Are electrical projects approved by building departments without transformers? How do they check the working of the electrical with no power? Just assume that if all the connections are correct that everything will work properly when power is applied?
Yes. The point of an electrical inspection for an (e.g.) new construction building is to ensure that when power is applied there won’t be a catastrophic failure that leads to a fire, severe injury or death. The building owner pays an electrician to do the work, and the inspecting agency verifies that the licensed electrician has done her job properly. (Where properly is defined by legislation and regulation, generally referencing
The National Electrical Safety Code® (NESC®) - IEEE Standards Association)
One can check the safety of an electrical installation prior to application of power by a reasoned analysis of the plans and by physical inspection of the connections and wiring. Adherence to the National Electrical Safety Code involves not just the actual physical safety of the wiring, but also the inspectability of the work. For example, grounding wires are marked green not because the laws of physics require them to be marked green, but (in part) because it makes the job of an inspector easier. (See:
Electrical Wiring Color Codes for AC & DC - NEC & IEC)
Low voltage continuity testing can also be used to gain confidence in the safety of the installation. A hand-held battery-powered multimeter can measure the resistance between a power conductor and safety ground. If the circuit was horribly miswired prior to the main fuse/circuit breaker then the resistance will be small, because there should be no direct connection between power and ground conductors.
On the other hand, if the mistake is made after the main fuse/circuit breaker then it is their job to stop the massive flow of energy through a downstream short circuit. Confidence in their ability to perform that function is built through testing and certification by organizations like Underwriter’s Laboratories (UL), plus obvious manufacturer legal product liability exposure.
The low-voltage continuity testing and verification of UL markings on the main fuse/circuit breaker can happen either at the time of the pre-energizing inspection, just before energizing (by the utility), or (preferably) both.
Modifications to an already-energized electrical system don’t always need inspection prior to being put into operation. For example, when the electricians put in my Tesla Wall Connector they tied into the already-energized breaker panel in my home. Similarly, adding a new transformer to an existing electrical utility distribution system qualifies as a modification—not a new installation—so the utility can install that new transformer under their normal policies and procedures using their own internal or contracted labor force.