the first ones to go when there's a power shortage. Price signals are still needed to ensure that people are sufficiently motivated to change their behavior. Leaving prices unchanged will basically mean consumer demand remains unchanged.
Charge rich people more than poor people. Cut them off sooner. They have bigger houses that use more energy and they have more blankets and supplies. Poor people with fewer options deserve the power more.