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To enhance the robustness of a power system state estimator to topology errors, bad critical measurements, multiple non-interacting, or interacting bad data (BD), this paper presents a new robust detection method by exploiting the temporal correlation and the statistical consistency of measurements. Particularly, we propose three innovation matrices to capture the measurement correlation and statistical consistency by processing the forecasted states/measurements and the interpolated reliable information from phasor measurement units. The latter is achieved by using a robust generalized maximum-likelihood estimator. We then propose to apply the projection statistics (PS) to the proposed innovation matrices for BD detection. Extensive Monte Carlo simulations and QQ-plots are carried out to obtain an analytical threshold of the statistical test of the PS. Because of the robustness of PS and the enhanced measurement redundancy by the innovations, the proposed method is able to handle various types of BD in both PMU observable and PMU partially observable power systems. Moreover, the proposed method is suitable for parallel implementation, and can be integrated with online applications. Comparison results with existing methods under different BD conditions on IEEE 14-bus, 118-bus, and Polish 2383-bus test systems demonstrate the effectiveness and robustness of the proposed method.