Warrington's Equations11/10/07

11/10/07 Warrington’s Equations

During the summer of 1978, I was asked by the WSU professor who taught the protective relaying class to work through the derivation of Warrington’s Equations for the phase distance relays. At the time, there was work being done to develop the micro processor based digital phase distance relays that are in wide use today. My understanding was that the reason for checking on the derivation of these equations was to determine if they were based on the usual stuff.

When I say “the usual stuff,” I am referring to the normal equivalent circuits and assumptions about power system parameters outlined in text books, etc. These assumptions are outlined at the beginning of the derivation. The conclusion I reached was that Warrington’s Equations are based on the usual considerations. I provided my conclusion and the derivation to WSU and, I assume, that is as far as it went. I doubt that anyone ever thought about it any further or used these equations for anything.

The interesting part is that I believe there are some errors in these equations. This added a little bit of emphasis to the question about what these equations may have been used for. Included at the end of the derivation is a letter from an engineer at GEC Measurements in the UK. The letter provides some reasons why the equations were probably never used.

Some references used for this effort are as follows:

- A.R. Van C. Warrington, Protective Relay's, Their Theory and Practice - Volume 1, Wiley/Chapman & Hall, 1968. This is the reference that contains the equations in questions.

- A. R. Van C. Warrington; Graphical Method for Estimating the Performance of Distance Relays During Faults and Power Swings; AIEE paper, 1949, Volume 68. This paper contains some earlier versions of the tables of equations that appear in Warrington’s book. Errors also appear in the paper.

- C. R. Mason; The Art and Science of Protective Relaying; Wiley, 1956

- W. D. Stevenson Jr.; Elements of Power Systems Analysis; 3rd edition

- Edith Clarke; Circuit Analysis of AC Power Systems, Volume I; Symmetrical and Related Components

The derivation is contained in the following JPEG files. I hope you can read my handwriting.

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Four pages of appendix follow:
Page 41, A1.
Page 42, A2.
Page 43, A3.
Page 44, A4.

The following tables are out of Warrington’s book. I have pointed out different aspects of the equations and where the errors are. In the book, these are fold out tables. I cut them up and scanned them in pieces so I could fit them into an 8.5 X 11” format.
Table 5.11 Page 1.
Table 5.11 Page 2.
Table 5.12 Page 1.
Table 5.12 Page 2.
Table 5.12 Page 3.

And finally, this is a letter from an engineer at GEC Measurements in the UK. As I said before, the letter provides some reasons why the equations were probably never used. The engineer also provides a check on my conclusions about the errors in Warrington’s equations.
GEC Page 1.
GEC Page 2.
GEC Page 3.
GEC Page 4.

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