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Wow. I nearly cried. It's difficult enough to explain the concept of energy and power to people without confusing them with unit abuse.
Energy is measured in Joules. The rate you use energy is Power and is measured in Joules per second, or Watts. People don't like multiplying the wattage of their equipment by the number of seconds it's on (to give Joules) as the numbers are big, so they multiply the wattage by the number of hours, to give kilowatt-hours. That's not kilowatts per hour, that's kilowatt hours (like pound-feet, or Newton-metres). One kilowatt-hour is 3.6 Megajoules.
Your energy company is only concerned how much energy you have used (in Joules, or Kilowatt-hours), not the rate at which you use it (in watts).
Don't feel too bad. Even eon (the large UK utility company) make mistakes. See the Scroby Sands Wind Farm display in Great Yarmouth, England talking about the number of homes per year that 60 Megawatts can supply.