Increasing deployment of information technologies and low-inertia renewable energy sources into smart grid fuel the uncertainties and reveal security and transient stability problems. Enhancing the stability margins of smart grids despite the cyber and physical disturbances emerges the need for cyber-aware robust controller design. Therefore, a distributed nonlinear robust controller is proposed to improve the transient stability margins of synchronous generators (SG) in the presence of excessive communication delay and cyber-physical disturbances. The proposed controller uses phasor measurement units to receive real-time measurements and actuates distributed storage systems to inject or absorb power in order to accelerate stabilization of frequency oscillations of SG following a disturbance. The communication dependency exposes time delay and cyber-security issues since latency is inherent and can be excessive during an attack such as denial of service. In addition, uncertainties in measurements challenge the stabilization process. Hence, the proposed controller is designed for robustness to delay and additive disturbances. A novel time delay compensation technique is developed to inject delay-free control signal into the closed-loop system. To guarantee that all tracking error signals are globally uniformly ultimately bounded, novel Lyapunov-Krasovskii functionals are used in the Lyapunov-based stability analysis. The simulation results validate the feasibility of the proposed control framework and robustness under cyber-physical practical limitations.