Even in summertime, assuming you covered your entire southern roof with photovoltaics, I rather doubt you'd be able to handle the starting surge of a large appliance like a central A/C unit. If you have an electrical stove, there's no starting surge since the heating element is resistive, but a 2 1/2 ton AC unit will have a normal running power consumption of about 3500 watts. Assuming you had a 30 X 15 array, you'd be getting about 4500 watts. No problem, right? Not so fast. That AC unit has a starting surge from 5-8 times normal running current, for a momentary power consumption of upwards of 20,000 watts. Whoops.
And again, remember, most photovoltaics only calculate USEABLE power for about 5 hours per day.
So smaller loads would be okay, but you're still going to need the grid power to kick in when you start or run large loads like your dryer, well pumps, furnace, water heaters (especially "on demand" water heaters), or air conditioners (even window units suck up well in excess of 12 KW/hr.)