State estimation plays a vital role to ensure safe and reliable operations in smart grid. Intelligent attackers can carefully design a destructive and stealthy false data injection attack (FDIA) sequence such that commonly used weighted least squares estimator combined with residual-based detection method is vulnerable to the FDIA. To effectively defend against an FDIA, in this paper, we propose a robust deviation-based detection method, in which an additional Kalman filter is introduced while retaining the original weighted least squares estimator, so that there are two state estimators. Moreover, an exponential weighting function is also applied to the introduced Kalman filter in our proposed method. When an FDIA occurs, the estimation results of weighted least squares estimator depend only on meter measurements at each time slot, but there is an adjustment process of estimated states for the Kalman filter based on historical states' transitions. Meanwhile, based on the exponential weighting function, estimated measurements in the Kalman filter can be adaptively suppressed for different attack strengths of FDIAs, and then the difference of the results of these two estimators increases. Subsequently, FDIAs can be effectively detected by checking the deviation of estimated measurements about the two estimators with a detection threshold. Experimental results validate the effectiveness of the proposed detection method against FDIAs. The impact of different attack strengths and noise on detection performance is also evaluated and analyzed.