On the Conditional Mutual Information in the Gaussian–Markov Structured Grids
 Hanie Sedghi,
 Edmond Jonckheere
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Abstract
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 widearea 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 socalled “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 1neighbor property of the bus phase angles is derived. The latter topological property refers to the conditional mutual information between two random variables being nonvanishing 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 walksummability 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.
 Abur, A., Exposito, A. (2004) Power System State Estimation, Theory and Implementation. Dekker, New York CrossRef
 Anandkumar, A., Tan, V., Huang, F., Willsky, A. (2012) Highdimensional Gaussian graphical model selection: walk summability and local separation criterion. J. Mach. Learn. Res. 13: pp. 22932337
 Banirazi, R., Jonckheere, E. (2010) Geometry of power flow in negatively curved power grid. 49th IEEE Conference on Decision and Control. pp. 62596264 CrossRef
 Bishop, C.M. (2006) Pattern Recognition and Machine Learning. Springer, New York
 Bolognani, S., Zampieri, S.: A distributed control strategy for reactive power compensation in smart microgrids. Preprint arXiv:1106.5626, Oct. 2012
 Borga, M.: Learning multidimensional signal processing. Tech. Rep. SE581 83, Linköping University, Sweden (1998)
 Friedman, J., Hastie, T., Tibshirani, R.: Sparse inverse covariance estimation with the graphical lasso. Biostatistics (2007)
 He, M., Zhang, J. (2011) A dependency graph approach for fault detection and localization towards secure smart grid. IEEE Trans. Smart Grid 2: pp. 342351 CrossRef
 Janzamin, M., Anandkumar, A.: Highdimensional covariance decomposition into sparse Markov and independence models. Preprint arXiv:1211.0919, Nov. 2012
 Jonckheere, E., Wu, B.F. (1992) Mutual Kolmogorov–Sinai entropy approach to nonlinear estimation. IEEE Conference on Decision and Control. pp. 22262232
 Larimore, W.E. Identification and filtering of nonlinear systems using canonical variate analysis. In: Casdagli, M., Eubank, S. eds. (1992) Nonlinear Modeling and Forecasting. Addison–Wesley, Reading
 Larimore, W.E., Baillieul, J. (1990) Identification and filtering of nonlinear systems using canonical variate analysis. 29th IEEE Conference on Decision and Control. pp. 635640 CrossRef
 Lauritzen, S. (1996) Graphical Models. Clarendon, Oxford
 Lofberg, J. (2004) YALMIP: a toolbox for modeling and optimization in MATLAB. IEEE International Symposium on Computer Aided Control Systems Design (CACSD).
 Luettgen, M., Karl, W., Willsky, A., Tenney, R. (1993) Multiscale representations of Markov random fields. IEEE Trans. Signal Process. 41: pp. 33773396 CrossRef
 Makarychev, K., Makarychev, Y., Romashchenko, A., Vereshchagin, N. (2002) A new class of nonShannontype inequalities for entropies. Commun. Inf. Syst. 2: pp. 147166
 Maniglia, S., Rhandi, A.: Gaussian measures on separable Hilbert spaces and applications. Tech. rep., Lecture Notes of the University of Lecce, Italy, Quaderno 1/2004 (2004). ISBN: 888305010X
 MurAmada, J., SallánArasanz, J. From turbine to wind farms—technical requirements and spinoff products. In: Krause, G. eds. (2011) Phase Transitions and Critical Phenomena. InTech, Rijeka, pp. 101132
 Prato, G.D. (2000) An Introduction to InfiniteDimensional Analysis. Springer, Berlin
 Ravikumar, P., Wainwright, M., Raskutti, G., Yu, B. (2011) Highdimensional covariance estimation by minimizing ℓ 1penalized logdeterminant divergence. Electron. J. Stat. 4: pp. 935980 CrossRef
 Rue, H., Held, L. (2005) Gaussian Markov Random Fields: Theory and Applications. CRC Press, Boca Raton CrossRef
 Shah, K., Jonckheere, E., Bohacek, S. (2007) Dynamic modeling of Internet traffic for intrusion detection. EURASIP J. Adv. Signal Process. 2007: CrossRef
 Teixeira, A., Dan, G., Sandberg, H., Johansson, K.H. (2011) A cyber security study of a SCADA energy management system: stealthy deception attacks on the state estimator. IFAC World Congress.
 Toh, K.C., Todd, M., Tutuncu, R.H. (1999) SDPT3—a MATLAB software package for semidefinite programming. Optim. Methods Softw. 11: pp. 545581 CrossRef
 Zimmerman, R.D., MurilloSánchez, C.E., Thomas, R.J. (2011) Matpower steadystate operations, planning and analysis tools for power systems research and education. IEEE Trans. Power Syst. 26: pp. 1219 CrossRef
 Title
 On the Conditional Mutual Information in the Gaussian–Markov Structured Grids
 Book Title
 Information and Control in Networks
 Book Part
 Part III
 Pages
 pp 277297
 Copyright
 2014
 DOI
 10.1007/9783319021508_9
 Print ISBN
 9783319021492
 Online ISBN
 9783319021508
 Series Title
 Lecture Notes in Control and Information Sciences
 Series Volume
 450
 Series ISSN
 01708643
 Publisher
 Springer International Publishing
 Copyright Holder
 Springer International Publishing Switzerland
 Additional Links
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 Industry Sectors
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 Editors

 Giacomo Como ^{(4)}
 Bo Bernhardsson ^{(5)}
 Anders Rantzer ^{(6)}
 Editor Affiliations

 4. Department of Automatic Control, Lund University
 5. Department of Automatic Control, Lund University
 6. Department of Automatic Control, Lund University
 Authors

 Hanie Sedghi ^{(7)}
 Edmond Jonckheere ^{(7)}
 Author Affiliations

 7. Department of Electrical Engineering, University of Southern California, Los Angeles, CA, 90089, USA
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