Is there a mathematical proof that reactive power regulates voltage?
Most explanations disregard reactive power as "useless" power or power that we don't want to pay for.


Such explanations are wrong. Sources of such information should probably be avoided.

"Reactive power" is a common term for reactive volt-amperes (VARs). Reactive volt-amperes have one very important use, they provide the magnetic fields for every electromagnetic mechanism. Induction motors represent most of the requirement for VARs, but transformers, AC solenoids and relays also require VARs.

Since VARs represent energy that is continually circulated back and forth between the source and load, the only power consumed is due to losses. That is a small amount and is measured as watts when it occurs on the consumer side of the meter. On the utility side of the meter, the losses are a concern for the utility. The utility is also concerned with the use of transmission capacity to carry current that is not delivering power.

Only synchronous generators and capacitors can provide VARs to magnetic loads. If capacitors are connected near the magnetic loads, that avoids the associated transmission losses and frees up generating, transmission and distribution capacity to provide more real power. However if too much capacitance is added to supply VARs, the utility must accept the extra VARs. Only synchronous generators and inductors can accept VARs from excess capacitance. Since utilities are not expecting excess capacitance, they are not prepared to add inductance. The synchronous generators may have difficulty with that kind of load. The result would be excess voltage.


  "In general terms, decreasing reactive power causes voltage to fall while increasing it causes voltage to rise." Well, aside from just believing it, i would like to see some type of mathematical proof that shows it?


Since the statement is not entirely correct, there is no proof. Understanding what can happen requires studying the "V curves" for synchronous generators. It is also necessary to understand the various limits on the safe operating conditions for synchronous generators and transmission system component.

More re increased voltage

In order to overcome voltage drops in the transmission system, the voltage at the generator must be higher (on a percentage basis) than the voltage seen by a customer. If the customer had been "consuming VARs," adding capacitors would reduce the current and voltage drop in the transmission lines. That would tend to increase the voltage at the meter, but the utility would adjust the generator to keep the customer voltage within limits. If the customer begins to supply vars, that would tend to make the voltage higher at the customer location. To some extent, VARs supplied by one customer may be absorbed by other customers. That would tend to make that customer's voltage higher than the voltage seen by other consumers.