Does it make sense to use MWh/h as unit of measure?
MW is a unit of power. MWh is a unit of energy. 

MWh/h is a unit of AVERAGE power during a certain period, so it is used to show ENERGY consumption. 

For example, I as a consumer can be buying power from the utility with a 1 MW power (that is, I pay to have a MAXIMUM DEMAND of 1 MW) but I only consume 12 MWh of energy per day, so I have an ENERGY consumption equivalent to 0.5 MW - flat.

To distinguish between one and the other, the first is referred to with MW, while the second one might be referred to as MWh/h.