**Inconsistent Decision System: Rough Set Data Mining Strategy to Extract Decision Algorithm of a Numerical Distance Relay – Tutorial**

Mohammad Lutfi Othman and Ishak Aris

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/50460

## **1. Introduction**

116 Advances in Data Mining Knowledge Discovery and Applications

Engineering &*Technology, pp. 383 – 387.*

Reinhold, New York.

*Sciences*, 4, pp. 494–498.

and Review, 34(4), 513-522.

Applications, 34, 780-787.

*Engineering, PowerTech Budapest 99.*

*20 (3), 435-446*

pp. 249 – 255

[32] Wasserman, P. D. (1989). Neural Computing: Theory and Practice, Van Nostrand-

[33] Marquez, L., Hill, T., O'Connor, M., & Remus, W. (1992). Neural network models for forecast a review. *IEEE proceedings of the 25th Hawaii International Conference on System* 

[34] Siraj, F., &Asman, H. (2002). Predicting Information Technology Competency Using Neural Networks.*Proceedings of the 7th Asia Pacific Decision Sciences Institute Conference*,

[35] Siraj, F. & Mohd Ali, A. (2004)*. Web-Based Neuro Fuzzy Classification for Breast Cancer.*  Proceedings of the Second International Conference on Artificial Intelligence in

[36] Zhang, D. & Zhou, L. (2004). Discovering Golden Nuggets: Data Mining in Financial Application. IEEE Transactions on Systems, Man, and Cybernetics, Part C: Applications

[37] Hung, C. & Tsai, C. F. (2008). Market segmentation based on hierarchical selforganizing map for markets of multimedia on demand. Expert Systems with

[38] Heravi, S., Osborn, D. R., &Brichernhall, C. R. (2004). Linear versus neural network forecasts for European industrial production series. *International Journal of Forecasting,* 

[39] Lam, M. (2004). Neural network techniques for financial performance prediction: integrating fundamental and technical analysis. *Decision Support System, 37 (4),567-581* [40] De Andre, J., Landajo, M., & Lorca P. (2005). Forecasting business profitability by using classification techniques: A comparative analysis based on a spanish case. *Electric Power*  Modern numerical protective relays being intelligent electronic devices (IED) are inevitably vulnerable to false tripping or failure of operation for faults in the power system [1]. With regular and rigorous analyses the performance reliability of the digital protective relays can be ascertained, their availability maximized and subsequently their misoperation risks minimized [2]. The precise relay operation analyses would normally be assessing the relay characteristics, evaluating the relay performance and identifying the relay-power system interactions so as to ensure that the protective relays operate in correspond to their predetermined settings [3,4].

Protection engineers would in practice resort to computing technologies for automating the analysis process when the gravity of event data exploration, manipulation and inferencing incapacitate human manageability. The voluminous amount of data to be processed has prompted the need to use intelligent data mining, an essential constituent in the Knowledge Discovery in Databases (KDD) process [5]. This has motivated the adoption of rough set theory to data mine the protective relay event report so as to discover its decision algorithm.

## **2. Problem statement and objective**

The following two pertinent problems are the attributing factors in driving this paper into studying the protective relay operation analysis:

 Inconsistencies in the device's event report particularly found when upon power system fault inception, a protective relay detects and invokes a common combination of tripping conditions in time succession but having two distinct tripping decisions

© 2012 Othman and Aris, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 Othman and Aris, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

(classifications). These distinct decisions are one, that upon relay pick-up, trip signal has not been asserted immediately after and the other is when a subsequent trip signal is asserted, after a preset time delay as set by the protection engineer.

Inconsistent Decision System: Rough Set Data Mining Strategy to Extract Decision Algorithm of a Numerical Distance Relay – Tutorial 119


*f*: *UQ V* called *information function* is a total function such that *f*(*t,q*) *Vq* for every *t* 

If a set of attributes *P Q = C D* and *f*(*tx,q*) = *f*(*ty,q*) where *tx*, *ty U*, then for every *q P*, *tx* and *ty* are *indiscernible* (indistinguishable) by the set of attributes *P* in *DS*. Thus, every *P Q* brings forth a binary relation on *U* called *P-indiscernibility relation* (or *equivalence relation*) which is denoted by *IND*(*P*). This suggests that there will be sets of relay events that are indiscernible based on any selected subset of attributes *P*. *UIND*(*P*) denotes the family of all *equivalence classes* of relation *IND*(*P*). *IND*(*P*) and *UIND*(*P*) can be

*UIND*(*P*) is also interchangeably referred as *P-basic knowledge* or *P-elementary sets* in *DS*. *P*elementary set including relay event *t* is denoted as [*t*]*IND*(*<sup>P</sup>*). The first step in classification with rough sets is the construction of elementary sets [11]. A description of *P*-elementary set

In the context of protective relay operations, consider *T U* as an arbitrary target set of relay events described (classified) by a particular trip assertion status that is needed to be represented by equivalence classes originating from attribute subset *P Q*. *P* could be a selected condition attribute set *P C* or all condition attributes *C* reflecting relay multifunctional protective elements while *T* could be the set of relay events indiscernible with respect to the decision attribute *D* = *Trip* having a domain value 'b' for pole-B tripping,

The idea of the rough set revolves around the concept of approximation [11]. Thus, by introducing a pair of sets, called the *lower* and *upper approximations* of the target set *T* using

only the information contained within *P*, the target set *T* can be *approximated*.

*X UIND*(*P*) in terms of values of attributes from *P* is denoted as *DesP*(*X*), i.e.

**3.2. Relay decision system set approximation** 

2 {, , }, *x y x y IND P t t U q P q t q t* (1)

*U IND P q P U IND q* { { } }, (2)

*A B X Y X A Y BX Y Ø* { , , }. (3)

*Des X q v f t q v t X q P <sup>P</sup>* {, : , , , } (4)

*U*, *q Q*. Any pair (*q,v*) is called *descriptor* in *DS*, where *q Q* and *v Vq*,.

*V* = *Uq<sup>Q</sup> Vq*, where *Vq* is a set of values (*domain*) of the attribute *q*.

**3.1. Relay decision system indiscernibility relation** 

for every *q Q*.

formulated as

where,

for example [7].

 Non-linear nature of relay operation that makes it very difficult to select a group of effective attributes to fully represent relay tripping behavior.

In the grueling manual analysis of relay event report [1,6], the selected attributes hardly provide adequate knowledge in accurately mapping the interclass boundary in the relay decision system due to inconsistency. This characterizes the interclass boundary to be usually "rough". Based on the selected attributes, some relay events close to the boundary are unclassifiable – trip or nontrip. The small overlaps between different relay events make the protective relay operation analysis to be actually a rough classification problem. Thus, rough set theory has been appropriately chosen to resolve this conflict [7].

## **3. Rough set data mining in dealing with inconsistent numerical distance relay decision system to extract decision algorithm – The fundamental concept**

Using rough set theory approach, relay decision rule extraction is naturally a byproduct of the data reduction process involved and easily understood. Rule extraction technique is inherent to the machine learning process of rough set theory. Thus, the inherent capability of rough set theory to discover fundamental patterns in relay data has essentially mooted this study. Using an approximation concept, rough set theory is able to remove data redundancies and consequently generate decision rules. In contrast to crisp sets, a rough set has boundary line cases – events that cannot be certainly classified either as members of the set or of its complement. Rough set theory is an alternative intelligent data analysis tool that can be employed to handle vagueness and inconsistencies [8].

An *information system* (*IS*) also alternatively known as *knowledge representation system* (*KRS*) is a tabulated data set, the rows of which are labeled by *objects* (events) of interest, columns labeled by *attributes*, and the entries are *attribute values* [8]. This data layout fits very well the protective relay event report that is characterized by its attributes of relay multifunctional elements versus sequence of time-stamped events [7].

In the protective relay event report, the *IS* manifestation is more appropriately referred as relay *decision table* or *decision system* (*DS*) as Huang et. al. [9] put it that decision table is characterized by disjoint sets of *condition attributes* (*C Q*) and *decision (action) attributes* (*D Q*). In this regard *Q* = *C D* and *C* ∩ *D* = Ø. This *DS* is a 4-tuple structure formulated as *DS* = *U*, *Q*, *V*, *f*, the elements of which are as follows [8,10,11]:

	- condition attributes (*ci C*) indicate the internally various multifunctional protective elements and analog measurands,

#### **3.1. Relay decision system indiscernibility relation**

If a set of attributes *P Q = C D* and *f*(*tx,q*) = *f*(*ty,q*) where *tx*, *ty U*, then for every *q P*, *tx* and *ty* are *indiscernible* (indistinguishable) by the set of attributes *P* in *DS*. Thus, every *P Q* brings forth a binary relation on *U* called *P-indiscernibility relation* (or *equivalence relation*) which is denoted by *IND*(*P*). This suggests that there will be sets of relay events that are indiscernible based on any selected subset of attributes *P*. *UIND*(*P*) denotes the family of all *equivalence classes* of relation *IND*(*P*). *IND*(*P*) and *UIND*(*P*) can be formulated as

$$\text{ADD}\left(P\right) \;= \left\{ \left(t\_{x'}t\_y\right) \in \mathcal{U}^2 \; \middle| \; \forall q \in P, q\left(t\_x\right) \;= q\left(t\_y\right) \right\} \;\tag{1}$$

$$\text{LU}\Big|\text{IND}\big(P\big) \ = \bigotimes \{q \in P\Big|\text{LU}\big|\text{IND}\big(\{q\}\big)\}\,\tag{2}$$

where,

118 Advances in Data Mining Knowledge Discovery and Applications

**concept** 

(classifications). These distinct decisions are one, that upon relay pick-up, trip signal has not been asserted immediately after and the other is when a subsequent trip signal is

Non-linear nature of relay operation that makes it very difficult to select a group of

In the grueling manual analysis of relay event report [1,6], the selected attributes hardly provide adequate knowledge in accurately mapping the interclass boundary in the relay decision system due to inconsistency. This characterizes the interclass boundary to be usually "rough". Based on the selected attributes, some relay events close to the boundary are unclassifiable – trip or nontrip. The small overlaps between different relay events make the protective relay operation analysis to be actually a rough classification problem. Thus,

**3. Rough set data mining in dealing with inconsistent numerical distance relay decision system to extract decision algorithm – The fundamental** 

Using rough set theory approach, relay decision rule extraction is naturally a byproduct of the data reduction process involved and easily understood. Rule extraction technique is inherent to the machine learning process of rough set theory. Thus, the inherent capability of rough set theory to discover fundamental patterns in relay data has essentially mooted this study. Using an approximation concept, rough set theory is able to remove data redundancies and consequently generate decision rules. In contrast to crisp sets, a rough set has boundary line cases – events that cannot be certainly classified either as members of the set or of its complement. Rough set theory is an alternative intelligent data analysis tool that

An *information system* (*IS*) also alternatively known as *knowledge representation system* (*KRS*) is a tabulated data set, the rows of which are labeled by *objects* (events) of interest, columns labeled by *attributes*, and the entries are *attribute values* [8]. This data layout fits very well the protective relay event report that is characterized by its attributes of relay multifunctional

In the protective relay event report, the *IS* manifestation is more appropriately referred as relay *decision table* or *decision system* (*DS*) as Huang et. al. [9] put it that decision table is characterized by disjoint sets of *condition attributes* (*C Q*) and *decision (action) attributes* (*D Q*). In this regard *Q* = *C D* and *C* ∩ *D* = Ø. This *DS* is a 4-tuple structure formulated as

 *U*, i.e. the *universe* denoted as *U* = {*t1*, *t2*, *t3*, …, *tm*}, is a finite set of relay events (*ti'*s). *Q = C D* is a non-empty finite union set of condition and decision *attributes*,


asserted, after a preset time delay as set by the protection engineer.

effective attributes to fully represent relay tripping behavior.

rough set theory has been appropriately chosen to resolve this conflict [7].

can be employed to handle vagueness and inconsistencies [8].

*DS* = *U*, *Q*, *V*, *f*, the elements of which are as follows [8,10,11]:

protective elements and analog measurands,

elements versus sequence of time-stamped events [7].

$$A \otimes B = \{ X \cap Y \Big| \forall X \in A, \forall Y \in B, X \cap Y \neq \mathcal{O} \}. \tag{3}$$

*UIND*(*P*) is also interchangeably referred as *P-basic knowledge* or *P-elementary sets* in *DS*. *P*elementary set including relay event *t* is denoted as [*t*]*IND*(*<sup>P</sup>*). The first step in classification with rough sets is the construction of elementary sets [11]. A description of *P*-elementary set *X UIND*(*P*) in terms of values of attributes from *P* is denoted as *DesP*(*X*), i.e.

$$\text{Des}\_p(X) = \{ (q, v) \colon f(t, q) \ = v, \forall t \in X, \forall q \in P \} \tag{4}$$

#### **3.2. Relay decision system set approximation**

In the context of protective relay operations, consider *T U* as an arbitrary target set of relay events described (classified) by a particular trip assertion status that is needed to be represented by equivalence classes originating from attribute subset *P Q*. *P* could be a selected condition attribute set *P C* or all condition attributes *C* reflecting relay multifunctional protective elements while *T* could be the set of relay events indiscernible with respect to the decision attribute *D* = *Trip* having a domain value 'b' for pole-B tripping, for example [7].

The idea of the rough set revolves around the concept of approximation [11]. Thus, by introducing a pair of sets, called the *lower* and *upper approximations* of the target set *T* using only the information contained within *P*, the target set *T* can be *approximated*.

Formally, with a given relay decision system *DS,* each target subset *T U* having equivalence relation *IND*(*P*) is related to two subsets of *T* as follows.

*P-lower approximation* of *T* expressed as,

$$\underline{P}T = \cup \{ X \in \mathcal{U} | \text{IND}(P) ; X \subseteq T \}, \tag{5}$$

Inconsistent Decision System: Rough Set Data Mining Strategy to Extract Decision Algorithm of a Numerical Distance Relay – Tutorial 121

*<sup>P</sup>*(*T*) 1. (Note: *card* 

*<sup>P</sup>*(*T*) =

**Figure 1.** Definition of approximation in rough set theory in the context of protective relay

Clearly, equal upper and lower approximations, i.e. empty boundary region and that

the number of events that can be *possibly* be placed in *P*. Noticeably 0

(cardinality) of a set is the number of events contained in the set [11]).

*Roughly definable T* in *U* given *PT* Ø and *PT U* (Ø denotes empty set)

i.e. the ratio of *P-correctly* approximated events to all events in the system.

 *Externally undefinable T* in *U* given *PT* Ø and *PT* = U *Internally undefinable T* in *U* given *PT* = Ø and *PT* U *Totally undefinable T* in *U* given *PT* = Ø and *PT* = U

The *quality of approximation* of a target set *T* is expressed as

approximation is empty.

definability [14,15]:

It provides a measure of how accurate the rough set is in approximating the target set of relay events *T* by comparing the number of events which can be *positively* placed in *P* with

1, would mean the target set *T* is said to be *definable* in *U* since it is perfectly approximated. Regardless of the size of the upper approximation, zero accuracy would mean the lower

In general, the set *T* can be defined in *U* according to one of the following four concepts of

( ) ( ) , ( ) *<sup>P</sup> PT card PT <sup>T</sup> U card U*

(9)

is defined as the union of all elementary sets in [*t*]*IND*(*P*) which are contained in *T*. For any relay event *ti* of the lower approximation of *T* with respect to the set of attributes *P* (i.e., *ti PT*), it *positively certain* belongs to *T.* 

*P-upper approximation* of *T* expressed as,

$$
\overline{P}T = \cup \{ X \in \mathcal{U} | \text{IND}(P) \colon X \cap T \neq \mathcal{Q} \} \tag{6}
$$

is defined as the union of elementary sets in [*t*]*IND*(*P*) which have a non-empty intersection with *T*. For any relay event *ti* of the upper approximation of *T* with respect to the set of attributes *P* (i.e., *ti PT*), it *may possibly* belong to *T.* 

*P-boundary* of set *T* expressed as,

$$\text{BN}\_P\left(T\right) \tag{7}$$

$$\text{BN}\_P\left(T\right) \tag{8}$$

is the difference between *PT* and *PT*. The set of elements *ti* which *cannot be certainly* classified as belonging to *T* using the set of attributes *P* [12].

The following three regions shall be derived from the lower- and upper-approximations as illustrated in Figure 1 [7,10,13].


#### **3.3. Approximation accuracy and quality**

*<sup>P</sup>*(*T*), the accuracy of the rough set representation of a target set of relay events *T*, is formulated as [10]

$$\alpha\_P(T) = \frac{\left| \underline{PT} \right|}{\left| \overline{P}T \right|} = \frac{card(\underline{PT})}{card(\overline{PT})} \,. \tag{8}$$

**Figure 1.** Definition of approximation in rough set theory in the context of protective relay

It provides a measure of how accurate the rough set is in approximating the target set of relay events *T* by comparing the number of events which can be *positively* placed in *P* with the number of events that can be *possibly* be placed in *P*. Noticeably 0 *<sup>P</sup>*(*T*) 1. (Note: *card*  (cardinality) of a set is the number of events contained in the set [11]).

Clearly, equal upper and lower approximations, i.e. empty boundary region and that *<sup>P</sup>*(*T*) = 1, would mean the target set *T* is said to be *definable* in *U* since it is perfectly approximated. Regardless of the size of the upper approximation, zero accuracy would mean the lower approximation is empty.

In general, the set *T* can be defined in *U* according to one of the following four concepts of definability [14,15]:


120 Advances in Data Mining Knowledge Discovery and Applications

*P-lower approximation* of *T* expressed as,

*PT*), it *positively certain* belongs to *T.* 

*P-boundary* of set *T* expressed as,

illustrated in Figure 1 [7,10,13].

formulated as [10]

*P-upper approximation* of *T* expressed as,

attributes *P* (i.e., *ti PT*), it *may possibly* belong to *T.* 

as belonging to *T* using the set of attributes *P* [12].

belonging to the complement of *T*).

Ø (empty set), which otherwise it is *rough*.

**3.3. Approximation accuracy and quality** 

classified with certainty in the approximated set *T*.

equivalence relation *IND*(*P*) is related to two subsets of *T* as follows.

Formally, with a given relay decision system *DS,* each target subset *T U* having

is defined as the union of all elementary sets in [*t*]*IND*(*P*) which are contained in *T*. For any relay event *ti* of the lower approximation of *T* with respect to the set of attributes *P* (i.e., *ti*

is defined as the union of elementary sets in [*t*]*IND*(*P*) which have a non-empty intersection with *T*. For any relay event *ti* of the upper approximation of *T* with respect to the set of

is the difference between *PT* and *PT*. The set of elements *ti* which *cannot be certainly* classified

The following three regions shall be derived from the lower- and upper-approximations as

*POSP*(*T*) = *PT*, described as *P-*positive region of *T*, is the set of relay events which can be

 *NEGP*(*T*) = *U* - *PT*, described as *P*-negative region of *T*, is the set of relay events which cannot be classified without ambiguity in the approximated set *T* (or classified as

 *BNP*(*T*) = *PT* - *PT*, described as *P*-boundary region of *T*, is the set of relay events in which none can be classified with certainty into *T* nor its complement *T* as far as the attributes *P* are concerned. The set *T* is *crisp* if there are no boundary sets, i.e. *BNP*(*T*) =

*<sup>P</sup>*(*T*), the accuracy of the rough set representation of a target set of relay events *T*, is

( ) ( ) . ( ) *<sup>P</sup> PT card PT <sup>T</sup> PT card PT*

*PT X U IND P X T* { : }, (5)

*PT X U IND P X T Ø* { : }, (6)

*BN T PT PT <sup>P</sup>* (7)

(8)

The *quality of approximation* of a target set *T* is expressed as

$$\mathcal{V}\_P(T) = \frac{\left| \underline{PT} \right|}{\underline{U}} = \frac{card(\underline{PT})}{card(\underline{U})},\tag{9}$$

i.e. the ratio of *P-correctly* approximated events to all events in the system.

## **3.4. The concept of reduct and core in reduction of protective relay attributes**

Dependencies between attributes are primarily important in the protective relay data analysis using rough sets approach. The set of attributes *R Q depends* on the set of attributes *P Q* in *IS* if and only if *IND*(*P*)  *IND*(*R*). This dependency is denoted as *P R*.

Inconsistent Decision System: Rough Set Data Mining Strategy to Extract Decision Algorithm of a Numerical Distance Relay – Tutorial 123

*UIND*(*D*) be *decision classes* in relay *DS* (a family of all *D*-elementary sets), denoted by

Then, *DesC*(*Xi*) *DesD*(*Yj*) is called relay *CD*-*decision rule*. For simplicity, *C D*. (As aforementioned, *DesP*(*X*) = {(*q*,*v*) : *f*(*x*,*q*) = *v*, *x X*, *q P*} which denotes a description of *P*-

The relay *CD*-decision rules are logical statements read as 'if *C…*then*…D*'. These rule correlate descriptions of condition attributes *C Q* (for internal multifunctional protective elements, voltages, currents and impedance measurements) to classes of decision attribute *D* 

*Decision algorithm* in *DS* is used to mean the set of decision rules for *all* decision classes, i.e. *CD-algorithm* [10,18]. In the context of protective relay operation characteristics, a decision algorithm is a collection of relay *CD*-decision rules, thus referred to as *relay CD-decision* 

Rules having the same conditions but different decisions are *inconsistent* (*nondeterministic, conflicting*); otherwise they are *consistent* (*certain, deterministic, nonconflicting*) [17]. When some conditions are satisfied, deterministic *DS uniquely* describes the decisions (actions) to be made. In a non-deterministic *DS*, decisions are not uniquely determined by the

The *degree of consistency* (or *degree of dependency*) between the set of attributes *C* and *D* of a

<sup>k</sup> | k , *<sup>C</sup> CD C*

(i.e. conceptually similar to the quality of approximation or classification) [10]. In other words, *D depends on C* in a degree of dependency *k* (0 *≤ k ≤* 1). All the values of attributes from *D depend totally* on (i.e. uniquely determined by) the values of attributes from *C* if *k* = 1,

set *C*. *C' is* a relative reduct called *D-reduct of C* if *C C* is a minimal subset of *C* and (*C*, *D*)

 

*D*

, *D*) is valid (i.e. similar in dependency). *REDD(C)* is used to mean the family of all *D*reducts of *C* [18]. Putting it simply, the minimal subsets of condition attributes that discern all decision equivalence classes of the relation *UIND*(*D*) discernable by the entire set of

relay *CD*-decision algorithm is denoted as *C <sup>k</sup> D* and can be formally defined as:

i.e. *C* <sup>1</sup> *D* or simply *C D*. *D depends partially* in a degree k on *C* if *k <* 1 [17].

attributes are called *D*-reducts [11]. The following notations are, thus, valid:

{ : , 1, , k} *ij C i Dj i j r Des X Des Y X Y Ø i* (10)

*PO*

*S D U*

(11)

called *relative reduct* and not on the entire

elementary set *X UIND*(*P*) in terms of values of attributes from *P*).

The set of decision rules for *each* decision class *Yj* (*j* = 1,…, *n*) is denoted by:

*Yj* (*j* = 1, …, *n*).

*Q* (i.e. type of trip assertions).

*algorithm* in this study.

= (*C*

conditions [9]. Formally, it is defined that:

Relay rule {*rij*} is *deterministic* in *DS* if and only if *Xi Yj*, and

Relay rule {*rij*} is *nondeterministic* in *DS*, otherwise.

It may happen that the set *D* depends on subset *C*

This so-called attribute reduction is so performed that the reduced set of attributes provides the same approximation quality as the original set of attributes. If a particular set of attributes is dependent, it is interesting to find *reducts* (all possible minimal subsets of attributes) that lead to the same number of elementary sets as in the case of the whole set of attributes and also to find *core* (the set of all indispensable attributes) [11]. By adopting the fundamental concepts of core and reduct, rough set theory minimizes the subsets of attributes in the relay database but still fully characterizes the inherent knowledge of relay operation behavior.

Reduct is essentially a sufficient set of features of a *DS*, which discerns (differentiates) all events discernible by the original *DS*. Reduct is a subset of attributes *RED P* (where *P Q*) such that:


Core is defined as the set of attributes found to be in common in all reducts. Core is a subset of attributes *CORE RED* (where *RED P* and *P Q*) such that:

 It consists of attributes which *cannot be removed* from the *DS* without causing collapse of the equivalence class structure. Formally, [*t*]*IND(RED-CORE)* ≠ [*t*]*IND(P)* where the above *ARED* in this case is *ACORE*.

A *discernibility matrix* with a symmetrical dimension *n n* is constructed to compute reducts and core. *n* denotes the number of elementary sets and each of the matrix's elements *dij* is defined as the set of all attributes which discern elementary sets [*t*]*IND(Pi)* and [*t*]*IND(Pj)* [17].

## **3.5. Decision rules interpreted from protective relay event report**

Relay *DS* analysis is considered as a supervised learning problem (classification) [13]. A *DS* determines a logical implication called *decision rule* when the conditions specified by condition attributes in each row of *DS* correlate what decisions (trip assertions) are to take effect [18]. Thus, in this study the logical implication is designated as *relay decision rule*. A complete set of relay decision rules can be derived from the relay decision table *DS*. Events in *DS*, i.e. {*t1*, *t2*, *t3*, …, *tm*} = *U*, identify as labels of relay decision rules.

#### Formally, let

 *UIND*(*C*) be *condition classes* in relay *DS* (a family of all *C*-elementary sets), denoted by *Xi* (*i* = 1, …, *k*),

 *UIND*(*D*) be *decision classes* in relay *DS* (a family of all *D*-elementary sets), denoted by *Yj* (*j* = 1, …, *n*).

Then, *DesC*(*Xi*) *DesD*(*Yj*) is called relay *CD*-*decision rule*. For simplicity, *C D*. (As aforementioned, *DesP*(*X*) = {(*q*,*v*) : *f*(*x*,*q*) = *v*, *x X*, *q P*} which denotes a description of *P*elementary set *X UIND*(*P*) in terms of values of attributes from *P*).

The relay *CD*-decision rules are logical statements read as 'if *C…*then*…D*'. These rule correlate descriptions of condition attributes *C Q* (for internal multifunctional protective elements, voltages, currents and impedance measurements) to classes of decision attribute *D Q* (i.e. type of trip assertions).

The set of decision rules for *each* decision class *Yj* (*j* = 1,…, *n*) is denoted by:

$$\left\{ r\_{\vec{\eta}} \right\} \, = \, \left\{ \text{Des}\_{\mathbb{C}} \left( X\_i \right) \Rightarrow \text{Des}\_D \left( Y\_j \right) ; X\_i \cap Y\_j \neq \mathcal{O} \,\middle| \, i = 1, \dots, k \right\} \tag{10}$$

*Decision algorithm* in *DS* is used to mean the set of decision rules for *all* decision classes, i.e. *CD-algorithm* [10,18]. In the context of protective relay operation characteristics, a decision algorithm is a collection of relay *CD*-decision rules, thus referred to as *relay CD-decision algorithm* in this study.

Rules having the same conditions but different decisions are *inconsistent* (*nondeterministic, conflicting*); otherwise they are *consistent* (*certain, deterministic, nonconflicting*) [17]. When some conditions are satisfied, deterministic *DS uniquely* describes the decisions (actions) to be made. In a non-deterministic *DS*, decisions are not uniquely determined by the conditions [9]. Formally, it is defined that:


122 Advances in Data Mining Knowledge Discovery and Applications

such that:

Formally, let

*Xi* (*i* = 1, …, *k*),

**3.4. The concept of reduct and core in reduction of protective relay attributes** 

Dependencies between attributes are primarily important in the protective relay data analysis using rough sets approach. The set of attributes *R Q depends* on the set of attributes *P Q* in *IS* if and only if *IND*(*P*)  *IND*(*R*). This dependency is denoted as *P R*.

This so-called attribute reduction is so performed that the reduced set of attributes provides the same approximation quality as the original set of attributes. If a particular set of attributes is dependent, it is interesting to find *reducts* (all possible minimal subsets of attributes) that lead to the same number of elementary sets as in the case of the whole set of attributes and also to find *core* (the set of all indispensable attributes) [11]. By adopting the fundamental concepts of core and reduct, rough set theory minimizes the subsets of attributes in the relay database

Reduct is essentially a sufficient set of features of a *DS*, which discerns (differentiates) all events discernible by the original *DS*. Reduct is a subset of attributes *RED P* (where *P Q*)

The reduced attribute set *RED* induces the same equivalence classes as those induced

 Attribute set *RED* is *minimal* in the sense that [*t*]*IND(RED-A)* [*t*]*IND(P)* for any attribute *ARED*. This suggests that no attribute can be dispensed from set *RED* without

Core is defined as the set of attributes found to be in common in all reducts. Core is a subset

 It consists of attributes which *cannot be removed* from the *DS* without causing collapse of the equivalence class structure. Formally, [*t*]*IND(RED-CORE)* ≠ [*t*]*IND(P)* where the above

A *discernibility matrix* with a symmetrical dimension *n n* is constructed to compute reducts and core. *n* denotes the number of elementary sets and each of the matrix's elements *dij* is defined as the set of all attributes which discern elementary sets [*t*]*IND(Pi)* and [*t*]*IND(Pj)* [17].

Relay *DS* analysis is considered as a supervised learning problem (classification) [13]. A *DS* determines a logical implication called *decision rule* when the conditions specified by condition attributes in each row of *DS* correlate what decisions (trip assertions) are to take effect [18]. Thus, in this study the logical implication is designated as *relay decision rule*. A complete set of relay decision rules can be derived from the relay decision table *DS*. Events

*UIND*(*C*) be *condition classes* in relay *DS* (a family of all *C*-elementary sets), denoted by

but still fully characterizes the inherent knowledge of relay operation behavior.

by full attribute set *P*. This is denoted as [*t*]*IND*(*RED*) = [*t*]*IND*(*<sup>P</sup>*).

of attributes *CORE RED* (where *RED P* and *P Q*) such that:

**3.5. Decision rules interpreted from protective relay event report** 

in *DS*, i.e. {*t1*, *t2*, *t3*, …, *tm*} = *U*, identify as labels of relay decision rules.

modifying the equivalence classes [*t*]*IND(P)* [16].

*ARED* in this case is *ACORE*.

The *degree of consistency* (or *degree of dependency*) between the set of attributes *C* and *D* of a relay *CD*-decision algorithm is denoted as *C <sup>k</sup> D* and can be formally defined as:

$$\mathbf{C} \Rightarrow\_{\mathbf{k}} D \mid \mathbf{k} \ = \mathcal{Y} \begin{pmatrix} \mathbf{C} \ \mathbf{D} \end{pmatrix} = \frac{\begin{vmatrix} POS\_{\bar{C}}D \end{vmatrix}}{\|\mathbf{U}\|} \tag{11}$$

(i.e. conceptually similar to the quality of approximation or classification) [10]. In other words, *D depends on C* in a degree of dependency *k* (0 *≤ k ≤* 1). All the values of attributes from *D depend totally* on (i.e. uniquely determined by) the values of attributes from *C* if *k* = 1, i.e. *C* <sup>1</sup> *D* or simply *C D*. *D depends partially* in a degree k on *C* if *k <* 1 [17].

It may happen that the set *D* depends on subset *C* called *relative reduct* and not on the entire set *C*. *C' is* a relative reduct called *D-reduct of C* if *C C* is a minimal subset of *C* and (*C*, *D*) = (*C*, *D*) is valid (i.e. similar in dependency). *REDD(C)* is used to mean the family of all *D*reducts of *C* [18]. Putting it simply, the minimal subsets of condition attributes that discern all decision equivalence classes of the relation *UIND*(*D*) discernable by the entire set of attributes are called *D*-reducts [11]. The following notations are, thus, valid:

	- If *POSC*(*D*) = *POS*(*C*-{*ci* })(*D*), an attribute *ci C* is *D-dispensable* in *C*. *ci* is *D*-superfluous if it exerts no influence on the lower approximation of *D.* Otherwise the attribute *ci* is *Dindispensable* in *C.*
	- If *C* is *D-independent*, then all attributes *ci C* are *D*-indispensable in *C* and called the *Dcore of C* which is denoted as *CORED(C)*.
	- The following property is also true for *DS* system as previously defined,

$$\text{CORE}\_D\text{(C)} = \ \frown \text{RED}\_D\text{(C)}\tag{12}$$

Inconsistent Decision System: Rough Set Data Mining Strategy to Extract Decision Algorithm of a Numerical Distance Relay – Tutorial 125

*time(U) ag bg cg Z1pu Z2pu Z3pu Z4pu Z1trp Z2trp Z3trp Z4trp Trip*  sec code zone zone zone logic logic logic logic logic logic logic logic pole 0.5066 *t7* 0 2 0 0 1 1 0 0 0 0 0 0 0.5498 *t8* 0 1 0 1 1 1 0 1 0 0 0 b 0.5510 *t9* 0 1 0 1 1 1 0 1 0 0 0 b 0.5522 *t10* 0 1 0 1 1 1 0 1 0 0 0 b 0.5534 *t11* 0 0 0 1 1 1 0 0 0 0 0 b 0.5546 *t12* 0 0 0 1 1 1 0 0 0 0 0 b 0.5558 *t13* 0 0 0 1 1 1 0 0 0 0 0 b 0.5966 *t14* 0 0 0 1 1 1 0 0 0 0 0 b 0.5978 *t15* 0 0 0 1 1 1 0 0 0 0 0 b 0.5990 *t16* 0 0 0 0 0 1 0 0 0 0 0 b 0.6002 *t17* 0 0 0 0 0 0 0 0 0 0 0 b 0.6014 *t18* 0 0 0 0 0 0 0 0 0 0 0 b 0.6026 *t19* 0 0 0 0 0 0 0 0 0 0 0 b 0.7347 *t20* 0 0 0 0 0 0 0 0 0 0 0 0 0.7359 *t21* 0 0 0 0 0 0 0 0 0 0 0 0 **Table 1.** Excerpt of an event report as a decision table *DS* of a protective distance relay (only ground

distance is considered for illustration)

*Vag*, *Vbg*, *Vcg*, = {1, 2, 3, 4}.

deduced as shown in Table 2.

The attribute names are described as follows:

*ag*, *bg,* and *cg* are A-G, B-G, and C-G fault detections.

VZ1pu, VZ2pu, VZ3pu, VZ4pu, VZ1trp, VZ2trp, VZ3trp, VZ4trp = {0, 1}.

**4.2. Protective relay decision table analysis** 

The sets of values (domains) of the particular attributes are as follows:

*VTrip* = {a, b, c, 0}, corresponding to tripping signals of phase A, B, C or none.

*U*

**Table 2.** Equivalence classes with respect to decision attribute *D* = {*Trip*}

 *Z1pu*, *Z2pu*, *Z3pu*, and *Z4pu* are zone 1, 2, 3, and 4 ground distance starts (pick-ups). *Z1trp*, *Z2trp*, *Z3trp*, and *Z4trp* are zone 1, 2, 3, and 4 ground distance trip signals.

From Table 1, the two elementary sets with respect to the decision attribute *D* = {*Trip*} can be

Six equivalence classes (elementary sets) can be deduced as shown in Table 3 when the full set of attributes *C* = {*ag*, *bg*, *cg*, *Z1pu*, *Z2pu*, *Z3pu*, *Z4pu*, *Z1trp*, *Z2trp*, *Z3trp*, *Z4trp*} is considered.

{*t1*, *t2*, *t3*, *t4*, *t5*, *t6*, *t7*, *t20*, *t21*} *= D1* 0 {*t8*, *t9*, *t10*, *t11*, *t12*, *t13*, *t14*, *t15*, *t16*, *t17*, *t18*, *t19*} *= D2* b

*D Trip* 

The previous definitions are valid if *D* = *C* [18].

 Using a slightly modified discernibility matrix called *D-discernibility matrix of C*, relative reducts can be computed. The set of all condition attributes which discern events *ti* and *tj* that do not belong to the same equivalence class of the relation *UIND*(*D*) defines the element of *D*-discernibility matrix of *C*. The set of all single elements of the *D*discernibility matrix of *C* is the *D-*core of C [10,11]. Rather than the ordinary reduct of C, *D-*reduct of *C* is very much the essence of this paper's study that aspires to derive the relay *CD*-decision rules (i.e. *C D*).

## **4. Discovering decision algorithm of numerical distance protective relay**

In order to fairly understand the indiscernibility relation and rules discovery from distance protective relay decision system *DS*, the following tutorial is presented.

## **4.1. Protective relay decision table**

Table 1 illustrates an example of a decision system *DS* = *U*, *Q*, *V*, *f* excerpted from an event report of a protective distance relay. The decision table is a presentation of information function *f*: *UQ V*. *C* = {*ag*, *bg*, *cg*, *Z1pu*, *Z2pu, Z3pu, Z4pu*, *Z1trp*, *Z2trp*, *Z3trp*, *Z4trp*} is the set of condition attributes representing the internal multifunctional protective elements. *D = Trip* is the decision attribute which, essentially, denotes the tripping signal asserted by the relay in response to a particular fault in the power system. The time codes are the events that are analyzed for equivalence relation on the basis of selected subset of attributes *P*, such that *P Q*. The finite set of the attribute *time*'s code forms the universe of interest *U* = {*t1*, *t2*, *t3*, *t4*, *t5*, *t6*, *t7*, *t8*, *t9*, *t10*, *t11*, *t12*, *t13*, *t14*, *t15*, *t16*, *t17*, *t18*, *t19*, *t20*, *t21*}.



**Table 1.** Excerpt of an event report as a decision table *DS* of a protective distance relay (only ground distance is considered for illustration)

The attribute names are described as follows:


The sets of values (domains) of the particular attributes are as follows:

*Vag*, *Vbg*, *Vcg*, = {1, 2, 3, 4}.

124 Advances in Data Mining Knowledge Discovery and Applications

*core of C* which is denoted as *CORED(C)*.

relay *CD*-decision rules (i.e. *C D*).

**4.1. Protective relay decision table** 

The previous definitions are valid if *D* = *C* [18].

})(*D*), an attribute *ci C* is *D-dispensable* in *C*. *ci* is *D*-superfluous if it

*CORE C RED C D D* (12)

exerts no influence on the lower approximation of *D.* Otherwise the attribute *ci* is *D-*

If *C* is *D-independent*, then all attributes *ci C* are *D*-indispensable in *C* and called the *D-*

 Using a slightly modified discernibility matrix called *D-discernibility matrix of C*, relative reducts can be computed. The set of all condition attributes which discern events *ti* and *tj* that do not belong to the same equivalence class of the relation *UIND*(*D*) defines the element of *D*-discernibility matrix of *C*. The set of all single elements of the *D*discernibility matrix of *C* is the *D-*core of C [10,11]. Rather than the ordinary reduct of C, *D-*reduct of *C* is very much the essence of this paper's study that aspires to derive the

**4. Discovering decision algorithm of numerical distance protective relay** 

In order to fairly understand the indiscernibility relation and rules discovery from distance

Table 1 illustrates an example of a decision system *DS* = *U*, *Q*, *V*, *f* excerpted from an event report of a protective distance relay. The decision table is a presentation of information function *f*: *UQ V*. *C* = {*ag*, *bg*, *cg*, *Z1pu*, *Z2pu, Z3pu, Z4pu*, *Z1trp*, *Z2trp*, *Z3trp*, *Z4trp*} is the set of condition attributes representing the internal multifunctional protective elements. *D = Trip* is the decision attribute which, essentially, denotes the tripping signal asserted by the relay in response to a particular fault in the power system. The time codes are the events that are analyzed for equivalence relation on the basis of selected subset of attributes *P*, such that *P Q*. The finite set of the attribute *time*'s code forms the universe of interest *U* = {*t1*, *t2*,

*time(U) ag bg cg Z1pu Z2pu Z3pu Z4pu Z1trp Z2trp Z3trp Z4trp Trip*  sec code zone zone zone logic logic logic logic logic logic logic logic pole 0.4982 *t1* 0 0 0 0 0 0 0 0 0 0 0 0 0.4994 *t2* 0 0 0 0 0 0 0 0 0 0 0 0 0.5006 *t3* 0 0 0 0 0 0 0 0 0 0 0 0 0.5018 *t4* 0 1 0 0 1 1 0 0 0 0 0 0 0.5030 *t5* 0 2 0 0 1 1 0 0 0 0 0 0 0.5054 *t6* 0 2 0 0 1 1 0 0 0 0 0 0

protective relay decision system *DS*, the following tutorial is presented.

*t3*, *t4*, *t5*, *t6*, *t7*, *t8*, *t9*, *t10*, *t11*, *t12*, *t13*, *t14*, *t15*, *t16*, *t17*, *t18*, *t19*, *t20*, *t21*}.

The following property is also true for *DS* system as previously defined,

If *POSC*(*D*) = *POS*(*C*-{*ci*

*indispensable* in *C.*


#### **4.2. Protective relay decision table analysis**

From Table 1, the two elementary sets with respect to the decision attribute *D* = {*Trip*} can be deduced as shown in Table 2.


**Table 2.** Equivalence classes with respect to decision attribute *D* = {*Trip*}

Six equivalence classes (elementary sets) can be deduced as shown in Table 3 when the full set of attributes *C* = {*ag*, *bg*, *cg*, *Z1pu*, *Z2pu*, *Z3pu*, *Z4pu*, *Z1trp*, *Z2trp*, *Z3trp*, *Z4trp*} is considered.

126 Advances in Data Mining Knowledge Discovery and Applications


**Table 3.** Equivalence classes with respect to condition attributes *C* = {*ag*, *bg*, *cg*, *Z1pu*, *Z2pu*, *Z3pu*, *Z4pu*, *Z1trp*, *Z2trp*, *Z3trp*, *Z4trp*}

Within the first equivalence class, {*t1*, *t2*, *t3*, *t17*, *t18*, *t19*, *t20*, *t21*}, the eight events are indiscernible among each other based on the available attributes. In the third and the fourth equivalence classes, {*t5*, *t6*, *t7*} and {*t8*, *t9*, *t10*}, the three events within them, based on the available attributes, cannot be distinguished from one another. Similarly, the five events within the fifth equivalence class are also indiscernible from one another. The remaining two events are each discernible (different) from all other events. [*t*]*IND*(*<sup>C</sup>*) or simply [*t*]*C* can denote these equivalence classes of the *C*-indiscernibility relation as aforementioned. Each row in Table 3 describes an individual elementary set, whereas the entire Table 3 describes the *DS* being studied. *UC* means that elementary sets of the universe *U* in the space *C* are being considered.

The calculations of the *C-*lower and *C-*upper approximations and accuracy of classification of *D*,

$$\underline{\mathbf{CD}}\_1 = \begin{Bmatrix} t\_4 \end{Bmatrix} \cup \begin{Bmatrix} t\_5, t\_{6'}, t\_7 \end{Bmatrix} \quad = \begin{Bmatrix} t\_{4'} t\_{5'} t\_{6'} t\_7 \end{Bmatrix} \tag{13}$$

Inconsistent Decision System: Rough Set Data Mining Strategy to Extract Decision Algorithm of a Numerical Distance Relay – Tutorial 127

(19)

*D* = {*Trip*} may remain in a certain domain value for a certain time-sequence of relay events after a particular relay trip trigger according to the protection engineer's preset time duration of signal assertion [7]. This may prevail even though the condition attributes have changed

> ( ) 4 9 0.45 12 7 ( ) 1 *i*

<sup>1</sup> ( ) 4 9 0.62 ( ) 9 12 *i*

*<sup>D</sup>* (20)

during this duration. This explains the inconsistency found in the *CD-*algorithm.

 

2

*i i*

*<sup>i</sup> card CD card U*

*d CD*

*D*-reducts and *D*-core of *C* can be discovered from the *D*-discernibility matrix of *C* by discerning relay events from different equivalence classes in the relation *UIND*(*D*) with respect to the condition attributes *C*. The *D*-discernibility matrix that is formed is illustrated in Table 4. It would suffice to consider only the lower diagonal part because of the matrix's symmericalness [11]. Note that even though relay events appearing in the same class in the *D*-space (for example *t1*, *t2*, *t3*, *t4*, *t5*, *t6*, *t7*, *t20*, *t21* ) are discernible in *C*-space, they are not discerned between each other with respect to the attributes *C*. Empty set (Ø) indicates

*U t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 t11 t12 t13 t14 t15 t16 t17 t18 t19 t20 t21*

*<sup>i</sup> card CD car*

The accuracy and quality of overall classification *D* are:

*D*

*C*

*C*

i.e. the overall classification with respect to *C* is rough.

indiscernibility between relay events.

*t1* Ø *t2* Ø Ø *t3* Ø Ø Ø *t4* Ø Ø Ø Ø *t5* Ø Ø Ø Ø Ø *t6* Ø Ø Ø Ø Ø Ø *t7* Ø Ø Ø Ø Ø Ø Ø

*t8*

*t9*

*t10*

*t11*

{*bg, Z1pu, Z2pu, Z3pu, Z1trp*}

{*bg, Z1pu, Z2pu, Z3pu, Z1trp*}

{*bg, Z1pu, Z2pu, Z3pu, Z1trp*}

{*Z1pu, Z2pu, Z3pu*}

{*bg, Z1pu, Z2pu, Z3pu, Z1trp*}

{*bg, Z1pu, Z2pu, Z3pu, Z1trp*}

{*bg, Z1pu, Z2pu, Z3pu, Z1trp*}

{*Z1pu, Z2pu, Z3pu*}

{*bg, Z1pu, Z2pu, Z3pu, Z1trp*}

{*bg, Z1pu, Z2pu, Z3pu, Z1trp*}

{*bg, Z1pu, Z2pu, Z3pu, Z1trp*}

{*Z1pu, Z2pu, Z3pu*}

{*Z1pu, Z1trp*}

{*Z1pu, Z1trp*}

{*Z1pu, Z1trp*}

{*bg, Z1pu*}

{*bg, Z1pu, Z1trp*}

{*bg, Z1pu, Z1trp*}

{*bg, Z1pu, Z1trp*}

{*bg, Z1pu*}

{*bg, Z1pu, Z1trp*}

{*bg, Z1pu, Z1trp*}

{*bg, Z1pu, Z1trp*}

{*bg, Z1pu*}

{*bg, Z1pu, Z1trp*}

{*bg, Z1pu, Z1trp*}

{*bg, Z1pu, Z1trp*}

{*bg,* 

Ø

Ø Ø

Ø Ø Ø

*Z1pu*} Ø Ø Ø Ø

$$\begin{aligned} \underline{\mathbf{C}} \mathbf{D}\_2 &= \{ \mathbf{t}\_{8\prime}, \mathbf{t}\_9, \mathbf{t}\_{10} \} \cup \{ \mathbf{t}\_{11\prime}, \mathbf{t}\_{12\prime}, \mathbf{t}\_{13\prime}, \mathbf{t}\_{14\prime}, \mathbf{t}\_{15} \} \cup \{ \mathbf{t}\_{16} \} \\ &= \{ \mathbf{t}\_{8\prime}, \mathbf{t}\_{9\prime}, \mathbf{t}\_{10\prime}, \mathbf{t}\_{11\prime}, \mathbf{t}\_{12\prime}, \mathbf{t}\_{13\prime}, \mathbf{t}\_{14\prime}, \mathbf{t}\_{15\prime}, \mathbf{t}\_{16} \} \end{aligned} \tag{14}$$

$$\begin{aligned} \text{CCD}\_1 &= \{ \mathbf{t}\_{1'}, \mathbf{t}\_{2'}, \mathbf{t}\_{3'}, \mathbf{t}\_{17'}, \mathbf{t}\_{18'}, \mathbf{t}\_{19'}, \mathbf{t}\_{20'}, \mathbf{t}\_{21} \} \cup \{ \mathbf{t}\_4 \} \cup \{ \mathbf{t}\_{5'}, \mathbf{t}\_{6'}, \mathbf{t}\_7 \} \\ &= \{ \mathbf{t}\_{1'}, \mathbf{t}\_{2'}, \mathbf{t}\_3, \mathbf{t}\_{4'}, \mathbf{t}\_5, \mathbf{t}\_{6'}, \mathbf{t}\_{7'}, \mathbf{t}\_{18'}, \mathbf{t}\_{19'}, \mathbf{t}\_{20'}, \mathbf{t}\_{21} \} \end{aligned} \tag{15}$$

$$\begin{aligned} \text{CD}\_2 &= \{ \mathbf{t}\_{1\prime}, \mathbf{t}\_{2\prime}, \mathbf{t}\_{3\prime}, \mathbf{t}\_{1\prime\prime}, \mathbf{t}\_{19\prime}, \mathbf{t}\_{19\prime}, \mathbf{t}\_{20\prime} \} \cup \{ \mathbf{t}\_{8\prime}, \mathbf{t}\_{9\prime}, \mathbf{t}\_{10} \} \cup \{ \mathbf{t}\_{11\prime}, \mathbf{t}\_{12\prime}, \mathbf{t}\_{13\prime}, \mathbf{t}\_{14\prime}, \mathbf{t}\_{15\prime} \} \cup \{ \mathbf{t}\_{16} \} \\ &= \{ \mathbf{t}\_{1\prime}, \mathbf{t}\_{2\prime}, \mathbf{t}\_{3\prime}, \mathbf{t}\_{8\prime}, \mathbf{t}\_{9\prime}, \mathbf{t}\_{10\prime}, \mathbf{t}\_{11\prime}, \mathbf{t}\_{12\prime}, \mathbf{t}\_{13\prime}, \mathbf{t}\_{14\prime}, \mathbf{t}\_{15\prime}, \mathbf{t}\_{16\prime}, \mathbf{t}\_{17\prime}, \mathbf{t}\_{18\prime}, \mathbf{t}\_{20\prime}, \mathbf{t}\_{21\prime} \} \end{aligned} \tag{16}$$

$$\log\_{\mathbb{C}}(D\_1) = \left| \frac{\underline{\mathbb{C}D}\_1}{\overline{\mathbb{C}D}\_1} \right| = \frac{card(\underline{\mathbb{C}D}\_1)}{card(\overline{\mathbb{C}D}\_1)} = \frac{4}{12} = 0.33\tag{17}$$

$$\log\_{\mathbb{C}}\left(D\_{2}\right) = \left|\frac{\underline{\mathbb{C}D}\_{2}}{\overline{\mathbb{C}D}\_{2}}\right| = \frac{\operatorname{card}(\underline{\mathbb{C}D}\_{2})}{\operatorname{card}(\overline{\mathbb{C}D}\_{2})} = \frac{9}{17} = 0.53\tag{18}$$

With classification accuracies of 0.33 and 0.53, the respective elementary sets *D1* and *D2* are roughly definable (vaguely classified) in the *DS*. This is rather expected. The decision attribute *D* = {*Trip*} may remain in a certain domain value for a certain time-sequence of relay events after a particular relay trip trigger according to the protection engineer's preset time duration of signal assertion [7]. This may prevail even though the condition attributes have changed during this duration. This explains the inconsistency found in the *CD-*algorithm.

The accuracy and quality of overall classification *D* are:

126 Advances in Data Mining Knowledge Discovery and Applications

*C ag bg cg Z1pu Z2pu Z3pu Z4pu Z1trp Z2trp Z3trp Z4trp* 

{*t1*, *t2*, *t3*, *t17*, *t18*, *t19*, *t20*, *t21*} 0 0 0 0 0 0 0 0 0 0 0 {*t4*} 0 1 0 0 1 1 0 0 0 0 0 {*t5*, *t6*, *t7*} 0 2 0 0 1 1 0 0 0 0 0 {*t8*, *t9*, *t10*} 0 1 0 1 1 1 0 1 0 0 0 {*t11*, *t12*, *t13*, *t14*, *t15*} 0 0 0 1 1 1 0 0 0 0 0 {*t16*} 0 0 0 0 0 1 0 0 0 0 0 **Table 3.** Equivalence classes with respect to condition attributes *C* = {*ag*, *bg*, *cg*, *Z1pu*, *Z2pu*, *Z3pu*, *Z4pu*,

Within the first equivalence class, {*t1*, *t2*, *t3*, *t17*, *t18*, *t19*, *t20*, *t21*}, the eight events are indiscernible among each other based on the available attributes. In the third and the fourth equivalence classes, {*t5*, *t6*, *t7*} and {*t8*, *t9*, *t10*}, the three events within them, based on the available attributes, cannot be distinguished from one another. Similarly, the five events within the fifth equivalence class are also indiscernible from one another. The remaining two events are each discernible (different) from all other events. [*t*]*IND*(*<sup>C</sup>*) or simply [*t*]*C* can denote these equivalence classes of the *C*-indiscernibility relation as aforementioned. Each row in Table 3 describes an individual elementary set, whereas the entire Table 3 describes the *DS* being studied. *UC*

The calculations of the *C-*lower and *C-*upper approximations and accuracy of classification

8 9 10 11 12 13 14 15 16

1 2 3 17 18 19 20 21 4 5 6 7 1 2 3 4 5 6 7 17 18 19 20 2

1 2 3 17 18 19 20 21 8 9 10 11 12 13 14 15 16

(16)

1 1

2

*d CD*

( ) <sup>4</sup> 0.33

( ) <sup>9</sup> 0.53

(17)

(18)

1 2 3 8 9 10 11 12 13 14 15 16 17 18 19 20 21 <sup>2</sup> {t , t , t , t , t , t , t , t } { , , } t , t , t , t , t {} {t , t , t , , , , t , t , t , t , t , , t , t , t , t , t }

{t , t , t , t , t , t , t , t } { } { , , } {t , t , t , , , , ,t , t , t , t , t }

{ , , } t , t , t , t , t { } { , , , t , t , t , t , t , } *CD t t t t ttt t* 

8 9 10 11 12 13 14 15 1

*t*

*CD*

( ) 12 ( ) *CD card CD <sup>D</sup> CD card CD*

*C t D*<sup>1</sup> *t ttt tt t* 4 567 4567 ,, ,,, (13)

(14)

(15)

6

1

*t ttt*

*ttt t*

means that elementary sets of the universe *U* in the space *C* are being considered.

*ttt*

*ttt t*

1

1

2 2

( ) 17 *<sup>C</sup> CD card CD CD car*

With classification accuracies of 0.33 and 0.53, the respective elementary sets *D1* and *D2* are roughly definable (vaguely classified) in the *DS*. This is rather expected. The decision attribute

2

*C*

2

C

1

2

*D*

1

*D*

*U*

*Z1trp*, *Z2trp*, *Z3trp*, *Z4trp*}

of *D*,

$$\alpha\_{\mathbb{C}}\left(D\right) = \frac{\sum\_{i=1}^{2} \text{card}(\underline{\mathbb{C}D}\_{i})}{\sum\_{i=1}^{2} \text{card}(\overline{\mathbb{C}D}\_{i})} = \frac{4+9}{12+17} = 0.45\tag{19}$$

$$\chi\_{\mathbb{C}}\left(D\right) = \frac{\sum\_{i=1}^{2} card(\underline{\mathbb{C}D}\_{i})}{card(\underline{\mathbb{U}})} = \frac{4+9}{9+12} = 0.62\tag{20}$$

i.e. the overall classification with respect to *C* is rough.

*D*-reducts and *D*-core of *C* can be discovered from the *D*-discernibility matrix of *C* by discerning relay events from different equivalence classes in the relation *UIND*(*D*) with respect to the condition attributes *C*. The *D*-discernibility matrix that is formed is illustrated in Table 4. It would suffice to consider only the lower diagonal part because of the matrix's symmericalness [11]. Note that even though relay events appearing in the same class in the *D*-space (for example *t1*, *t2*, *t3*, *t4*, *t5*, *t6*, *t7*, *t20*, *t21* ) are discernible in *C*-space, they are not discerned between each other with respect to the attributes *C*. Empty set (Ø) indicates indiscernibility between relay events.



Inconsistent Decision System: Rough Set Data Mining Strategy to Extract Decision Algorithm of a Numerical Distance Relay – Tutorial 129

= *bg* ((*bgZ1puZ3pu* + *bgZ3puZ1trp*) + (*Z1puZ3pu*) + (*Z1puZ3puZ1trp*)) + *Z2pu* ((*bgZ1puZ3pu* + *bgZ3puZ1trp*) + (*Z1puZ3pu*) + (*Z1puZ3puZ1trp*)) = (*bgZ1puZ3pu*) + (*bgZ3puZ1trp*) + (*bgZ1puZ3pu*) + (*bgZ1puZ3puZ1trp*) + (*bgZ1puZ2pu Z3pu*) + (*bgZ2puZ3puZ1trp*) + (*Z1puZ2puZ3pu*) +

Absorption law and eventual expression multiplication are implemented to solve the

Normalization in its final normal form, the last Boolean expression *fC*(*D*) is recognized as Disjunctive Normal Form (DNF). DNF is analogous to Sum Of Product (SOP) boolean algebra in digital electronics logic. *fC*(*D*) in DNF form is an alternative representation of the *DS* in which all its constituents are the *D-*reducts of *C* (i.e. *REDD*(*C*)) [11,17]*.* Either one of the set of reducts can be used to represent exactly the same data classification as that depicted by the entire set of attributes *C*. The following *REDD*(*C*) of the above final *fC*(*D*) reveals that either one of the *D*-reducts of *C* can be used alternatively to represent exactly the same equivalence relation *UIND*(*D*) of the *DS* as that represented by the whole set of attributes

, 1 , 3 , 1 , , 1 , 2 , 3 , , 2 , 3 , 1 , 1 , 2 , 3 , 1 , 2 , 3 , 1

, 1 , 3 , , 3 , 1 , , 1 , 3 ,

(21)

 

 Identifying all the single attribute entries in the *D*-discernibility matrix of C [11], which from Table 4, attribute *Z3pu* is the only single attribute entry and thus *CORED*(*C*) =

Hence, *Z3pu* is the most characteristic attribute that is indispensible in *DS* without reducing

*CORED*(*C*) = *Z3pu* does not seem to signify any significance in the behavior of the relay under analysis. Had the reduct analysis been worked out based only on the whole condition attributes *C* (as per the equivalence relation in Table 3, where decision attribute *D* is excluded such as in the case of *IS* instead of *DS*), the core of *C* (i.e. the core of the

This implies the protective relay has been subjected to B-G fault. In reality this fault occurred in distance zone 1 operation characteristic and was picked up by the zone 1 distance element. However, the *D*-core of *C* discovers the indispensability of the condition attribute *Z3pu* as being the core when the decision attribute *D* is considered for the *DS*

*CORE C bg U C* (22)

Taking intersection of all *D*-reducts of *C*, i.e. *CORED*(*C*) = ∩*REDD*(*C*) = *Z3pu*

the approximation quality of equivalence relation *UC* with respect to *D*.

equivalence relation *UC* with respect to *C*) would have been,

*RED C bg Z pu Z pu bg Z pu Z trp bg Z pu Z pu <sup>D</sup> bg Z pu Z pu Z trp bg Z pu Z pu Z pu bg Z pu Z pu Z trp Z pu Z pu Z pu Z pu Z pu Z pu Z trp*

(*Z1puZ2puZ3puZ1trp*)

The *D*-core of *C* can be figured out by either:

∩*REDD*(*C*) = *Z3p*, or

Boolean expression of *fC*(*D*) [19].

*C*, i.e.,

The final Disjunctive Normal Form (DNF) of *fC*(*D*)

**Table 4.** *D*-discernibility matrix of *C* 

The discovery of the desired reduct(s) is possible via the formulation of the so-called *discernibility function f(P)* that calculates according to Boolean function operation in which each attribute acts as a Boolean variable [11]. Using the technique introduced by Pawlak [17], a Boolean discernibility function is deduced right off the discernibility matrix in Table 4, i.e.:

*fC(D )* 


Absorption law and eventual expression multiplication are implemented to solve the Boolean expression of *fC*(*D*) [19].

Normalization in its final normal form, the last Boolean expression *fC*(*D*) is recognized as Disjunctive Normal Form (DNF). DNF is analogous to Sum Of Product (SOP) boolean algebra in digital electronics logic. *fC*(*D*) in DNF form is an alternative representation of the *DS* in which all its constituents are the *D-*reducts of *C* (i.e. *REDD*(*C*)) [11,17]*.* Either one of the set of reducts can be used to represent exactly the same data classification as that depicted by the entire set of attributes *C*. The following *REDD*(*C*) of the above final *fC*(*D*) reveals that either one of the *D*-reducts of *C* can be used alternatively to represent exactly the same equivalence relation *UIND*(*D*) of the *DS* as that represented by the whole set of attributes *C*, i.e.,

$$\text{RED}\_D(\text{C}) = \{\text{bg}, \text{Z1}pu, \text{Z3}pu\}, \{\text{bg}, \text{Z3}pu, \text{Z1}tr\}, \{\text{bg}, \text{Z1}pu, \text{Z3}pu\},$$

$$\{\text{bg}, \text{Z1}pu, \text{Z3}pu, \text{Z1}tr\}, \{\text{bg}, \text{Z1}pu, \text{Z2}pu, \text{Z3}pu\}, \{\text{bg}, \text{Z2}pu, \text{Z3}pu, \text{Z1}tr\}\}, \tag{21}$$

$$\{\text{Z1}pu, \text{Z2}pu, \text{Z3}pu\}, \{\text{Z1}pu, \text{Z2}pu, \text{Z3}pu, \text{Z1}tr\}$$

The *D*-core of *C* can be figured out by either:

128 Advances in Data Mining Knowledge Discovery and Applications

{*bg, Z1pu*}

{*bg, Z1pu*}

{*bg, Z1pu*}

{*bg, Z1pu*}

{*bg, Z2pu*}

{*bg, Z2pu, Z3pu*}

{*bg, Z2pu, Z3pu*}

{*bg, Z2pu, Z3pu*}

{*bg, Z1pu*}

{*bg, Z1pu*}

{*bg, Z1pu*}

{*bg, Z1pu*}

{*bg, Z2pu*}

{*bg, Z2pu, Z3pu*}

{*bg, Z2pu, Z3pu*}

{*bg, Z2pu, Z3pu*}

{*bg,* 

{*bg,* 

{*bg,* 

{*bg,* 

{*bg,* 

{*bg, Z2pu, Z3pu*}

{*bg, Z2pu, Z3pu*}

{*bg, Z2pu, Z3pu*}

> {*bg, Z1pu, Z2pu, Z3pu, Z1trp*}

> {*bg, Z1pu, Z2pu, Z3pu, Z1trp*}

× (*bg*+*Z1pu*+*Z2pu*+*Z3pu*+*Z1trp*) × (*Z1pu*+*Z2pu*+*Z3pu*) × *Z3pu*

{*bg, Z1pu, Z2pu, Z3pu, Z1trp*}

{*bg, Z1pu, Z2pu, Z3pu, Z1trp*}

{*bg, Z1pu, Z2pu, Z3pu, Z1trp*}

{*bg, Z1pu, Z2pu, Z3pu, Z1trp*}

The discovery of the desired reduct(s) is possible via the formulation of the so-called *discernibility function f(P)* that calculates according to Boolean function operation in which each attribute acts as a Boolean variable [11]. Using the technique introduced by Pawlak [17], a Boolean discernibility function is deduced right off the discernibility matrix in Table

= (*bg*+*Z1pu*+*Z2pu*+*Z3pu*+*Z1trp*) (*Z1pu*+*Z2pu*+*Z3pu*) (*Z3pu*) × (*Z1pu*+*Z1trp*) (*bg*+*Z1trp*) (*bg*+*Z2pu*) (*bg*+*Z2pu*+*Z3pu*) × (*bg*+*Z1pu*+*Z1trp*) (*bg*+*Z1pu*) (*bg*+*Z2pu*) (*bg*+*Z2pu*+*Z3pu*)

= *Z3pu* × (*Z1pu*+*Z1trp*) (*bg*+*Z1pu*) (*bg*+*Z2pu*) × (*bg*+*Z1pu*) (*bg*+*Z2pu*) × (*bg*+*Z1pu*+*Z2pu*+*Z3pu*+*Z1trp*) × (*Z1pu*+*Z2pu*+*Z3pu*) × *Z3pu*  The final Conjunctive

= (*bg*(*Z1puZ3pu* + *Z3puZ1trp*) + *Z1pu*(*Z1puZ3pu* + *Z3puZ1trp*)) (*bg*+*Z2pu*) = ((*bgZ1puZ3pu* + *bgZ3puZ1trp*) + (*Z1puZ3pu*) + (*Z1puZ3puZ1trp*)) (*bg*+*Z2pu*)

{*Z1pu, Z2pu, Z3pu*}

{*Z1pu, Z2pu, Z3pu*}

{*Z1pu, Z2pu, Z3pu*}

{*Z1pu, Z2pu, Z3pu*}

*t12*

*t13*

*t14*

*t15*

{*Z1pu, Z2pu, Z3pu*}

{*Z1pu, Z2pu, Z3pu*}

{*Z1pu, Z2pu, Z3pu*}

{*Z1pu, Z2pu, Z3pu*}

{*Z1pu, Z2pu, Z3pu*}

{*Z1pu, Z2pu, Z3pu*}

{*Z1pu, Z2pu, Z3pu*}

{*Z1pu, Z2pu, Z3pu*}

*t16* {*Z3pu*} {*Z3pu*} {*Z3pu*} {*bg,* 

*t17* Ø Ø Ø

*t18* Ø Ø Ø

*t19* Ø Ø Ø

4, i.e.:

*fC(D )* 

{*Z1pu, Z2pu, Z3pu*}

{*Z1pu, Z2pu, Z3pu*}

{*Z1pu, Z2pu, Z3pu*}

{*Z1pu, Z2pu, Z3pu*}

{*bg, Z1pu*}

{*bg, Z1pu*}

{*bg, Z1pu*}

{*bg, Z1pu*}

*Z2pu*}

{*bg, Z2pu, Z3pu*}

{*bg, Z2pu, Z3pu*}

{*bg, Z2pu, Z3pu*}

*t20* Ø Ø Ø Ø Ø Ø Ø

*t21* Ø Ø Ø Ø Ø Ø Ø

**Table 4.** *D*-discernibility matrix of *C* 

Normal Form (CNF)

= *Z3pu* (*Z1pu*+*Z1trp*) (*bg*+*Z1pu*) (*bg+Z2pu*) = (*Z1puZ3pu* + *Z3puZ1trp*) (*bg*+*Z1pu*) (*bg*+*Z2pu*)

*U t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 t11 t12 t13 t14 t15 t16 t17 t18 t19 t20 t21*

*Z1pu*} Ø Ø Ø Ø Ø Ø

*Z1pu*} Ø Ø Ø Ø Ø Ø Ø

*Z1pu*} Ø Ø Ø Ø Ø Ø Ø Ø

*Z2pu*} Ø Ø Ø Ø Ø Ø Ø Ø Ø

Ø Ø Ø Ø Ø Ø Ø Ø Ø Ø

Ø Ø Ø Ø Ø Ø Ø Ø Ø Ø Ø

Ø Ø Ø Ø Ø Ø Ø Ø Ø Ø Ø Ø

{*Z1pu, Z2pu, Z3pu*}

{*Z1pu, Z2pu, Z3pu*}

{*Z1pu, Z2pu, Z3pu*}

{*Z1pu, Z2pu, Z3pu*}

{*Z1pu, Z2pu, Z3pu*}

{*Z1pu, Z2pu, Z3pu*}

{*Z3pu*} Ø Ø Ø Ø

{*Z3pu*} Ø Ø Ø Ø Ø

*Z1pu*} Ø Ø Ø Ø Ø


Hence, *Z3pu* is the most characteristic attribute that is indispensible in *DS* without reducing the approximation quality of equivalence relation *UC* with respect to *D*.

*CORED*(*C*) = *Z3pu* does not seem to signify any significance in the behavior of the relay under analysis. Had the reduct analysis been worked out based only on the whole condition attributes *C* (as per the equivalence relation in Table 3, where decision attribute *D* is excluded such as in the case of *IS* instead of *DS*), the core of *C* (i.e. the core of the equivalence relation *UC* with respect to *C*) would have been,

$$\text{CORE}\_{\text{ul}\{\text{C}\}}\text{(C)}\,=\,\,\{\text{bg}\}\tag{22}$$

This implies the protective relay has been subjected to B-G fault. In reality this fault occurred in distance zone 1 operation characteristic and was picked up by the zone 1 distance element. However, the *D*-core of *C* discovers the indispensability of the condition attribute *Z3pu* as being the core when the decision attribute *D* is considered for the *DS*

analysis. Actually, this attribute is entirely insignificant based on the understanding of the manner the distance relay functions. This is simply because of the concurrent nature of the distance relay quadrilateral operation characteristic whereby zone 1 element is encapsulated in zone 2 element and subsequently zone 2 element is encapsulated in zone 3 element. Zone 4 element is on its own separate entity not encapsulated in any zone elements [7]. Thus, by merely considering the exertion of the zone 1 element in case of fault and correspondingly disregarding zones 2, 3 and 4 operation is principally correct. Figure 2 illustrates that a fault occurring in zone 1 is also concurrently shown as present in zones 2 and 3 as well.

Inconsistent Decision System: Rough Set Data Mining Strategy to Extract Decision Algorithm of a Numerical Distance Relay – Tutorial 131

The absence of relay trip assertion signal in attributes *Z2trp*, *Z3trp*, and *Z4trp* which is represented by the attribute value "0" further justifies the necessity of disregarding attributes *Z2pu*, *Z3pu*, and *Z4pu* for fault in zone 1. This is because, for example, the assertion of attribute *Z1pu* (value of "1") must always be accompanied by the assertion (after and for a preset time duration, i.e. sequence of consecutive events) of the corresponding attribute *Z1trp* in order to be taken into consideration in the analysis. However, in the above example, it is highly likely that attribute *Z2trp* will assert (after and for a preset number of events) in lieu of the attribute *Z1trp* as shown in Table 5 if the relay failed to operate in

Taking into account the proposition, the *DS* system in Table 1 is then modified prior to

Time (*U*) *ag bg cg Z1pu Z4pu Z1trp Z2trp Z3trp Z4trp Trip t1* 0 0 0 0 0 0 0 0 0 0 *t2* 0 0 0 0 0 0 0 0 0 0 *t3* 0 0 0 0 0 0 0 0 0 0 *t4* 0 1 0 0 0 0 0 0 0 0 *t5* 0 2 0 0 0 0 0 0 0 0 *t6* 0 2 0 0 0 0 0 0 0 0 *t7* 0 2 0 0 0 0 0 0 0 0 *t8* 0 1 0 1 0 1 0 0 0 b *t9* 0 1 0 1 0 1 0 0 0 b *t10* 0 1 0 1 0 1 0 0 0 b *t11* 0 0 0 1 0 0 0 0 0 b *t12* 0 0 0 1 0 0 0 0 0 b *t13* 0 0 0 1 0 0 0 0 0 b *t14* 0 0 0 1 0 0 0 0 0 b *t15* 0 0 0 1 0 0 0 0 0 b *t16* 0 0 0 0 0 0 0 0 0 b *t17* 0 0 0 0 0 0 0 0 0 b *t18* 0 0 0 0 0 0 0 0 0 b *t19* 0 0 0 0 0 0 0 0 0 b *t20* 0 0 0 0 0 0 0 0 0 0 *t21* 0 0 0 0 0 0 0 0 0 0

asserting the attribute *Z1trp* when the attribute *Z1pu* is asserted.

**Table 6.** Modified decision table *DS* to reflect protective relay operation behavior

same as shown in Table 2.

the whole attributes *C* considered (Table 3).

From Table 6, the elementary sets with respect to the decision attribute *D* = {*Trip*} are still the

However, the elementary sets with respect to the shrunk condition = {*ag*, *bg*, *cg*, *Z1pu*, *Z4pu*, *Z1trp*, *Z2trp*, *Z3trp*, *Z4trp*} as shown in Table 7 are slightly different from those found with

reanalysis using rough set as shown in Table 6.

To simplify and make the analysis process more sense, an attribute priority of the distance relay operation has to be formulated so that the relay *DS* can be modified as shown Table 5.

**Figure 2.** Distance protective relay operation characteristic with impedance measurement


+ denotes value of attribute equal to "1", i.e. *Vci* = 1 where attribute *ciC*.

\* denotes the attribute's value of "1" occurring at possibly different events (rows).

**Table 5.** Condition attribute priority of the distance relay operation

The absence of relay trip assertion signal in attributes *Z2trp*, *Z3trp*, and *Z4trp* which is represented by the attribute value "0" further justifies the necessity of disregarding attributes *Z2pu*, *Z3pu*, and *Z4pu* for fault in zone 1. This is because, for example, the assertion of attribute *Z1pu* (value of "1") must always be accompanied by the assertion (after and for a preset time duration, i.e. sequence of consecutive events) of the corresponding attribute *Z1trp* in order to be taken into consideration in the analysis. However, in the above example, it is highly likely that attribute *Z2trp* will assert (after and for a preset number of events) in lieu of the attribute *Z1trp* as shown in Table 5 if the relay failed to operate in asserting the attribute *Z1trp* when the attribute *Z1pu* is asserted.

130 Advances in Data Mining Knowledge Discovery and Applications

analysis. Actually, this attribute is entirely insignificant based on the understanding of the manner the distance relay functions. This is simply because of the concurrent nature of the distance relay quadrilateral operation characteristic whereby zone 1 element is encapsulated in zone 2 element and subsequently zone 2 element is encapsulated in zone 3 element. Zone 4 element is on its own separate entity not encapsulated in any zone elements [7]. Thus, by merely considering the exertion of the zone 1 element in case of fault and correspondingly disregarding zones 2, 3 and 4 operation is principally correct. Figure 2 illustrates that a fault

To simplify and make the analysis process more sense, an attribute priority of the distance relay operation has to be formulated so that the relay *DS* can be modified as shown Table 5.

occurring in zone 1 is also concurrently shown as present in zones 2 and 3 as well.

**Figure 2.** Distance protective relay operation characteristic with impedance measurement

zone 2 zone 1

zone 4

zone 3

Trajectory of impedance measurement

+ denotes value of attribute equal to "1", i.e. *Vci* = 1 where attribute *ciC*. \* denotes the attribute's value of "1" occurring at possibly different events (rows). **Table 5.** Condition attribute priority of the distance relay operation

Condition Attributes*, ciC* 

Most significant attribute

*Z1pu Z2pu Z3pu Z4pu Z1trp Z2trp Z3trp Z4trp*

Case 1 + + + + *Z1pu* Case 1´ + + + \* *Z2pu* Case 2 + + \* *Z2pu* Case 2´ + + \* *Z3pu* Case 3 + \* *Z3pu* Case 4 + \* *Z4pu*

Cases of concurrence Taking into account the proposition, the *DS* system in Table 1 is then modified prior to reanalysis using rough set as shown in Table 6.


**Table 6.** Modified decision table *DS* to reflect protective relay operation behavior

From Table 6, the elementary sets with respect to the decision attribute *D* = {*Trip*} are still the same as shown in Table 2.

However, the elementary sets with respect to the shrunk condition = {*ag*, *bg*, *cg*, *Z1pu*, *Z4pu*, *Z1trp*, *Z2trp*, *Z3trp*, *Z4trp*} as shown in Table 7 are slightly different from those found with the whole attributes *C* considered (Table 3).

132 Advances in Data Mining Knowledge Discovery and Applications


Inconsistent Decision System: Rough Set Data Mining Strategy to Extract Decision Algorithm of a Numerical Distance Relay – Tutorial 133

The Boolean discernibility function is formulated from the discernibility matrix as follows:

= (*Z1pu*) (*Z1pu*+*Z1trp*) (*bg*) (*bg*) (*bg*+*Z1pu*+*Z1trp*) *Z1pu* = (*Z1pu*) (*bg*) The final Disjunctive Normal Form (DNF) of *fC*(*D*)

*Z1pu*}. In this case, the *D*-core of *C* is similar to *D*-reduct of *C*.

patterns such as that presented by the attribute *Trip* shown all along.

**Table 9.** Equivalent decision table with respect to *REDD*(*C*) = {*bg*, *Z1pu*}

= (*bg*+*Z1pu*+*Z1trp*) (*Z1pu*) (*Z1pu*+*Z1trp*) (*bg*+*Z1pu*) (*bg*) (*bg*+*Z1pu*+*Z1trp*) (bg+*Z1pu*)

There is only one *D*-reduct of *C*, *REDD*(*C*) = {*bg*, *Z1pu*}. As shown in Table 9, it can alternatively be used to represent exactly similar equivalence relation *UIND*(*D*) of the down scaled *DS* as that represented by the whole set of attributes *C*. The *D*-core of *C* is the set of all single entries of the *D*-discernibility matrix, (or *CORED*(*C*) = ∩*REDD*(*C*)), i.e. {*bg*,

As previously discussed, the possibility of the core inferring the power system state the relay has been subjected to is really prominently singled out now by the new *CORED*(*C*) = {*bg*, *Z1pu*}. Due the very characteristic of indispensability of core, it is undoubtedly identified that a A-G fault has occurred and consequently the relay's Z1 ground distance element has picked up to get rid of it. This eventually translates into the trip decision having

Time (*U*) *bg Z1pu Trip t1* 0 0 0 *t2* 0 0 0 *t3* 0 0 0 *t4* 1 0 0 *t5* 2 0 0 *t6* 2 0 0 *t7* 2 0 0 *t8* 1 1 b *t9* 1 1 b *t10* 1 1 b *t11* 0 1 b *t12* 0 1 b *t13* 0 1 b *t14* 0 1 b *t15* 0 1 b *t16* 0 0 b *t17* 0 0 b *t18* 0 0 b *t19* 0 0 b *t20* 0 0 0 *t21* 0 0 0

*fC*(*D*)

(*bg*) (*bg*+*Z1pu*+ *Z1trp*) *Z1pu*

**Table 7.** Equivalence classes with respect to modified condition attributes *C* = {*ag*, *bg*, *cg*, *Z1pu*, *Z4pu*, *Z1trp*, *Z2trp*, *Z3trp*, *Z4trp*}

The new *D*-discernibility matrix of *C* as in Table 8 will result in new *D*-reducts and *D*-core of *C* when events are discerned with respect to the modified condition attributes *C* between different equivalence classes in the relation *UIND*(*D*). As before, similar consideration is taken in discerning events appearing only in different classes in *D*-space.


**Table 8.** *D*-discernibility matrix of modified *C* 

The Boolean discernibility function is formulated from the discernibility matrix as follows:

*fC*(*D*)

132 Advances in Data Mining Knowledge Discovery and Applications

*C ag bg cg Z1pu Z4pu Z1trp Z2trp Z3trp Z4trp* 

{*t1*, *t2*, *t3*, *t16*, *t17*, *t18*, *t19*, *t20*, *t21*} 0 0 0 0 0 0 0 0 0 {*t4*} 0 1 0 0 0 0 0 0 0 {*t5*, *t6*, *t7*} 0 2 0 0 0 0 0 0 0 {*t8*, *t9*, *t10*} 0 1 0 1 0 1 0 0 0 {*t11*, *t12*, *t13*, *t14*, *t15*} 0 0 0 1 0 0 0 0 0 **Table 7.** Equivalence classes with respect to modified condition attributes *C* = {*ag*, *bg*, *cg*, *Z1pu*, *Z4pu*,

The new *D*-discernibility matrix of *C* as in Table 8 will result in new *D*-reducts and *D*-core of *C* when events are discerned with respect to the modified condition attributes *C* between different equivalence classes in the relation *UIND*(*D*). As before, similar consideration is

*U t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 t11 t12 t13 t14 t15 t16 t17 t18 t19 t20 t21*

taken in discerning events appearing only in different classes in *D*-space.

*U*

*Z1trp*, *Z2trp*, *Z3trp*, *Z4trp*}

*t1* Ø *t2* Ø Ø *t3* Ø Ø Ø *t4* Ø Ø Ø Ø *t5* Ø Ø Ø Ø Ø *t6* Ø Ø Ø Ø Ø Ø *t7* Ø Ø Ø Ø Ø Ø Ø

*t8*

*t9*

*t10*

{*bg, Z1pu, Z1trp*}

{*bg, Z1pu, Z1trp*}

{*bg, Z1pu, Z1trp*}

{*bg, Z1pu, Z1trp*}

{*bg, Z1pu, Z1trp*}

{*bg, Z1pu, Z1trp*}

*t11* {*Z1pu*} {*Z1pu*} {*Z1pu*} {*bg,* 

*t12* {*Z1pu*} {*Z1pu*} {*Z1pu*} {*bg,* 

*t13* {*Z1pu*} {*Z1pu*} {*Z1pu*} {*bg,* 

*t14* {*Z1pu*} {*Z1pu*} {*Z1pu*} {*bg,* 

*t15 Z1pu*} *Z1pu*} *Z1pu*} {*bg,* 

{*bg, Z1pu, Z1trp*}

{*bg, Z1pu, Z1trp*}

{*bg, Z1pu, Z1trp*}

{*Z1pu, Z1trp*}

{*Z1pu, Z1trp*}

{*Z1pu, Z1trp*}

*Z1pu*}

*Z1pu*}

*Z1pu*}

*Z1pu*}

*Z1pu*}

*t20* Ø Ø Ø Ø Ø Ø Ø

*t21* Ø Ø Ø Ø Ø Ø Ø

**Table 8.** *D*-discernibility matrix of modified *C* 

{*bg, Z1pu, Z1trp*}

{*bg, Z1pu, Z1trp*}

{*bg, Z1pu, Z1trp*}

{*bg, Z1pu*}

{*bg, Z1pu*}

{*bg, Z1pu*}

{*bg, Z1pu*}

{*bg, Z1pu*}

{*bg, Z1pu, Z1trp*}

{*bg, Z1pu, Z1trp*}

{*bg, Z1pu, Z1trp*}

{*bg, Z1pu*}

{*bg, Z1pu*}

{*bg, Z1pu*}

{*bg, Z1pu*}

{*bg, Z1pu*}

{*bg, Z1pu, Z1trp*}

{*bg, Z1pu, Z1trp*}

{*bg, Z1pu, Z1trp*}

{*bg,* 

{*bg,* 

{*bg,* 

{*bg,* 

{*bg,* 

*t16* Ø Ø Ø {*bg*} {*bg*} {*bg*} {*bg*} Ø Ø Ø Ø Ø Ø Ø Ø Ø *t17* Ø Ø Ø {*bg*} {*bg*} {*bg*} {*bg*} Ø Ø Ø Ø Ø Ø Ø Ø Ø Ø *t18* Ø Ø Ø {*bg*} {*bg*} {*bg*} {*bg*} Ø Ø Ø Ø Ø Ø Ø Ø Ø Ø Ø *t19* Ø Ø Ø {*bg*} {*bg*} {*bg*} {*bg*} Ø Ø Ø Ø Ø Ø Ø Ø Ø Ø Ø Ø

> {*bg, Z1pu, Z1trp*}

> {*bg, Z1pu, Z1trp*}

{*bg, Z1pu, Z1trp*}

{*bg, Z1pu, Z1trp*}

{*bg, Z1pu, Z1trp*}

{*bg, Z1pu, Z1trp*}

Ø

Ø Ø

Ø Ø Ø

*Z1pu*} Ø Ø Ø Ø

*Z1pu*} Ø Ø Ø Ø Ø

*Z1pu*} Ø Ø Ø Ø Ø Ø

*Z1pu*} Ø Ø Ø Ø Ø Ø Ø

*Z1pu*} Ø Ø Ø Ø Ø Ø Ø Ø

{*Z1pu*} {*Z1pu*} {*Z1pu*} {*Z1pu*} {*Z1pu*} Ø Ø Ø Ø Ø

{*Z1pu*} {*Z1pu*} {*Z1pu*} {*Z1pu*} {*Z1pu*} Ø Ø Ø Ø Ø Ø


There is only one *D*-reduct of *C*, *REDD*(*C*) = {*bg*, *Z1pu*}. As shown in Table 9, it can alternatively be used to represent exactly similar equivalence relation *UIND*(*D*) of the down scaled *DS* as that represented by the whole set of attributes *C*. The *D*-core of *C* is the set of all single entries of the *D*-discernibility matrix, (or *CORED*(*C*) = ∩*REDD*(*C*)), i.e. {*bg*, *Z1pu*}. In this case, the *D*-core of *C* is similar to *D*-reduct of *C*.

As previously discussed, the possibility of the core inferring the power system state the relay has been subjected to is really prominently singled out now by the new *CORED*(*C*) = {*bg*, *Z1pu*}. Due the very characteristic of indispensability of core, it is undoubtedly identified that a A-G fault has occurred and consequently the relay's Z1 ground distance element has picked up to get rid of it. This eventually translates into the trip decision having patterns such as that presented by the attribute *Trip* shown all along.


**Table 9.** Equivalent decision table with respect to *REDD*(*C*) = {*bg*, *Z1pu*}

#### **4.3. Protective relay decision algorithm discovery**

As aforementioned, a relay decision algorithm in *DS* called *CD*-decision algorithm manifests as a *CD*-decision table. It comprises a finite set of relay *CD*-decision rules or instructions. The event report of a protective distance relay in the form of a *DS* is a manifestation of relay decision algorithm. In protection system, protection engineers relate relay decision algorithm as relay operation logic. It is envisaged that with rough set theory, the relay operation logic knowledge can be discovered. Later it can be transformed into a knowledge base of a decision support system for determining anticipated relay behavior out of a new test *DS* [7].

Checking whether or not all the relay operation logics (decision rules) are true would enable us to check whether or not a relay decision algorithm is consistent. As aforementioned, consistency is measured by the degree of dependency *k* (or alternatively, dependency is measured by the degree of consistency) [10]. Thus, it is well understood that with the degree of consistency given in Equation (10),

$$k = \frac{card\,POS(C, D)}{card\,(CD - decision\,algorithm)}\tag{23}$$

Inconsistent Decision System: Rough Set Data Mining Strategy to Extract Decision Algorithm of a Numerical Distance Relay – Tutorial 135

rule 14: *ag0 bg0 cg0 Z1pu1 Z4pu0 Z1trp0 Z2trp0 Z3trp0 Z4trp0 Tripb* rule 15: *ag0 bg0 cg0 Z1pu1 Z4pu0 Z1trp0 Z2trp0 Z3trp0 Z4trp0 Tripb* rule 16: *ag0 bg0 cg0 Z1pu0 Z4pu0 Z1trp0 Z2trp0 Z3trp0 Z4trp0 Tripb* rule 17: *ag0 bg0 cg0 Z1pu0 Z4pu0 Z1trp0 Z2trp0 Z3trp0 Z4trp0 Tripb* rule 18: *ag0 bg0 cg0 Z1pu0 Z4pu0 Z1trp0 Z2trp0 Z3trp0 Z4trp0 Tripb* rule 19: *ag0 bg0 cg0 Z1pu0 Z4pu0 Z1trp0 Z2trp0 Z3trp0 Z4trp0 Tripb* rule 20: *ag0 bg0 cg0 Z1pu0 Z4pu0 Z1trp0 Z2trp0 Z3trp0 Z4trp0 Trip0* rule 21: *ag0 bg0 cg0 Z1pu0 Z4pu0 Z1trp0 Z2trp0 Z3trp0 Z4trp0 Trip0*

**4.4. Protective relay decision algorithm simplification** 

 *CD*-decision algorithm is reduct of *CD*-decision algorithm. The set of all reducts of *CD*-decision algorithm is called *RED*(*C*,*D*)

Therefore, the following terms are valid:

*CORED*(*C*) = ∩*REDD*(*C*)).

duration of time [7].

reducing *DS*.

The two sets of relay decision rules, i.e. rules 1, 2, 3 and rules 16, 17, 18, 19, altogether totaling 7 rules, are inconsistent (false). The positive region of the *CD*-decision algorithm consists of only consistent decision rules 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 20 and 21 (i.e. *card POS*(*C*,*D*) = 14) and, hence, the degree of dependency is *k* = 14/21= 0.67. Since there are decision rules in the algorithm that are consistent only by the degree of 0.67 (i.e. false), the relay *CD*-decision algorithm is said to be inconsistent. The decision classes are not all uniquely discernible by conditions of all decision rules in the *CD*-decision algorithm. In other words, there are at least two decision rules having the same conditions but different implications in the decision. This phenomenon is certainly anticipated especially as shown by rules 16, 17, 18, and 19 whereby the decision attribute *Trip* remains in the value "*b*" reflecting the actual distance relay operation behavior. Technically speaking, irrespective of the presence or otherwise of the fault (assertion via attribute "*bg*") and zone 1 element pickup (assertion via attribute "*Z1pu*"), the relay trip signal remains asserted for a certain preset

Algorithm reduction results in simplification of the *CD*-decision algorithm. This is done by investigating whether all condition attributes are necessary to make decisions. Therefore, reducing *CD*-decision algorithm is essentially closely related to the previous discussion on

The subset of condition attributes *C C* is called a reduct of *C* in the *CD*-decision algorithm if the *CD*-decision algorithm is independent and consistent, i.e. *POS*(*C*,*D*) = *POS*(*C*,*D*).

 The set of all indispensible condition attributes in the *CD*-decision algorithm is called the core of the of the *CD*-decision algorithm and, similarly like before, takes on the expression, *CORE*(*C*,*D*) = ∩*RED*(*C*,*D*). (In principle it is similar to the expression

a relay *CD*-decision algorithm has a degree *k*, i.e. the degree of dependency between condition attributes *C =* {*ag, bg, cg, Z1pu, Z4pu, Z1trp, Z2trp, Z3trp, Z4trp*} and decision attributes *D =* {*Trip*}.

The relay *CD*-decision rules (*C D*) are:

rule 1: *ag0 bg0 cg0 Z1pu0 Z4pu0 Z1trp0 Z2trp0 Z3trp0 Z4trp0 Trip0* rule 2: *ag0 bg0 cg0 Z1pu0 Z4pu0 Z1trp0 Z2trp0 Z3trp0 Z4trp0 Trip0* rule 3: *ag0 bg0 cg0 Z1pu0 Z4pu0 Z1trp0 Z2trp0 Z3trp0 Z4trp0 Trip0* rule 4: *ag0 bg1 cg0 Z1pu0 Z4pu0 Z1trp0 Z2trp0 Z3trp0 Z4trp0 Trip0* rule 5: *ag0 bg2 cg0 Z1pu0 Z4pu0 Z1trp0 Z2trp0 Z3trp0 Z4trp0 Trip0* rule 6: *ag0 bg2 cg0 Z1pu0 Z4pu0 Z1trp0 Z2trp0 Z3trp0 Z4trp0 Trip0* rule 7: *ag0 bg2 cg0 Z1pu0 Z4pu0 Z1trp0 Z2trp0 Z3trp0 Z4trp0 Trip0* rule 8: *ag0 bg1 cg0 Z1pu1 Z4pu0 Z1trp1 Z2trp0 Z3trp0 Z4trp0 Tripb* rule 9: *ag0 bg1 cg0 Z1pu1 Z4pu0 Z1trp1 Z2trp0 Z3trp0 Z4trp0 Tripb* rule 10: *ag0 bg1 cg0 Z1pu1 Z4pu0 Z1trp1 Z2trp0 Z3trp0 Z4trp0 Tripb* rule 11: *ag0 bg0 cg0 Z1pu1 Z4pu0 Z1trp0 Z2trp0 Z3trp0 Z4trp0 Tripb* rule 12: *ag0 bg0 cg0 Z1pu1 Z4pu0 Z1trp0 Z2trp0 Z3trp0 Z4trp0 Tripb* rule 13: *ag0 bg0 cg0 Z1pu1 Z4pu0 Z1trp0 Z2trp0 Z3trp0 Z4trp0 Tripb*

rule 14: *ag0 bg0 cg0 Z1pu1 Z4pu0 Z1trp0 Z2trp0 Z3trp0 Z4trp0 Tripb* rule 15: *ag0 bg0 cg0 Z1pu1 Z4pu0 Z1trp0 Z2trp0 Z3trp0 Z4trp0 Tripb* rule 16: *ag0 bg0 cg0 Z1pu0 Z4pu0 Z1trp0 Z2trp0 Z3trp0 Z4trp0 Tripb* rule 17: *ag0 bg0 cg0 Z1pu0 Z4pu0 Z1trp0 Z2trp0 Z3trp0 Z4trp0 Tripb* rule 18: *ag0 bg0 cg0 Z1pu0 Z4pu0 Z1trp0 Z2trp0 Z3trp0 Z4trp0 Tripb* rule 19: *ag0 bg0 cg0 Z1pu0 Z4pu0 Z1trp0 Z2trp0 Z3trp0 Z4trp0 Tripb* rule 20: *ag0 bg0 cg0 Z1pu0 Z4pu0 Z1trp0 Z2trp0 Z3trp0 Z4trp0 Trip0* rule 21: *ag0 bg0 cg0 Z1pu0 Z4pu0 Z1trp0 Z2trp0 Z3trp0 Z4trp0 Trip0*

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of consistency given in Equation (10),

test *DS* [7].

attributes *D =* {*Trip*}.

The relay *CD*-decision rules (*C* 

**4.3. Protective relay decision algorithm discovery** 

As aforementioned, a relay decision algorithm in *DS* called *CD*-decision algorithm manifests as a *CD*-decision table. It comprises a finite set of relay *CD*-decision rules or instructions. The event report of a protective distance relay in the form of a *DS* is a manifestation of relay decision algorithm. In protection system, protection engineers relate relay decision algorithm as relay operation logic. It is envisaged that with rough set theory, the relay operation logic knowledge can be discovered. Later it can be transformed into a knowledge base of a decision support system for determining anticipated relay behavior out of a new

Checking whether or not all the relay operation logics (decision rules) are true would enable us to check whether or not a relay decision algorithm is consistent. As aforementioned, consistency is measured by the degree of dependency *k* (or alternatively, dependency is measured by the degree of consistency) [10]. Thus, it is well understood that with the degree

> (, ) ( )

a relay *CD*-decision algorithm has a degree *k*, i.e. the degree of dependency between condition attributes *C =* {*ag, bg, cg, Z1pu, Z4pu, Z1trp, Z2trp, Z3trp, Z4trp*} and decision

*card CD decisionalgorithm* (23)

*card POS C D <sup>k</sup>*

rule 1: *ag0 bg0 cg0 Z1pu0 Z4pu0 Z1trp0 Z2trp0 Z3trp0 Z4trp0 Trip0* rule 2: *ag0 bg0 cg0 Z1pu0 Z4pu0 Z1trp0 Z2trp0 Z3trp0 Z4trp0 Trip0* rule 3: *ag0 bg0 cg0 Z1pu0 Z4pu0 Z1trp0 Z2trp0 Z3trp0 Z4trp0 Trip0* rule 4: *ag0 bg1 cg0 Z1pu0 Z4pu0 Z1trp0 Z2trp0 Z3trp0 Z4trp0 Trip0* rule 5: *ag0 bg2 cg0 Z1pu0 Z4pu0 Z1trp0 Z2trp0 Z3trp0 Z4trp0 Trip0* rule 6: *ag0 bg2 cg0 Z1pu0 Z4pu0 Z1trp0 Z2trp0 Z3trp0 Z4trp0 Trip0* rule 7: *ag0 bg2 cg0 Z1pu0 Z4pu0 Z1trp0 Z2trp0 Z3trp0 Z4trp0 Trip0* rule 8: *ag0 bg1 cg0 Z1pu1 Z4pu0 Z1trp1 Z2trp0 Z3trp0 Z4trp0 Tripb* rule 9: *ag0 bg1 cg0 Z1pu1 Z4pu0 Z1trp1 Z2trp0 Z3trp0 Z4trp0 Tripb* rule 10: *ag0 bg1 cg0 Z1pu1 Z4pu0 Z1trp1 Z2trp0 Z3trp0 Z4trp0 Tripb* rule 11: *ag0 bg0 cg0 Z1pu1 Z4pu0 Z1trp0 Z2trp0 Z3trp0 Z4trp0 Tripb* rule 12: *ag0 bg0 cg0 Z1pu1 Z4pu0 Z1trp0 Z2trp0 Z3trp0 Z4trp0 Tripb* rule 13: *ag0 bg0 cg0 Z1pu1 Z4pu0 Z1trp0 Z2trp0 Z3trp0 Z4trp0 Tripb*

 *D*) are:

The two sets of relay decision rules, i.e. rules 1, 2, 3 and rules 16, 17, 18, 19, altogether totaling 7 rules, are inconsistent (false). The positive region of the *CD*-decision algorithm consists of only consistent decision rules 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 20 and 21 (i.e. *card POS*(*C*,*D*) = 14) and, hence, the degree of dependency is *k* = 14/21= 0.67. Since there are decision rules in the algorithm that are consistent only by the degree of 0.67 (i.e. false), the relay *CD*-decision algorithm is said to be inconsistent. The decision classes are not all uniquely discernible by conditions of all decision rules in the *CD*-decision algorithm. In other words, there are at least two decision rules having the same conditions but different implications in the decision. This phenomenon is certainly anticipated especially as shown by rules 16, 17, 18, and 19 whereby the decision attribute *Trip* remains in the value "*b*" reflecting the actual distance relay operation behavior. Technically speaking, irrespective of the presence or otherwise of the fault (assertion via attribute "*bg*") and zone 1 element pickup (assertion via attribute "*Z1pu*"), the relay trip signal remains asserted for a certain preset duration of time [7].

#### **4.4. Protective relay decision algorithm simplification**

Algorithm reduction results in simplification of the *CD*-decision algorithm. This is done by investigating whether all condition attributes are necessary to make decisions. Therefore, reducing *CD*-decision algorithm is essentially closely related to the previous discussion on reducing *DS*.

The subset of condition attributes *C C* is called a reduct of *C* in the *CD*-decision algorithm if the *CD*-decision algorithm is independent and consistent, i.e. *POS*(*C*,*D*) = *POS*(*C*,*D*). Therefore, the following terms are valid:


The modified *DS* in Table 6 had found its only reduct of condition attributes, *RED*(*C*,*D*) = *REDD*(*C*) = {*bg*, *Z1pu*}, in the relay *CD*-decision algorithm. The core had a similar set as that of the reduct, i.e. *CORE*(*C*,*D*) = ∩*RED*(*C*,*D*) = {*bg*, *Z1pu*}. The resulting equivalent *DS* with respect to *RED*(*C*,*D*) = {*bg*, *Z1pu*} in Table 9 produces a rather simplified version of relay *CD*decision algorithm, i.e.,

Inconsistent Decision System: Rough Set Data Mining Strategy to Extract Decision Algorithm of a Numerical Distance Relay – Tutorial 137

From the apparently inconsistent rules 1 and 6, i.e. similar conditions but dissimilar decisions, the simplified relay *CD*-decision algorithm reveals pronouncedly its inconsistent nature. This inconsistency may not be desirable in some information system analysis. However, in as far as protective relay operation is concerned, it is interesting to know,

 the time delay between relay pick-up and relay trip signal assertion – traced by identifying the translated time sequence (*DS* row label) from rule 4 (*t4*) to rule 8 (*t8*), the lapsed time of the relay assertion in instructing the circuit breaker to open its contacts – traced by identifying the translated time sequence from rule 8 (*t8*) to rule 15

Decision rules 2, 3, 4, and 5 are the consistent ones that constitute the positive region of

The core *CORE*(*C*,*D*) = {*bg*, *Z1pu*} can be justified why it is so. By dropping the attributes *bg* or *Z1pu,* one step at a time, their indispensability can be seen and whether the positive region that consists of the consistent rules changes can be checked. Different positive region

the affected pole(s) – determined from the decision attribute *Trip* value.

Likewise, the positive region can be changed as well by removing attribute Z1pu:

rule 2 (4): *bg*<sup>1</sup> *Z1pu*<sup>0</sup> *Trip*<sup>0</sup> rule 3 (5, 6, 7): *bg*<sup>2</sup> *Z1pu*<sup>0</sup> *Trip*<sup>0</sup> rule 4 (8, 9, 10): *bg*<sup>1</sup> *Z1pu*<sup>1</sup> *Trip*<sup>b</sup> rule 5 (11, 12, 13, 14, 15): *bg*<sup>0</sup> *Z1pu*<sup>1</sup> *Trip*<sup>b</sup> rule 6 (16, 17, 18, 19): *bg*<sup>0</sup> *Z1pu*<sup>0</sup> *Trip*<sup>b</sup>

(*t15*) and eventually to rule 19(*t19*), and

the CD-decision algorithm.

is obtained by removing attribute *bg*:

rule 1 (1, 2, 3, 20, 21): *Z1pu*<sup>0</sup> *Trip*<sup>0</sup>

rule 2 (4): *Z1pu*<sup>0</sup> *Trip*<sup>0</sup>

rule 3 (5, 6, 7): *Z1pu*<sup>0</sup> *Trip*<sup>0</sup>

rule 4 (8, 9, 10): *Z1pu*<sup>1</sup> *Trip*<sup>b</sup>

rule 5 (11, 12, 13, 14, 15): *Z1pu*<sup>1</sup> *Trip*<sup>b</sup>

rule 6 (16, 17, 18, 19): *Z1pu*<sup>0</sup> *Trip*<sup>b</sup>

rule 1 (1, 2, 3, 20, 21): *bg*<sup>0</sup> *Trip*<sup>0</sup>

rule 2 (4): *bg*<sup>1</sup> *Trip*<sup>0</sup>

rule 3 (5, 6, 7): *bg*<sup>2</sup> *Trip*<sup>0</sup>

rule 4 (8, 9, 10): *bg*<sup>1</sup> *Trip*<sup>b</sup>

among others:


Each relay *CD*-decision rule designation corresponds to the row label in the *DS;* for example, rule 9 corresponding to row label *t9*.

The relay *CD*-decision algorithm can be cut down by removing duplicate relay *CD*-decision rules,

rule 1 (1, 2, 3, 20, 21): *bg*<sup>0</sup> *Z1pu*<sup>0</sup> *Trip*<sup>0</sup>

rule 2 (4): *bg*<sup>1</sup> *Z1pu*<sup>0</sup> *Trip*<sup>0</sup> rule 3 (5, 6, 7): *bg*<sup>2</sup> *Z1pu*<sup>0</sup> *Trip*<sup>0</sup> rule 4 (8, 9, 10): *bg*<sup>1</sup> *Z1pu*<sup>1</sup> *Trip*<sup>b</sup> rule 5 (11, 12, 13, 14, 15): *bg*<sup>0</sup> *Z1pu*<sup>1</sup> *Trip*<sup>b</sup> rule 6 (16, 17, 18, 19): *bg*<sup>0</sup> *Z1pu*<sup>0</sup> *Trip*<sup>b</sup>

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decision algorithm, i.e.,

rule 1: *bg*<sup>0</sup> *Z1pu*<sup>0</sup> *Trip*<sup>0</sup> rule 2: *bg*<sup>0</sup> *Z1pu*<sup>0</sup> *Trip*<sup>0</sup> rule 3: *bg*<sup>0</sup> *Z1pu*<sup>0</sup> *Trip*<sup>0</sup> rule 4: *bg*<sup>1</sup> *Z1pu*<sup>0</sup> *Trip*<sup>0</sup> rule 5: *bg*<sup>2</sup> *Z1pu*<sup>0</sup> *Trip*<sup>0</sup> rule 6: *bg*<sup>2</sup> *Z1pu*<sup>0</sup> *Trip*<sup>0</sup> rule 7: *bg*<sup>2</sup> *Z1pu*<sup>0</sup> *Trip*<sup>0</sup> rule 8: *bg*<sup>1</sup> *Z1pu*<sup>1</sup> *Trip*<sup>b</sup> rule 9: *bg*<sup>1</sup> *Z1pu*<sup>1</sup> *Trip*<sup>b</sup> rule 10: *bg*<sup>1</sup> *Z1pu*<sup>1</sup> *Trip*<sup>b</sup> rule 11: *bg*<sup>0</sup> *Z1pu*<sup>1</sup> *Trip*<sup>b</sup> rule 12: *bg*<sup>0</sup> *Z1pu*<sup>1</sup> *Trip*<sup>b</sup> rule 13: *bg*<sup>0</sup> *Z1pu*<sup>1</sup> *Trip*<sup>b</sup> rule 14: *bg*<sup>0</sup> *Z1pu*<sup>1</sup> *Trip*<sup>b</sup> rule 15: *bg*<sup>0</sup> *Z1pu*<sup>1</sup> *Trip*<sup>b</sup> rule 16: *bg*<sup>0</sup> *Z1pu*<sup>0</sup> *Trip*<sup>b</sup> rule 17: *bg*<sup>0</sup> *Z1pu*<sup>0</sup> *Trip*<sup>b</sup> rule 18: *bg*<sup>0</sup> *Z1pu*<sup>0</sup> *Trip*<sup>b</sup> rule 19: *bg*<sup>0</sup> *Z1pu*<sup>0</sup> *Trip*<sup>b</sup> rule 20: *bg*<sup>0</sup> *Z1pu*<sup>0</sup> *Trip*<sup>0</sup> rule 21: *bg*<sup>0</sup> *Z1pu*<sup>0</sup> *Trip*<sup>0</sup>

The modified *DS* in Table 6 had found its only reduct of condition attributes, *RED*(*C*,*D*) = *REDD*(*C*) = {*bg*, *Z1pu*}, in the relay *CD*-decision algorithm. The core had a similar set as that of the reduct, i.e. *CORE*(*C*,*D*) = ∩*RED*(*C*,*D*) = {*bg*, *Z1pu*}. The resulting equivalent *DS* with respect to *RED*(*C*,*D*) = {*bg*, *Z1pu*} in Table 9 produces a rather simplified version of relay *CD*-

Each relay *CD*-decision rule designation corresponds to the row label in the *DS;* for

The relay *CD*-decision algorithm can be cut down by removing duplicate relay *CD*-decision

example, rule 9 corresponding to row label *t9*.

rule 1 (1, 2, 3, 20, 21): *bg*<sup>0</sup> *Z1pu*<sup>0</sup> *Trip*<sup>0</sup>

rules,

From the apparently inconsistent rules 1 and 6, i.e. similar conditions but dissimilar decisions, the simplified relay *CD*-decision algorithm reveals pronouncedly its inconsistent nature. This inconsistency may not be desirable in some information system analysis. However, in as far as protective relay operation is concerned, it is interesting to know, among others:


Decision rules 2, 3, 4, and 5 are the consistent ones that constitute the positive region of the CD-decision algorithm.

The core *CORE*(*C*,*D*) = {*bg*, *Z1pu*} can be justified why it is so. By dropping the attributes *bg* or *Z1pu,* one step at a time, their indispensability can be seen and whether the positive region that consists of the consistent rules changes can be checked. Different positive region is obtained by removing attribute *bg*:


Likewise, the positive region can be changed as well by removing attribute Z1pu:


rule 5 (11, 12, 13, 14, 15): *bg*<sup>0</sup> *Trip*<sup>b</sup> rule 6 (16, 17, 18, 19): *bg*<sup>0</sup> *Trip*<sup>b</sup>

Thus, when one by one the said condition attributes is removed, the changes incurred in the positive region of the relay *CD*-decision algorithm concur with the core attributes' indispensability. Thus, the core having both attributes {*bg, Z1pu*} is correct.

Inconsistent Decision System: Rough Set Data Mining Strategy to Extract Decision Algorithm of a Numerical Distance Relay – Tutorial 139

*bg1 Z1pu0* 

*bg2 Trip0*

*Z1pu1 Tripb*

*Z1pu1* 

*bg0 Z1pu0*

*bg0 Z1pu0* 

i.e.

i.e.

or,

and

 *Trip0*

 *Tripb* 

 *bg0 Z1pu0* 

*Z1pu*0 (*bg*<sup>0</sup> *bg*1) *bg*2 *Trip*<sup>0</sup>

 *Z1pu1 Tripb* 

(*Z1pu =* 0), or

of the trip assertion is taking place.

*bg*<sup>0</sup> *Z1pu*<sup>0</sup> *bg*<sup>1</sup> *Z1pu*<sup>0</sup> *bg*2 *Trip*<sup>0</sup>

 *Tripb* 

*bg*<sup>0</sup> *Z1pu*<sup>0</sup> *bg*<sup>1</sup> *Z1pu*<sup>0</sup> *bg*2 *Trip*<sup>0</sup>

The combined form of the *minimal CD*-decision algorithm is

For decision attribute *Trip* = *a*, one minimal set of decision rules is obtained from

The final form of *CD*-decision algorithm can now be easily interpreted as follows:

i. when no fault occurs (*bg =* 0) and no relay pick-up (*Z1pu =* 0), or

IF *Z1pu =* 0 AND either *bg =* 0 OR *bg =* 1 OR IF *bg =* 2, THEN *Trip =* 0.

ii. when a A-G fault occurs in zone 1(*bg =* 1) and no relay pick-up (*Z1pu =* 0), or

IF *Z1pu =* 0 AND *bg =* 0 OR IF *Z1pu =* 1, THEN *Trip =* b. The trip assertion (*Trip =* b) is imminent with either one of the following situations:

i. when there is no more fault indication (*bg =* 0) and relay pick-up element has reset

Item i. indicates the fact that trip assertion *Trip =* b is still present in the face of the fault and relay pick-up resets (i.e. *bg =* 0 and *Z1pu =* 0) suggests that the preset time duration

The non-trip assertion (*Trip =* 0) is imminent with either one of the following situations:

The decision rule *Z1pu*0 (*bg*<sup>0</sup> *bg*1) *bg*<sup>2</sup> *Trip*0 is interpreted as,

iii. when a A-G fault occurs in zone 2 (*bg =* 2)

The decision rule *bg*<sup>0</sup> *Z1pu*<sup>0</sup> *Z1pu*<sup>1</sup> *Trip*b is interpreted as,

ii. when relay pick-up element remains asserted (*Z1pu =* 1)

#### **4.5. Protective relay decision algorithm minimization**

It is subsequently desirable to further minimize the decision rules in the relay *CD*-decision algorithm after the above simplification via reduction of the set of condition attributes. This is achieved by removal of any possibly superfluous decision rules which essentially involves reducing the superfluous values of attributes. In other words, the unnecessary conditions have to be separately removed leaving only the core attribute in each decision rule of the algorithm [10].

The tabulated version of the above simplified relay *CD*-decision algorithm is shown in Table 10.


**Table 10.** *DS* of simplified *CD*-decision algorithm

In Table 11 the condition attribute of each decision rule in Table 10 is removed one by one. In each removal the resultant rule is cross checked with other rules to find whether they are in conflict (inconsistent). This cross reference with other rules is to figure out whether the remaining condition attribute's value is the same but implication on the decision attribute is different. This process discovers the core attribute(s) that when eliminated causes the corresponding decision rule, or in general the *CD*-decision algorithm, inconsistent and consequently invalid (albeit not necessarily in the relay analysis perspective).

In summary, Table 12 contains cores of each decision rule. The condition attribute having eliminated value can be said as having no effect whatsoever on the *CD*-decision algorithm and may be termed as "don't care". It can be assigned with a value or otherwise. Combining duplicate rules and demarcating separate decision classes, Table 13 is obtained.

For decision attribute *Trip* = 0, one *minimal* set of decision rules is obtained from

*bg0 Z1pu0 Trip0*

```
bg1 Z1pu0  Trip0
```
*bg2 Trip0*

i.e.

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**Table 10.** *DS* of simplified *CD*-decision algorithm

Thus, when one by one the said condition attributes is removed, the changes incurred in the positive region of the relay *CD*-decision algorithm concur with the core attributes'

It is subsequently desirable to further minimize the decision rules in the relay *CD*-decision algorithm after the above simplification via reduction of the set of condition attributes. This is achieved by removal of any possibly superfluous decision rules which essentially involves reducing the superfluous values of attributes. In other words, the unnecessary conditions have to be separately removed leaving only the core attribute in each decision rule of the

The tabulated version of the above simplified relay *CD*-decision algorithm is shown in Table

In Table 11 the condition attribute of each decision rule in Table 10 is removed one by one. In each removal the resultant rule is cross checked with other rules to find whether they are in conflict (inconsistent). This cross reference with other rules is to figure out whether the remaining condition attribute's value is the same but implication on the decision attribute is different. This process discovers the core attribute(s) that when eliminated causes the corresponding decision rule, or in general the *CD*-decision algorithm, inconsistent and

In summary, Table 12 contains cores of each decision rule. The condition attribute having eliminated value can be said as having no effect whatsoever on the *CD*-decision algorithm and may be termed as "don't care". It can be assigned with a value or otherwise. Combining

consequently invalid (albeit not necessarily in the relay analysis perspective).

duplicate rules and demarcating separate decision classes, Table 13 is obtained. For decision attribute *Trip* = 0, one *minimal* set of decision rules is obtained from

*U bg Z1pu Trip* 

 (1, 2, 3, 20, 21) 0 0 0 (4) 1 0 0 (5, 6, 7) 2 0 0 (8, 9, 10) 1 1 a (11, 12, 13, 14, 15) 0 1 a (16,17,18,19) 0 0 a

indispensability. Thus, the core having both attributes {*bg, Z1pu*} is correct.

**4.5. Protective relay decision algorithm minimization** 

rule 5 (11, 12, 13, 14, 15): *bg*<sup>0</sup> *Trip*<sup>b</sup> rule 6 (16, 17, 18, 19): *bg*<sup>0</sup> *Trip*<sup>b</sup>

algorithm [10].

*bg0 Z1pu0* 

 *Trip0*

10.

*bg*<sup>0</sup> *Z1pu*<sup>0</sup> *bg*<sup>1</sup> *Z1pu*<sup>0</sup> *bg*2 *Trip*<sup>0</sup>

For decision attribute *Trip* = *a*, one minimal set of decision rules is obtained from

```
Z1pu1  Tripb
bg0 Z1pu0  Tripb
```
i.e.

*Z1pu1 bg0 Z1pu0 Tripb* 

The combined form of the *minimal CD*-decision algorithm is

```
bg0 Z1pu0  bg1 Z1pu0  bg2  Trip0
```
or,

*Z1pu*0 (*bg*<sup>0</sup> *bg*1) *bg*2 *Trip*<sup>0</sup>

and

*bg0 Z1pu0 Z1pu1 Tripb* 

The final form of *CD*-decision algorithm can now be easily interpreted as follows:

The decision rule *Z1pu*0 (*bg*<sup>0</sup> *bg*1) *bg*<sup>2</sup> *Trip*0 is interpreted as,

IF *Z1pu =* 0 AND either *bg =* 0 OR *bg =* 1 OR IF *bg =* 2, THEN *Trip =* 0.

The non-trip assertion (*Trip =* 0) is imminent with either one of the following situations:


IF *Z1pu =* 0 AND *bg =* 0 OR IF *Z1pu =* 1, THEN *Trip =* b.

The trip assertion (*Trip =* b) is imminent with either one of the following situations:


Item i. indicates the fact that trip assertion *Trip =* b is still present in the face of the fault and relay pick-up resets (i.e. *bg =* 0 and *Z1pu =* 0) suggests that the preset time duration of the trip assertion is taking place.


Inconsistent Decision System: Rough Set Data Mining Strategy to Extract Decision Algorithm of a Numerical Distance Relay – Tutorial 141

The *D*-core of *C* (i.e. *CORED*(*C*) = {*bg*, *Z1pu*}), determined as the set of all single entries of the *D*-discernibility matrix, provides us with a novel technique in inferring the power system state where the relay has been subjected to. The core, because of its indispensability nature, draws our attention undoubtedly to the fact that an B-G fault has occurred and consequently the relay's Z1 ground distance element has picked up to eliminate it. This eventually translates into the trip decision having patterns such as that presented by the

The degree of dependency *k* < 1 of the relay *CD*-decision algorithm justifies our anticipation of rough classification in the distance relay data. This is evidently shown in some of the rules that have the decision attribute *Trip* remain asserted with the value "b" for a certain preset duration of time. This is irrespective of the presence or absence of the fault via the assertion of attribute "*bg*" and zone 1 element pick-up via the assertion of attribute "*Z1pu*"). The *RED*(*C*,*D*) = {*bg*, *Z1pu*} provides us with the discovery of the relay *CD*-decision algorithm in a simple form. By eliminating any possible superfluous decision rules, isolating condition attributes, one value at a time, further minimization of the algorithm can be

This work was supported by the Ministry of Higher Education Malaysia under the 2011 Fundamental Research Grant Scheme with the project code FRGS/1/11/TK/UPM/03/4 and

[1] Bakar A H A (2001) Disturbance Analysis in TNB Transmission System. Developments

[2] Kumm J J, Weber M S, Schweitzer E O and Hou D (1994) Philosophies For Testing Protective Relays. 48th Annual Georgia Tech Protective Relaying Conference, Atlanta,

[3] Kezunovic, M (2001) Section II: Equipment characteristics. IEEE Tutorial on Automated

in Power System Protection Conference. IEE Publication No.479: 339-442

Fault Analysis. Texas A&M University, College Station, USA, July, pp 5-9.

*Department of Electrical and Electronic Engineering, Faculty of Engineering,* 

*Department of Electrical and Electronic Engineering, Faculty of Engineering,* 

attribute *Trip*.

performed.

Ishak Aris

**Author details** 

Mohammad Lutfi Othman

**Acknowledgement** 

project file code 1057FR.

**6. References** 

Georgia.

*Universiti Putra Malaysia, Serdang, Malaysia* 

*Universiti Putra Malaysia, Serdang, Malaysia* 

**Table 11.** Eliminating unnecessary condition attribute in decision rules


**Table 12.** Cores of decision rules


**Table 13.** Cores of decision rules

#### **5. Conclusion**

Rough set theory has been proven to be an essentially useful mathematical tool in intelligent data mining analysis of inconsistent and vague protective relay data pattern as evident in the rough classification involved in the assertion of the trip decision attribute. The adoption of rough set theory is managed under supervised learning.

A single *D*-reduct of *C* (i.e. *REDD*(*C*) = {*bg*, *Z1pu*}) has been discovered after formulating the attribute priority of the distance relay operation to trim the *DS*. *REDD*(*C*) can alternatively be used to represent exactly the same equivalence relation *UIND*(*D*) represented by the whole set of attributes *C*. Relying on the reduced number of condition attributes represented by *REDD*(*C*), relay analysis that can be achieved at ease.

The *D*-core of *C* (i.e. *CORED*(*C*) = {*bg*, *Z1pu*}), determined as the set of all single entries of the *D*-discernibility matrix, provides us with a novel technique in inferring the power system state where the relay has been subjected to. The core, because of its indispensability nature, draws our attention undoubtedly to the fact that an B-G fault has occurred and consequently the relay's Z1 ground distance element has picked up to eliminate it. This eventually translates into the trip decision having patterns such as that presented by the attribute *Trip*.

The degree of dependency *k* < 1 of the relay *CD*-decision algorithm justifies our anticipation of rough classification in the distance relay data. This is evidently shown in some of the rules that have the decision attribute *Trip* remain asserted with the value "b" for a certain preset duration of time. This is irrespective of the presence or absence of the fault via the assertion of attribute "*bg*" and zone 1 element pick-up via the assertion of attribute "*Z1pu*").

The *RED*(*C*,*D*) = {*bg*, *Z1pu*} provides us with the discovery of the relay *CD*-decision algorithm in a simple form. By eliminating any possible superfluous decision rules, isolating condition attributes, one value at a time, further minimization of the algorithm can be performed.

## **Author details**

140 Advances in Data Mining Knowledge Discovery and Applications

Removed attribute

*bg Z1pu*

*U bg Z1pu Trip*  1 0 0 0 2 1 0 0 3 2 - 0 4 -1 b 5 -1 b 6 00 b

*U bg Z1pu Trip*  1 0 0 0 2 1 0 0 3 2 - 0 4 (4,5) - 1 b 6 0 0 b

Rough set theory has been proven to be an essentially useful mathematical tool in intelligent data mining analysis of inconsistent and vague protective relay data pattern as evident in the rough classification involved in the assertion of the trip decision attribute. The adoption

A single *D*-reduct of *C* (i.e. *REDD*(*C*) = {*bg*, *Z1pu*}) has been discovered after formulating the attribute priority of the distance relay operation to trim the *DS*. *REDD*(*C*) can alternatively be used to represent exactly the same equivalence relation *UIND*(*D*) represented by the whole set of attributes *C*. Relying on the reduced number of condition attributes represented by

Resultant rule to check

*Z1pu*0 *Trip*<sup>0</sup> rule 6: *Z1pu*0 *Trip*<sup>b</sup>

*Z1pu*0 *Trip*<sup>0</sup> rule 6: *Z1pu*<sup>0</sup> *Trip*<sup>b</sup>

*Z1pu*0 *Trip*<sup>0</sup> rule 6: *Z1pu*0 *Trip*<sup>b</sup>

*Z1pu*0 *Trip*<sup>b</sup> rule 1: *Z1pu*<sup>0</sup> *Trip*<sup>0</sup>

At least one other rule in conflict

} *bg*, *Z1pu bg*<sup>0</sup> *Trip*<sup>0</sup> rule 5: *bg*<sup>0</sup> *Trip*<sup>b</sup>

} *bg*, *Z1pu bg*1 *Trip*<sup>0</sup> rule 4: *bg*<sup>1</sup> *Trip*<sup>b</sup>

} *bg*, *Z1pu bg*0 *Trip*<sup>b</sup> rule 1: *bg*<sup>0</sup> *Trip*<sup>0</sup>

} *bg bg*2 *Trip*<sup>0</sup> none

*Z1pu*1 *Trip*<sup>b</sup> none } *Z1pu bg*1 *Trip*<sup>b</sup> rule 2: *bg*<sup>1</sup> *Trip*<sup>0</sup>

*Z1pu*1 *Trip*<sup>b</sup> none } *Z1pu bg*0 *Trip*<sup>b</sup> rule 1: *bg*0 *Trip*<sup>0</sup>

Core attribute

*CD*-decision algorithm

rule 1 (1, 2, 3, 20, 21): *bg*<sup>0</sup> *Z1pu*0 *Trip*0 {

rule 2 (4): *bg*<sup>1</sup> *Z1pu*<sup>0</sup> *Trip*0 {

rule 3 (5, 6, 7): *bg*<sup>2</sup> *Z1pu*0 *Trip*0 {

rule 4 (8, 9, 10): *bg*<sup>1</sup> *Z1pu*1 *Trip*a {

rule 5 (11, 12, 13, 14, 15): *bg*<sup>0</sup> *Z1pu*1 *Trip*b {

rule 6 (16, 17, 18, 19): *bg*<sup>0</sup> *Z1pu*0 *Trip*b {

**Table 12.** Cores of decision rules

**Table 13.** Cores of decision rules

**5. Conclusion** 

**Table 11.** Eliminating unnecessary condition attribute in decision rules

of rough set theory is managed under supervised learning.

*REDD*(*C*), relay analysis that can be achieved at ease.

Mohammad Lutfi Othman

*Department of Electrical and Electronic Engineering, Faculty of Engineering, Universiti Putra Malaysia, Serdang, Malaysia* 

Ishak Aris *Department of Electrical and Electronic Engineering, Faculty of Engineering, Universiti Putra Malaysia, Serdang, Malaysia* 

## **Acknowledgement**

This work was supported by the Ministry of Higher Education Malaysia under the 2011 Fundamental Research Grant Scheme with the project code FRGS/1/11/TK/UPM/03/4 and project file code 1057FR.

## **6. References**

	- [4] Kezunovic, M (2001) Section III: Scope of Analysis. IEEE Tutorial on Automated Fault Analysis. Texas A&M University, College Station, USA, July, pp 10-13.

**Chapter 6** 

© 2012 Wen and Yang, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2012 Wen and Yang, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**An Unsupervised Classification Method** 

**Based on Spectral Data Mining** 

Xingping Wen and Xiaofeng Yang

http://dx.doi.org/10.5772/50135

**1. Introduction** 

Additional information is available at the end of the chapter

**for Hyperspectral Remote Sensing Image** 

Hyperspectral remote sensing is one of the most significant recent breakthroughs in remote sensing. It obtains image in a large number (usually more than 40), narrow (typically 10 to 20 nm spectral resolution) and contiguous spectral bands to enable the extraction of spectral information at a pixel scale, so it can produce data with sufficient spectral resolution for the direct recognition those materials with diagnostic spectral features [1]. Usually classification method of hyperspectral remote sensing data are divided into two categories [2]: using subpixel classification techniques [3] and spectral matching techniques [4]. In the former, the images should not need to atmospheric correction, however, due to higher dimension of hyperspectral image, it will lead to dimensionality disaster and Hughes phenomenon [5, 6] which refer to the fact that with the number of spectral bands increased the sample size required for training set grows exponentially. The solution methods usually are increasing sample size, thus this will cost a lot of human and material resources. Another simple but sometimes effective way to solve this problem is dimension reduction of hyperspectral data, but some useful information will be lost. Furthermore, it is hard to solve mixed pixels. In the latter, matched filtering method is successfully used in information extraction from hyperspectral remote sensing image. It classifies by computing the similarity of the pixel spectrum and the reference spectrum, and it needs no sample data but the image data should be atmospheric corrected beforehand. These methods based on the hypothesis that dark currents of the sensor and path radiation are removed and all spectra data have been calibrated to apparent reflectance. However, it is only the ideal condition for these effects are hard to be removed completely, so some mistakes will be caused due to atmospheric influence, especially for low reflectivity ground objects. This chapter proposed an unsupervised classification for hyperspectral remote sensing image. It can effectively extract

