This paper develops a novel approach to track power system state evolution based on the maximum correntropy criterion, due to its robustness against non-Gaussian errors. It includes the temporal aspects on the estimation process within a maximum-correntropy-based extended Kalman filter (MCEKF), which is able to deal with both nonlinear supervisory control and data acquisition (SCADA) and phasor measurement unit (PMU) measurement models. By representing the behavior of the state variables with a nonparametric model within the kernel density estimation, it is possible to include abrupt state transitions as part of the process noise with non-Gaussian characteristics. Also, a novel strategy to update the size of Parzen windows in the kernel estimation is proposed to suppress the effects of suspect samples. By properly adjusting the kernel bandwidth, the proposed MCEKF keeps its accuracy during sudden load changes and contingencies, or in the presence of bad data. Simulations with IEEE test systems and the Brazilian interconnected system are carried out. The results show that the method deals with non-Gaussian noises in both the process and measurement, and provides accurate estimates of the system state under normal and abnormal conditions.