$(5$ points each$)$ A $3-$phase, $240V,35kW$ generator is being driven by a water turbine, and you want to extract power to sell back to the power company. The generator is connected to the grid through an active rectifier, so that power can be provided as the turbine speed varies.

a$.$ The turbine is producing $30$ kW $.$ The gearbox losses are $2$ kW $.$ The generator power factor is $0\mathrm{.}8.$ Find the generator current if the generator voltage is $210V_{rms}.$

b$.$ Suppose the inverter can correct the power factor to the grid. Find the minimum grid connection size required, in kVA. Assume the turbine will never produce more than $30$ kW $,$ and the gearbox losses will not change. The other components are lossless.

c$.$ Find the minimum current required from the inverter to the grid. The grid voltage is $240V_{rms}.$

d$.$ The turbine was formerly supplying power to the main driving wheel in a mill, which uses roughly $600$ kWh per week. This will now be provided from the generator. The new installation can extract power continuously, at an average of $13$ kW $.$ The installation will cost $$20k.$ If electricity costs $$\frac{0\mathrm{.}20}{k}Wh,$ how long will it take for the system to be paid off in power savings?