The secure operation of smart grids is closely linked to state estimates that accurately reflect the physical characteristics of the grid. However, well-designed false data injection attacks (FDIAs) can manipulate the process of state estimation by injecting malicious data into the measurement data while bypassing the detection of the security system, ultimately causing the results of state estimation to deviate from secure values. Since FDIAs tampering with the measurement data of some buses will lead to error offset, this paper proposes an attack-detection algorithm based on statistical learning according to the different characteristic parameters of measurement error before and after tampering. In order to detect and classify false data from the measurement data, in this paper, we report the model establishment and estimation of error parameters for the tampered measurement data by combining the the k-means++ algorithm with the expectation maximization (EM) algorithm. At the same time, we located and recorded the bus that the attacker attempted to tamper with. In order to verify the feasibility of the algorithm proposed in this paper, the IEEE 5-bus standard test system and the IEEE 14-bus standard test system were used for simulation analysis. Numerical examples demonstrate that the combined use of the two algorithms can decrease the detection time to less than 0.011883 s and correctly locate the false data with a probability of more than 95%.

The Phasor Measurement Unit (PMU) with a GPS signal receiver is a synchronized sensor widely used for power system state estimation. While the GPS receiver ensures time accuracy, it is vulnerable to network attacks. GPS spoofing attacks can alter the phase angle of PMU measurement signals and manipulate system states. This paper derives a power system state model based on PMUs under GPS spoofing attacks, according to the characteristics of changes in bus voltages and branch currents after GSA. Based on the characteristics of this model, a detection and correction algorithm for attacked data is proposed to detect GSA and correct attacked measurements. The corrected measurements can be used for power system state estimation. Simulation results on the IEEE 14-bus system show that the proposed algorithm improves the accuracy of state estimation under one or multiple GSAs, especially when multiple GSAs are present, compared to classical Weighted Least Squares Estimation (WLSE) and Alternating Minimization (AM) algorithms. Further research indicates that this algorithm is also applicable to large-scale networks.

Outliers can be generated in the power system due to aging system equipment, faulty sensors, incorrect line connections, etc. The existence of these outliers will pose a threat to the safe operation of the power system, reduce the quality of the data, affect the completeness and accuracy of the data, and thus affect the monitoring analysis and control of the power system. Therefore, timely identification and treatment of outliers are essential to ensure stable and reliable operation of the power system. In this paper, we consider the problem of detecting and localizing outliers in power systems. The paper proposes a Minorization–Maximization (MM) algorithm for outlier detection and localization and an estimation of unknown parameters of the Gaussian mixture model (GMM). To verify the performance of the method, we conduct simulation experiments by simulating different test scenarios in the IEEE 14-bus system. Numerical examples show that in the presence of outliers, the MM algorithm can detect outliers better than the traditional algorithm and can accurately locate outliers with a probability of more than $95\%$. Therefore, the algorithm provides an effective method for the handling of outliers in the power system, which helps to improve the monitoring analyzing and controlling ability of the power system and to ensure the stable and reliable operation of the power system.