An accurate state estimation (SE) considering increased uncertainty by the high penetration of renewable energy systems (RESs) is more and more important to enhance situational awareness, and the optimal and resilient operation of the renewable-rich power grids. However, it is anticipated that adversaries who plan to manipulate the target power grid will generate attacks that inject inaccurate data to the SE using the vulnerabilities of the devices and networks. Among potential attack types, false data injection attack (FDIA) is gaining popularity since this can bypass bad data detection (BDD) methods implemented in the SE systems. Although numerous FDIA detection methods have been recently proposed, the uncertainty of system configuration that arises by the continuously increasing penetration of RESs has been been given less consideration in the FDIA algorithms. To address this issue, this paper proposes a new FDIA detection scheme that is applicable to renewable energy-rich power grids. A deep learning framework is developed in particular by synergistically constructing a Bidirectional Long Short-Term Memory (Bi-LSTM) with modern smart grid characteristics. The developed framework is evaluated on the IEEE 14-bus system integrating several RESs by using several attack scenarios. A comparison of the numerical results shows that the proposed FDIA detection mechanism outperforms the existing deep learning-based approaches in a renewable energy-rich grid environment.