Elements of Fuzzy Logic in Protective Relays - Digital Signal Processing in Power System Protection and Control - page 229

0 0 0 1 1 1 0 0 0.2 0.4 0.6 0.8 1 1

Boolean Logic Multi-valued Logic

Fig. 11.8 Boolean versus Fuzzy logic—an illustration of the concept

Unlike the two-valued Boolean logic, FL is multi-valued. It deals with degrees
of membership and degrees of truth. FL uses the continuum of logical values
between 0 (completely false) and 1 (completely true). Instead of just black and
white, it employs the spectrum of colors (or color shades, Fig. 11.8 ), accepting that
things can be partly true and partly false at the same time. For instance, one can
say that a glass is half-empty or half-full, depending on the context of this state-
ment, which in turn may lead to different implications. By analogy, one may
express the fulfillment of given protection criterion as:

• fully met (the criterion signal is far beyond the decision threshold),
• partially met (the criterion signal exceeds the threshold just slightly and/or
oscillates around it),
• partially excluded (the measured value is slightly below the threshold, may
exceed it for a moment during transient),
• completely excluded (the signal is well below the threshold value).
All the above is closely dependent on the adopted conception of truth. As
opposed to the Aristotelian conception of truth, FL allows for a gradual transition
between TRUE and FALSE. By comparison to Fig. 14.6 , where overlapping of the
decision regions were interpreted as superimposing of the conditional probability
density functions, here the fuzzy border between two classes of events is defined
(Fig. 11.9 ), where a smooth curve substitutes for a crisp threshold value. One
should say that in case (b) no crisp threshold is appropriate for effective decision-
making, while with fuzzy approach the degree of setting exceeding can be spec-
ified, which can lead to the final protection decision.
Decision-making with FL can be compared to classification of the object to one
of two or more sets, where the border between them is not sharp, but is specified
with some fuzziness. An element is classified as belonging to given set with some
degree between zero and unity. When the degree of membership is ''high'' (also a
fuzzy term) the final decision can be made, otherwise it is either denied or post-
poned until sound difference between the membership grades for given sets is
observed. When

j d 1 d 2 j [ D ; ð 11 : 22 Þ

one of the hypotheses (H 1 or H 2 ) is accepted, the one for which the fuzzy support
value d 1 or d 2 is the highest, with D being certain threshold representing the
decision confidence margin.