Smart grids are vulnerable to cyber-attacks, which can cause significant damage and huge economic losses. Generally, state estimation (SE) is used to observe the operation of the grid. State estimation of the grid is vulnerable to false data injection attack (FDIA), so diagnosing this type of malicious attack has a major impact on ensuring reliable operation of the power system. In this paper, we present an effective FDIA detection method based on residual recurrent neural network (R2N2) prediction model and adaptive judgment threshold. Specifically, considering the data contains both linear and nonlinear components, the R2N2 model divides the prediction process into two parts: The first part uses the linear model to fit the state data; the second part predicts the nonlinearity of the residuals of the linear prediction model. The adaptive judgment threshold is inferred through fitting the Weibull distribution with the sum of squared errors between the predicted values and observed values. The thorough simulation results demonstrate that our scheme performs better than other prediction based FDIA detection schemes.