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.

No, it doesn't. h/h = 1, so it's MW. But power companies never made sense anyway :)

Maybe they think the target audience doesn't understand watts but have a vagues idea about what MWh is and know how to think of magnitudes per hour.

(And fingrid? What shame. ;)

agreed, it is power companies trying to keep their end simple consistent.

As people are generally billed in kWh or (MWh for big business?) it is simpler to indicate the a unit of power per time period.
* kWh and MWh are alternatives to joules for representing an amount of energy, * kWh/h and MWh/h are alternatives to kW and MW for representing power (or energy production/consumption).

As appliances as generally rated in W (or kW) telling people they have been charged for 100MJ of electricity might be confusing to the end user, so instead they have opted for kWh as the unit for energy. The kWh/h and MWh/h is just an extension of that, to provide end users with units that have meaning to them.

For example Fingrid is now generating enough power to supply a 1,000,000 fridges (assuming 1kW fridge). So it does make sense, in weird non scientific kind of way