“How much power does a 10kW solar system produce per day?”
Power is an instantaneous measure of output, so the amount of power produced in a day isn’t really a thing. Further, power is measured in watts (or kilowatts), so the power a 10kW solar photovoltaic (PV) system would be expected to produce is…10kW.
You could certainly ask what is the maximum power a nominal 10kW system might produce. In my own 10kW PV array, the recorded peak is 10,109 W. At night, I’ve seen the same set of solar panels produce tenths or even a few hundredths of a watt. I guess starlight and moonlight are enough to produce a measurable wattage. Today is cloudy and at the moment it’s producing about 320 Watts.
If you did mean energy instead of power…. My solar panels are in East Tennessee. It’s not the most ideal location for producing electricity from sunlight. We have no shortage of cloudy days and clouds certainly cut into the maximum wattage. Further, the noonday sun is pretty low in the sky from November through the end of January. My solar panels are fixed (they can’t track the sun) and thus don’t produce much in the late fall and winter months. They are much more productive during the long days of summer. If that momentary 320 Watts were to remain constant for an hour (it won’t) that’s only 0.32 of a kilowatt-hour.
Since October 2013, my solar panels have produced 70,133 kilowatt-hours of energy. I don’t remember the exact date I turned it on, but that’s a smidgen over 26 kilowatt-hours per day on average.