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In a power grid, the grid itself has a certain amount of inertia from all the spinning loads and generators. If at a given moment in time the production and consumption of active power does not match, the power mismatch will be provided/absorbed by the spinning energy stored in the grid, and the frequency of the whole grid will raise or lower in lockstep to compensate. Eventually the generators' governors and the load/frequency relationship will arrest the frequency drop, and it will arrive at a new steady-state value, but it's not something that happens immediately.

Does it work similarly for reactive power? If a capacitor bank supplying reactive power suddenly trips, does the voltage immediately drop to its new steady-state value? Or is there a period of voltage decline as the missing reactive power is siphoned from somewhere?

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Nothing ever happens instantaneously. There will always be some parasitic components (inductances, capacitances) that limit the speed of change.

Take a simple lightbulb switch for example. When you switch off the light, you can often see a "blue light" coming from the inside of the switch. This light comes from a spark that occurs between switch contacts, which happens because of the wire inductance. This inductance is very small, but it is finite and prevents breaking the current from happening instantaneously.

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