IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 23, NO. 2, MAY 2008 817

[3] N. G. Bretas and J. B. A. London, Measurement placement designand reinforcement for state estimation purposes, in Proc. 2001 IEEEPower Tech Conf., Sep. 2001, vol. 3, pp. 1013.

[4] J. B. A. London, L. F. C. Alberto, and N. G. Bretas, Network ob-servability: Identification of the measurements redundancy level, inProc. 2000 Int. Conf. Power System Technology, Dec. 2000, vol. 2, pp.577582.

[5] G. Peters and J. H. Wilkinson, The least-squares problem and pseudoinverses, Comput. J., vol. 13, no. 4, pp. 309316, Aug. 1970.

Closure to Discussion on Placement of PMUs to EnableBad Data Detection in State Estimation

Jian Chen, Student Member, IEEE, and Ali Abur, Fellow, IEEE

The authors are grateful for the discussion which allows clarificationof some important issues.

In order not to crowd the reference list, reference [3] is used, since itcontains descriptions and/or references to the methods followed in ourpaper. Specifically, the numerical method that is used to identify thecritical measurements in the paper is described in detail on pages ofreference [3, pp. 9092]the book entitled Power System State Esti-mation: Theory and Implementation. Please also see reference [1] onGeneralized Observability Analysis and Measurement Classificationby A. Gomez-Exposito and A. Abur for more details on the identifica-tion of critical measurements. Section III-A closely follows the methoddeveloped in [1].

In the same section of the book, the PetersWilkinson decomposi-tion is also used and the relevant reference is given. This method, whichis proposed in [2], can be considered as an extension of Gaussian elim-ination. It is quite useful in solving the normal equations. In this tech-nique, a mxn matrix H can be decomposed into a unit lower trapezoidal

Manuscript received July 25, 2007.J. Chen is with the Department of Electrical Engineering, Texas A&M Uni-

versity, College Station, TX 77843 USA (e-mail: jianchen@ee.tamu.edu).A. Abur is with the Department of Electrical and Computer Engineering,

Northeastern University, Boston, MA 02115 USA (e-mail: abur@ece.neu.edu).Digital Object Identifier 10.1109/TPWRS.2008.920736

mxn matrix L, and an upper triangular nxn matrix U. More details canbe found in the Appendix of the paper as well. Reference [2] does notcontain any method related to observability analysis and our paper doesnot make any claims to that effect. The decomposition is used in step1 of the algorithm in section III-A with proper reference to [2] for de-composition of the H matrix.

The discussors state that the method to determine the critical mea-surements by checking the null columns ofKred

is given without proof.The reason is that the proof is already available for the interested readerin [3, p. 91], where the redundant measurements are expressed in termsof the essential measurements via the matrix T. Please note that this ma-trix T in [3, eq. (4.66)] is the same as Kred

in the paper.The main objective of this paper is to enable bad data detection for all

measurements by strategically placing a minimum number of PMUs.An algorithm is developed and presented for this purpose. An exten-sion of the algorithm is also presented where the local redundancy isincreased to some desired level by adding more PMUs. The extensionof the algorithm transforms some of the critical pairs so that errors inthem can be identified; however, as indicated clearly at the bottom ofSection III-E2 of the paper, there is no guarantee that all critical pairswill be transformed.

As for the example in Fig. 2 of the discussion, the measurementsystem after the placement of the injection is more reliable and de-sirable than without it. Without the injection, no matter which mea-surement is in error, bad data can never be detected. However, withthe injection measurement, irrespective of which of the three measure-ments are in error, bad data will be detected. Creation of critical pairsvia placement of the injection is a small price to pay for having the ca-pability to detect bad data which is otherwise not detectable.

We thank the discussers again for giving us a chance to elaborate onthe above issues.

REFERENCES

[1] A. G. Expsoito and A. Abur, Generalized observability analysis andmeasurement classification, IEEE Trans. Power Syst., vol. 13, no. 3,pp. 10901096, Aug. 1998.

[2] G. Peters and J. H. Wilkinson, The least-squares problem and psudeoinverses, Comput. J., vol. 13, no. 4, pp. 309316, Aug. 1970.

[3] A. Abur and A. Gomez Exposito, Power System State EstimationTheory and Implementation. New York: Marcel Dekker, 2004.

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