Given an AC circuit, the instantaneous power in the circuit can be determined by the formula $$P(t)=\frac {VmIm}2cos(\theta)+\frac {VmIm}2cos(\theta)cos(wt)-\frac {VmIm}2sin(\theta)sin(wt)$$ where \$ \theta \$ is the power factor angle, or the phase difference between voltage and current.
Complex power is defined as $$\vec S = P + jQ$$ where $$P=\frac {VmIm}2cos(\theta), Q =\frac {VmIm}2sin(\theta)$$ where \$ P\$ is real power, \$ Q \$ is reactive power, and \$ |\vec S|\$ is apparent power.
I completely understand the uses for complex power, and how it works nicely mathematically with voltage, current and impedance in AC.
What I don't understand is, why the "apparent" power is what we need to cater for when providing power and e.g. designing a power source for a circuit, as opposed to the maximum instantaneous power, which equals \$|\vec S| + P\$. After all, there will be a moment in time where the source does need to provide this amount of power, even if it isn't "useful".
What I also don't really understand is how these mathematical formula ultimately only end up involving \$ \vec S \$, and instantaneous power (i.e. \$ P(t) \$) is completely left out of the picture. For example, why is the real component of \$ \vec S \$ independent of the constant shift amount \$ \frac {VmIm}2cos\theta \$ for instantaneous power? Also, why is the \$ Q \$ term not negative in complex power, like it is in instantaneous power?
There are just a few gaps in my intuition which make it uncomfortable for me to properly understand complex power.