1. Orthogonal Transforms (Haar, Hartley) Algorithm Student: Payman Dehghanian Instructor: Prof. Mladen Kezunovic Course: ECEN679-Computer Relaying March, 03, 2014 Email: payman.dehghanian@tamu.edupayman.dehghanian@tamu.edu

2. Outline  Transformer Protection Principles  Theory of Orthogonal Haar Transform  Application of Orthogonal Haar Transform  Advantages/Disadvantages  References 2

3. Transformer Protection Principles Basic Idea  Differential Protection is the common approach!  Comparison of primary and secondary currents.  Internal faults recognition! Trip signal issued if substantially different from pre-defined/threshold value False Tripping  Inrush, saturation and overexcitation happens  Relay misunderstanding and false trip Solution  Harmonic content analysis of differential current samples  Inrush current incorporates higher percentage of harmonics (2 nd )  Comparison with a pre-defined percentage of the fundamental 3

4. Theory of Orthogonal Haar Transform What is an Orthogonal transform?  Consider a set of real –valued functions: {u n (t)} = {u 0 (t), u 1 (t), u 2 (t),…} defined on the interval (t, t+T). {un(t)} is said to be Orthogonal if:  Orthogonal transforms (in terms of a lot of algorithms) are used for Harmonic Analysis. 4

5. Theory of Orthogonal Haar Transform Why an Orthogonal transform?  Ease of SIGNAL REPRESENTATION using predefined functions and corresponding weights/coefficients  DECORRELATION of naturally correlated signals; due to cross correlation of parent functions resulting in ZERO  ENERGY COMPACTION pack large fraction of average energy of signal into relatively fewer components of transform coefficients  INFORMATION PRESERVATION 5

6. Theory of Orthogonal Haar Transform What is a Haar Function?  Belongs to NON-SINUSOIDAL family of orthogonal functions  Defined in t Є [0,1) 6

7. Theory of Orthogonal Haar Transform 7

8. Application of Orthogonal Haar Transform  Haar coefficients calculated from differential current samples are used to estimate Fourier coefficients for fundamental, 2 nd and 5 th harmonics. ………… 8

9. Application of Orthogonal Haar Transform  Fourier coefficients are used to estimate ratio of sum of 2 nd and 5 th harmonics to fundamental.  The relay logic then determines the operating and restraining signal.  Trip signal issued if ratio is below a threshold 9

10. Advantages/Disadvantages Advantages  Accurate signal representation independent of the wave shape using only few terms  Excellent filtering characteristics  Superior convergence features (e.g., compared with Walsh)  Speed unaffected by predominant harmonics/ decaying DC noise Disadvantages  Have to use 2 n samples/cycle only; restricted sampling frequencies.  Computationally complex with many +/- operations and a few * operations (if modified or approximate Haar algorithms cannot be used) 10

11. References [1] M. Rahman and B. Jeyasurya, “A State-of-the Art Review of Transformer Protection Algorithms”, IEEE Transactions on Power Delivery, vol. 3, no. 2, pp. 534-544, April 1988. [2] M. Habib and M. Marin, “A Comparative Analysis of Digital Relaying Algorithms for the Differential Protection of Three Phase Transformers”, IEEE Transactions on Power Systems, vol.3, no.3, pp. 460-466, August 1988. [3] D. B. Fakruddin, K. Parthsarathy, L. Jenkins and B. W. Hogg, “Applications of Haar functions for transmission line and transformer differential protection”, Electrical Power & Energy Systems, vol. 6, no. 3, pp. 169-180, July 1984. [4] R. R. Larson, A. J. Flechsig, E. O. Schweitzer, “The design and test of a digital relay for transformer protection”, IEEE Transactions on Power Apparatus & Systems, vol. PAS-98 (1979) pp. 795-804. [5] N. Ahmed, K. R. Rao (1975). Orthogonal Transforms for Digital Signal Processing. New York: Springer. 11

12. E-mail: Payman.Dehghanian@tamu.eduPayman.Dehghanian@tamu.edu Thank You! 12