On the Conditional Mutual Information in the Gaussian–Markov Structured Grids

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The Supervisory Control and Data Acquisition (SCADA) State Estimator (SE) and the Phasor Measurement Units (PMUs) network constitute the communication infrastructures meant to provide the “smart grid” dispatcher with wide-area bus phase angles and other data from which the operational status of the grid can be assessed—if the measurements are not compromised somewhere along their way to the SCADA dispatch and/or the PMU concentrator. Unfortunately, this is precisely what happens under the so-called “false data injection.” In this chapter, we develop a fast test for measurement data integrity, based on the Gaussian Markov Random Field (GMRF) assumption on the PMU data. This assumption, fundamental to this chapter, is supported by (i) the many fluctuating generations and variable loads justifying the Gaussian distribution assumption and, as more specifically addressed in this chapter, (ii) the DC power flow equations from which an approximate 1-neighbor property of the bus phase angles is derived. The latter topological property refers to the conditional mutual information between two random variables being non-vanishing if and only if the nodes at which they are observed are linked in the edge set of the corresponding graph. Under the Gaussian distribution assumption, the conditional mutual information is easily computable from the conditional covariance. Then it is shown that Conditional Covariance Test (CCT) together with the walk-summability and the local separation property of grid graph allows the reconstruction of the grid graph from uncompromised measurement data. On the other hand, with corrupted data, CCT reconstructs only a proper subset of the edge set of the grid graph, hence triggering the alarm.