Fourier Transform Algorithm in
Power System Relaying
Ahmad Abdullah
ECEN679: Computer Relays for
Electric Power Systems
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0
United States License.
What Do I want you to take home?
• Difference between Continuous Fourier
(CFT)transform/series and the discrete
version(DFT)
• How is DFT is used in distance relaying?
• What are the advantages and disadvantages?
Fourier Transform: Continuous VS.
Discrete
• Fourier Series applied to continuous signals
decomposes a “Periodic” signal into its sin and
cosine “components” by calculating the
coefficients of those components at integral
multiples of the fundamental frequency defined
by the period of the function.
• Continuous Fourier transform ALSO calculates a
function from which coefficients can be
calculated at each frequency. This time the
coefficient can be calculated at any frequency.
Example
Images taken from http://www.mathworks.com/matlabcentral/fileexchange/43590-fourier-series-of-squarewave-demo-of-gibbs-phenomenon-with-overshoot-calculation
Example Continued
Images taken from http://www.thefouriertransform.com/transform/fourier.php
What’s the difference?
• Enumerability!
• In the series case, the coefficients can be
maps in a one to one fashion to the set of
positive integers while in the transform case
this is not possible.
What is DFT?
• Name is very misleading because it is not like
CFT at all!!
• Basically DFT is an approximation to Fourier
coefficients for periodic signals!
• Approximation is based on samples not
functions!
• Always Periodic (since it’s approximation to
Fourier series based on samples!)
DFT Definition
Equation taken from http://www.cs.ioc.ee/~khoros2/linear/dft-pulse-example/front-page.html
Basic Algorithm for distance relaying
• Get 𝐶1 and 𝐶2 for both fundamental voltage
and current, divide both then calculate the
impedance for each sample.
• Do this recursively for 3 or 4 samples
• If distance in the circle, issue trip signal.
Advantages
• Very Fast as Fast Fourier Transform can be used if
even number of samples exist which can be done
by suitably choosing sampling frequency and data
window length.
• It can be proved that DFT is the best fit for the
samples if the exponential term caused by the
fault is removed
• Very widely used because of it simplicity
• Symmetrical Component version exist which
smoothen the samples thus reducing the error
Disadvantages
• Waveform has to be known in advance (FACTS
devices?)
• Used for relaying, it suffers from the same
problem found in distance relaying
• A transient monitor function has to be used to
monitor the transient behavior of the waveform
as a secure decision is not possible in transient
time.
• A more precise decision requires longer data
window.
Data Window Problem
Image taken from Computer Relaying for Power Systems
References
1. Mann, B.J.; Morrison, I. F., "Digital Calculation of
Impedance for Transmission Line Protection," Power
Apparatus and Systems, IEEE Transactions on ,
vol.PAS-90, no.1, pp.270,279, Jan. 1971
doi: 10.1109/TPAS.1971.292966
2. Phadke, A.G.; Hlibka, T.; Ibrahim, M.; Adamiak, M.G.,
"A Microcomputer Based Symmetrical Component
Distance Relay," Power Industry Computer
Applications Conference, 1979. PICA-79. IEEE
Conference Proceedings , vol., no., pp.47,55, 15-18
May 1979
doi: 10.1109/PICA.1979.720045
References
3. Phadke, A.G.; Thorp, J.S.; Adamiak, M.G., "A New
Measurement Technique for Tracking Voltage Phasors,
Local System Frequency, and Rate of Change of
Frequency," Power Apparatus and Systems, IEEE
Transactions on , vol.PAS-102, no.5, pp.1025,1038,
May 1983
doi: 10.1109/TPAS.1983.318043
4. Thorp, J.S.; Phadke, A.G.; Horowitz, S.H.; Beehler, J.E.,
"Limits to Impedance Relaying," Power Apparatus and
Systems, IEEE Transactions on , vol.PAS-98, no.1,
pp.246,260, Jan. 1979
doi: 10.1109/TPAS.1979.319525
References
5. Phadke, A.G.; Hlibka, T.; Adamiak, M.G.;
Ibrahim, M.; Thorp, J.S., "A Microcomputer
Based Ultra-High-Speed Dostance Relay: Field
Tests," Power Apparatus and Systems, IEEE
Transactions on , vol.PAS-100, no.4,
pp.2026,2036, April 1981
doi: 10.1109/TPAS.1981.316480
6. Phadke, A.G. and Thorp, J.S. “Computer
Relaying for Power Systems”, Wiley , 2009