Your grid-connected solar power systems must by law shut down if the grid loses power. The reason is that linesmen repairing faulty electricity wires must be able to do so safely, without any solar power travelling back into the grid during maintenance and repair. Your electricity if fed back into the line could create a danger to the service personnel. Therefore when there is a black out you will also be without power. Your solar power system will automatically turn on during daylight hours when the power comes back to the grid.
However, if you have a battery backup system connected to your solar system, then depending on the wiring, some internal services like the fridge could still be operated by the batteries. In the next few years smaller size energy storage systems will become available, as many solar system owners prefer to store the electricity they generate and use it during rainy days or at night, rather than sell it to the energy retailers.