On the conditional mutual information in Gauss-Markov structured grids
Abstract:
Over the past few years, the topology of the “smart grid” has become a popular research subject. One motivation for a serious study of the geometric topology, i.e., the curvature, of the grid is line overloading. Here, however, we are primarily concerned with the detection of false data injection—one of the most dreaded attacks on the power grid—from Phasor Measurements Units (PMU’s) providing the dispatcher with the phase angles Xi, i = 1, 2, ..., at the various buses. Synchronous PMU’s with GPS time stamps are indeed being massively deployed across the grid and are considered the most reliable sensing information to monitor the state of “health” of the grid. Given a graph G = (V,E), Gaussian random variables Xi, i ∈ V = {1, 2, ...}, with probability distribution f(X) ∼ exp(−1/2X^T JX +h^TX) for J = JT ≻ 0, are said to have the Gaussian Markov Random Field (GMRF) property corresponding to G if the following property holds: Jij = 0 ⇔ ij not in E. More specifically here, we use Anandkumer et al’s approach of Conditional Covariance Test (CCT) to reconstruct the Markov graph of the measurements. Given phase angles Xi’s computed from the DC power flow equations Pi = sum j \in N(i) b_{ij}(X_i − X_j), where the bij ’s are inverse of line reactance, N(i) the neighbors of i for the bij graph, and the P_i’s the powers injected at the buses, the first and fundamental question addressed in this paper is whether the X_i’s have the GMRF property relative to the grid graph G_g = (V,E_b), where (i, j) ∈ E_b iff b_{ij} \notin 0. By several examples, e.g., infinite chains and lattices, we show that the answer is “no,” in general. For example, if the powers Pi’s injected at the buses are normal and independent, the DC power flow equations of an infinite chain imply that E(X_i | X_j : j \not= i) = E(X_i | X_j : j ∈ {N(i) ∪ N(N(i))}), which falls short of the GMRF property E(X_i | X_j : j \not= i) = E(X_i | X_j : j ∈ N(i)) for the N(i) neighboring property. However, the grid graph is walk-summable and the correlation coefficients, which should vanish at any distance greater than 1 neighbor, are tapering off fairly quickly. A second contribution of the paper is to show that for an infinite chain this tapering off is in fact the tapering off of some Fourier coefficients associated with some Toeplitz operator. Third, despite the lack of GMRF property relative to G_g, but with the fast tapering off of the correlations and the correct set up of the threshold, CCT reconstructs—under normal grid conditions—the correct line topology. Finally, we show that when we apply CCT method to PMU angle measurements, in case of the recent stealth deception attack, the Markov graph is missing some links that are present in the grid graph and this triggers the alarm. It should be noted that this attack assumes knowledge of bus-branch model of the system and is capable of fooling the state estimator and no remedies have been suggested for it so far.