Only when the sun is not shining. A transfer switch will isolate you from the Grid Electricity during an emergency outage (excess electricity from your PV system during the summer months etc will reduce your overall annual electricity bill when the electrical grid is connected to your premises). That 4 KW generator on top of the roof will allow you to keep other household appliances to keep working, i.e. high efficiency refrigeration, which can cope without being powered for a day or so at a time. The internal temperature of the refrigerator will just oscillate over a greater temperature range. Washing machines could be powered during those sunny days.

If you have a hot water system i.e. a electrical immersion heating system, even the excess electrical power of the 4KW PV Generator on the roof can be used to heat the water, which is useful for washing up and even a hot shower or bath.

A battery system based on LiFePo could be specified to keep the low wattage LED lighting system i.e. < 200 Watts running through out the property during an emergency.

Realistically this would work for 6-9 months during the year. This is mainly a function of geographical latitude.

There are a lot advantages to a Solar PV system. The main advantage is that the Solar PV system will actually pay for itself. The pay back time is a matter of economics, PV solar calculation, your power company KW/Hr rates and even Government Agenda 21 energy policies i.e. the closure of much of the Power station electrical grid network in the future. i.e. rationing and brown outs. It has been reported that the UK will loose 10% of its generation capacity this month.

Installed Gennie sets especially the larger ones will quickly burn fuel at very high rates and resupply of LPG, Diesel, Petrol etc may be of short supply during a long time emergency energy outage situation.

$15,000 gives you a lot of options, putting your funds into a single system that might give you a continuation of your current lifestyle for a week or two at the expense of a renewable energy system for a longer term emergency problem might be a little short sighted.