The diversity of converter-based sources and loads in renewable energy power systems (REPS) results in various sub- and super-synchronous oscillations (SSOs). Universal stabilizing of various SSOs needs to be considered for robustly stable operation of REPS. In view of this problem, a novel design principle for universal stabilization of SSOs is proposed based on the concept of generator-network interaction energy (GIE) injection. Firstly, the general small signal stability of RPES considering GIE injection is analyzed based on the energy network-based damping contribution assessment method. The analysis results show that GIE injection is general for various kinds of RPES and SSO modes, thus the universal stabilization is theoretically achievable. Then, with the proposed principle, the self-synchronized power converter-based universal stabilizer is designed to inject controllable GIE into REPS in practical applications. In this design, H-infinity based harmonic current tracking method is utilized to realize accurate GIE injection in wide frequency band. And the GIE injection is decoupled with synchronization dynamic of converter. Finally, both probabilistic small signal stability analysis and time-domain simulation are conducted to validate the damping ability of universal stabilizer. The results show that the private stabilizer for each converter-based source in large-scaled REPS is not necessary. And various SSOs under both small and large system disturbance are simultaneously damped by a universal stabilizer.