In this study, static state estimation problem in smart grid is considered for non-Gaussian environments. The noise model in state estimation is widely assumed to possess Gaussian distribution. However, in some real-world applications, noise may also possess an impulsive distribution. Method of least squares (LS) is generally used for state estimation in systems where noise is modeled by using Gaussian distribution. In this study, noise which contains impulsive components is modeled by alpha-stable distributions. Robust filters are chosen for static state estimation and performances of these filters are compared against the performance of LS. In addition, cumulative sum (CUSUM) technique is employed and its performance is investigated under alpha-stable distributed noise for quickest detection of malicious data injection attacks which might be launched at measurement values of buses in smart grids. In quickest detection, there is a trade-off between detection speed and detection reliability. The chosen threshold value for CUSUM determines the probability of detection for malicious attacks. In this study, impact of the threshold value on detection rate, false alarm rate, and average run length is examined in detail for different alpha values.