Power supply variation will likely change the output amplitude and frequency of the oscillator. What simulator are you using?
To see the steady state sensitivity ( d(freq)/d(supply voltage) and d(amplitude)/d(supply voltage) ), simulate your oscillator at two supply voltages (for example Vnom and Vnom-5%). This will show you the sensitivity for slow power supply variations.
Very likely you want to run the amplitude and frequency response to a supply voltage step change. From the amplitude and frequency change versus time, you can assess the amplitude and frequency sensitivity for power supply variations in the frequency domain.
So if you simulated 10 kHz frequency change due to a supply voltage variation of 100 mVp at 100 Hz, and the actual supply ripple at 100 Hz is 10 mV, you can see that your oscillator is frequency modulated with 1 kHz. This frequency modulation adds to the phase noise (use first order Bessel functions).