Invited Paper: Detection of False Data Injection Attacks in Power Systems Using a Secured-Sensors and Graph-Based Method

Abstract

False data injection (FDI) attacks pose a significant threat to the reliability of power system state estimation (PSSE). Recently, graph signal processing (GSP)-based detectors have been shown to enable the detection of well-designed cyber attacks named unobservable FDI attacks. However, current detectors, including GSP-based detectors, do not consider the impact of secured sensors on the detection process; thus, they may have limited power, especially in the low signal-to-noise ratio (SNR) regime. In this paper, we propose a novel FDI attack detection method that incorporates both knowledge of the locations of secured sensors and the GSP properties of power system states (voltages). We develop the secured-sensors-and-graph-Laplacian-based generalized likelihood ratio test (SSGL-GLRT) that integrates the secured data and the graph smoothness properties of the state variables. Furthermore, we introduce a generalization of the method that allows the use of different high-pass GSP filters together with prior knowledge of the locations of the secured sensors. Then, we develop the SSGL-GLRT for a distributed PSSE based on the alternating direction method of multipliers (ADMM). Numerical simulations demonstrate that the proposed method significantly improves the probability of detecting FDI attacks compared to existing GSP-based detectors, achieving an increase of up to 30% in the detection probability for the same false alarm rate by integrating secured sensor location information.

Appendix: Concavity of \(\mathcal {Q}(\boldsymbol{\theta },\mathbf{{a}})\)

In order to show that the function $\mathcal {Q}(\boldsymbol{\theta },\mathbf{{a}})$ from (8) is a concave function w.r.t $\boldsymbol{\theta }$ and $\mathbf{{a}}$, we need to show that the Hessian matrix of the second-order partial derivatives of $-\mathcal {Q}(\boldsymbol{\theta },\mathbf{{a}})$ is a positive semidefinite matrix. It can be seen that the Hessian matrix of $-\mathcal {Q}(\boldsymbol{\theta },\mathbf{{a}})$ w.r.t. the vector $[\boldsymbol{\theta }^T,\mathbf{{a}}^T]^T$ is

$$\begin{aligned} \begin{pmatrix} \mathbf{{H}}^T\mathbf{{R}}^{-1}\mathbf{{H}}+\mathbf{{B}}&{} \mathbf{{H}}^T\mathbf{{R}}^{-1} \\ \mathbf{{R}}^{-1}\mathbf{{H}}&{} \mathbf{{R}}^{-1}+{\mathbf{{M}}} \end{pmatrix} = \begin{pmatrix} \mathbf{{H}}^T\mathbf{{R}}^{-1}\mathbf{{H}}&{} \mathbf{{H}}^T\mathbf{{R}}^{-1} \\ \mathbf{{R}}^{-1}\mathbf{{H}}&{} \mathbf{{R}}^{-1} \end{pmatrix} + \begin{pmatrix} \mathbf{{B}}&{} \mathbf{{0}}\\ \mathbf{{0}}&{} {\mathbf{{M}}} \end{pmatrix}. \end{aligned}$$

The Hessian is a sum of two matrices. In the following, we show that each one of these matrices is positive semidefinite, which implies that the Hessian is a positive semidefinite matrix. First, it can be seen that the matrix $\begin{pmatrix} \mathbf{{B}}&{} \mathbf{{0}}\\ \mathbf{{0}}&{} {\mathbf{{M}}} \end{pmatrix}$ is a positive semidefinite matrix because it is a block diagonal matrix of two positive semidefinite matrices (see the definitions of $\mathbf{{B}}$ and ${\mathbf{{M}}}$ in (1) and (8), respectively). Second, the matrix $\begin{pmatrix} \mathbf{{H}}^T\mathbf{{R}}^{-1}\mathbf{{H}}&{} \mathbf{{H}}^T\mathbf{{R}}^{-1} \\ \mathbf{{R}}^{-1}\mathbf{{H}}&{} \mathbf{{R}}^{-1} \end{pmatrix}$ is a positive semidefinite matrix since it can be verified that its Schur complement,

$$\begin{aligned} \mathbf{{H}}^T\mathbf{{R}}^{-1}\mathbf{{H}}-\mathbf{{H}}^T\mathbf{{R}}^{-1}\mathbf{{R}}\mathbf{{R}}^{-1}\mathbf{{H}}=\mathbf{{0}}, \end{aligned}$$

is a positive semidefinite matrix [13].