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

1. Introduction

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 (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.

• 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]:

• U, i.e. the universe denoted as U = {t₁, t₂, t₃, …, t_m}, is a finite set of relay events (t_i’s).

• Q = C ∪ D is a non-empty finite union set of condition and decision attributes,

  1. condition attributes (c_i ⊂ C) indicate the internally various multifunctional protective elements and analog measurands,

  2. decision attribute (d_i ⊂ D) indicates the trip output of the relay, such that q: U → V_q for every q ∈ Q.

• V = U_q∈Q V_q, where V_q is a set of values (domain) of the attribute q.

• f: U×Q → V called information function is a total function such that f(t,q) ∈ V_q for every t ∈ U, q ∈ Q. Any pair (q,v) is called descriptor in DS, where q ∈ Q and v ∈ V_q,.

3.1. Relay decision system indiscernibility relation

If a set of attributes P ⊆ Q = C ∪ D and f(t_x,q) = f(t_y,q) where t_x, t_y ∈ U, then for every q ∈ P, t_x and t_y 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. U│IND(P) denotes the family of all equivalence classes of relation IND(P). IND(P) and U│IND(P) can be formulated as

$$IND\left(P\right)=\{\left(t_x,t_y\right)\in U^{\text{2}}|\forall q\in P,q\left(t_x\right)=q\left(t_y\right)\},$$E1

$$U|IND\left(P\right)=\otimes \{q\in P\left|U\right|IND\left(\left\{q\right\}\right)\},$$E2

where,

$$A\otimes B=\{X\cap Y|\forall X\in A,\forall Y\in B,X\cap Y\ne Ø\}.$$E3

U│IND(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(P). The first step in classification with rough sets is the construction of elementary sets [11]. A description of P-elementary set X ∈ U│IND(P) in terms of values of attributes from P is denoted as Des_P(X), i.e.

$$De{s}_P\left(X\right)=\{\left(q,v\right):f\left(t,q\right)=v,\forall t\in X,\forall q\in P\}$$E4

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,

$$\underset{\_}{P}T=\cup \{X\in U|IND\left(P\right):X\subseteq T\},$$E5

is defined as the union of all elementary sets in [t]_IND(P) which are contained in T. For any relay event t_i of the lower approximation of T with respect to the set of attributes P (i.e., t_i ∈ $\overline{\text{P}}$T), it positively certain belongs to T.

P-upper approximation of T expressed as,

$$\overline{P}T=\cup \{X\in U|IND\left(P\right):X\cap T\ne Ø\},$$E6

is defined as the union of elementary sets in [t]_IND(P) which have a non-empty intersection with T. For any relay event t_i of the upper approximation of T with respect to the set of attributes P (i.e., t_i ∈ $\overline{\text{P}}$T), it may possibly belong to T.

P-boundary of set T expressed as,

$$B{N}_P\left(T\right)=\overline{P}T-\underset{\_}{P}T$$E7

is the difference between $\overline{\text{P}}$T and $\overline{\text{P}}$T. The set of elements t_i 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].

• POS_P(T) = $\overline{\text{P}}$T, described as P-positive region of T, is the set of relay events which can be classified with certainty in the approximated set T.

• NEG_P(T) = U - $\overline{\text{P}}$T, 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 belonging to the complement of T).

• BN_P(T) = $\overline{\text{P}}$T - $\overline{\text{P}}$T, 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 $\overline{\text{P}}$ as far as the attributes P are concerned. The set T is crisp if there are no boundary sets, i.e. BN_P(T) = Ø (empty set), which otherwise it is rough.

3.3. Approximation accuracy and quality

α_P(T), the accuracy of the rough set representation of a target set of relay events T, is formulated as [10]

$${\alpha }_P(T) = \frac{\left|\underset{\_}{P}T\right|}{\left|\overline{P}T\right|}=\frac{card(\underset{\_}{P}T)}{card(\overline{P}T)} .$$E8

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 ≤ α_P(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 α_P(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]:

• Roughly definable T in U given $\overline{\text{P}}$T ≠ Ø and $\overline{\text{P}}$T ≠ U (Ø denotes empty set)

• Externally undefinable T in U given $\overline{\text{P}}$T ≠ Ø and $\overline{\text{P}}$T = U

• Internally undefinable T in U given $\overline{\text{P}}$T = Ø and $\overline{\text{P}}$T ≠ U

• Totally undefinable T in U given $\overline{\text{P}}$T = Ø and $\overline{\text{P}}$T = U

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

$${\gamma }_P(T) = \frac{\left|\underset{\_}{P}T\right|}{U} = \frac{card(\underset{\_}{P}T)}{card(U)},$$E9

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.

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:

• The reduced attribute set RED induces the same equivalence classes as those induced by full attribute set P. This is denoted as [t]_IND(RED) = [t]_IND(P).

• Attribute set RED is minimal in the sense that [t]_IND(RED-A) ≠ [t]_IND(P) for any attribute A∈RED. This suggests that no attribute can be dispensed from set RED without modifying the equivalence classes [t]_IND(P) [16].

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 A∈RED in this case is A∈CORE.

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 d_ij 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, …, t_m} = U, identify as labels of relay decision rules.

Formally, let

• U│IND(C) be condition classes in relay DS (a family of all C-elementary sets), denoted by X_i (i = 1, …, k),

• U│IND(D) be decision classes in relay DS (a family of all D-elementary sets), denoted by Y_j (j = 1, …, n).

Then, Des_C(X_i) ⇒ Des_D(Y_j) is called relay CD-decision rule. For simplicity, C ⇒ D. (As aforementioned, Des_P(X) = {(q,v) : f(x,q) = v, $\forall$x ∈ X, $\forall$q ∈ P} which denotes a description of P-elementary set X ∈ U│IND(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 Y_j (j = 1,…, n) is denoted by:

$$\left\{r_{ij}\right\}=\{De{s}_C\left(X_i\right)\Rightarrow De{s}_D\left(Y_j\right):X_i\cap Y_j\ne Ø,i=\text{ 1},\dots ,\text{ k}\}$$E10

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:

• Relay rule {r_ij} is deterministic in DS if and only if X_i ⊆ Y_j, and

• Relay rule {r_ij} is nondeterministic in DS, otherwise.

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 ⇒_k D and can be formally defined as:

$$C{\Rightarrow }_{\text{k}}D|\text{ k }=\gamma \left(C,D\right)=\frac{\left|PO{S}_{C}D\right|}{\left|U\right|}$$E11

(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 ⇒₁ 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 γ(C, D) = γ(C′, D) is valid (i.e. similar in dependency). RED_D(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 U│IND(D) discernable by the entire set of attributes are called D-reducts [11]. The following notations are, thus, valid:

• If POS_C(D) = POS_(C-{ci})(D), an attribute c_i ∈ C is D-dispensable in C. c_i is D-superfluous if it exerts no influence on the lower approximation of D. Otherwise the attribute c_i is D-indispensable in C.

• If C is D-independent, then all attributes c_i ∈ C are D-indispensable in C and called the D-core of C which is denoted as CORE_D(C).

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

  $$COR{E}_D\left(C\right)=\cap RE{D}_D\left(C\right)$$E12

  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 t_i and t_j that do not belong to the same equivalence class of the relation U│IND(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: U×Q → 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}.

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
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)

The attribute names are described as follows:

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

• 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.

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

• V_ag, V_bg, V_cg, = {1, 2, 3, 4}.

• V_Z1pu, V_Z2pu, V_Z3pu, V_Z4pu, V_Z1trp, V_Z2trp, V_Z3trp, V_Z4trp = {0, 1}.

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

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.

U│D	Trip
{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

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.

U│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, 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(C) 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. U│C 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,

$$\underset{\_}{C}{D}_1=\left\{t_4\right\}\cup \left\{t_5,t_6,t_7\right\}=\left\{t_4,t_5,t_6,t_7\right\}$$E13

$$\begin{array}{c}\underset{\_}{C}{D}_2=\{t_{\text{8}},t_{\text{9}},t_{\text{1}0}\}\cup \left\{{\text{t}}_{\text{11}},{\text{ t}}_{\text{12}},{\text{ t}}_{\text{13}},{\text{ t}}_{\text{14}},{\text{ t}}_{\text{15}}\right\}\cup \left\{t_{\text{16}}\right\}\\ =\{t_{\text{8}},t_{\text{9}},t_{\text{1}0},{\text{ t}}_{\text{11}},{\text{ t}}_{\text{12}},{\text{ t}}_{\text{13}},{\text{ t}}_{\text{14}},{\text{ t}}_{\text{15}},t_{\text{16}}\}\end{array}$$E14

$$\begin{array}{c} \overline{C}{D}_1=\{{\text{t}}_{\text{1}},{\text{ t}}_{\text{2}},{\text{ t}}_{\text{3}},{\text{ t}}_{\text{17}},{\text{ t}}_{\text{18}},{\text{ t}}_{\text{19}},{\text{ t}}_{\text{2}0},{\text{ t}}_{\text{21}}\}\cup \left\{t_{\text{4}}\right\}\cup \{t_{\text{5}},t_{\text{6}},t_{\text{7}}\}\\ =\{{\text{t}}_{\text{1}},{\text{ t}}_{\text{2}},{\text{ t}}_{\text{3}},t_{\text{4}},t_{\text{5}},t_{\text{6}},t_{\text{7}},{\text{t}}_{\text{17}},{\text{ t}}_{\text{18}},{\text{ t}}_{\text{19}},{\text{ t}}_{\text{2}0},{\text{ t}}_{\text{21}}\}\end{array}$$E15

$$\begin{array}{c}\overline{C}{D}_2=\{{\text{t}}_{\text{1}},{\text{ t}}_{\text{2}},{\text{ t}}_{\text{3}},{\text{ t}}_{\text{17}},{\text{ t}}_{\text{18}},{\text{ t}}_{\text{19}},{\text{ t}}_{\text{2}0},{\text{ t}}_{\text{21}}\}\cup \{t_{\text{8}},t_{\text{9}},t_{\text{1}0}\}\cup \left\{{\text{t}}_{\text{11}},{\text{ t}}_{\text{12}},{\text{ t}}_{\text{13}},{\text{ t}}_{\text{14}},{\text{ t}}_{\text{15}}\right\}\cup \left\{t_{\text{16}}\right\}\\ =\{{\text{t}}_{\text{1}},{\text{ t}}_{\text{2}},{\text{ t}}_{\text{3}},t_{\text{8}},t_{\text{9}},t_{\text{1}0},{\text{ t}}_{\text{11}},{\text{ t}}_{\text{12}},{\text{ t}}_{\text{13}},{\text{ t}}_{\text{14}},{\text{ t}}_{\text{15}},t_{\text{16}},{\text{ t}}_{\text{17}},{\text{ t}}_{\text{18}},{\text{ t}}_{\text{19}},{\text{ t}}_{\text{2}0},{\text{ t}}_{\text{21}}\}\end{array}$$E16

$${\alpha }_{\text{C}}(D_{\text{1}}) =\left|\frac{\underset{\_}{C}{D}_1}{\overline{C}{D}_1}\right|=\frac{card(\underset{\_}{C}{D}_1)}{card(\overline{C}{D}_1)}=\frac{4}{12}=0.33$$E17

$${\alpha }_C\left(D_2\right)=\left|\frac{\underset{\_}{C}{D}_2}{\overline{C}{D}_2}\right|=\frac{card(\underset{\_}{C}{D}_2)}{card(\overline{C}{D}_2)}=\frac{9}{17}=0.53$$E18

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:

$${\alpha }_C\left(D\right) =\frac{{{\displaystyle \sum }}_{i=1}^{2}card(\underset{\_}{C}{D}_i)}{{{\displaystyle \sum }}_{i=1}^{2}card(\overline{C}{D}_i)}=\frac{4 + 9}{12 + 17}=0.45$$E19

$${\gamma }_C\left(D\right)=\frac{{{\displaystyle \sum }}_{i=1}^{2}card(\underset{\_}{C}{D}_i)}{card(U)}=\frac{4 + 9}{9 + 12}=0.62$$E20

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 U│IND(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.

U

t1

t2

t3

t4

t5

t6

t7

t8

t9

t10

t11

t12

t13

t14

t15

t16

t17

t18

t19

t20

t21

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

{bg, Z1pu, Z2pu, Z3pu, Z1trp}

{bg, Z1pu, Z2pu, Z3pu, Z1trp}

{bg, Z1pu, Z2pu, Z3pu, Z1trp}

{Z1pu, Z1trp}

{bg, Z1pu, Z1trp}

{bg, Z1pu, Z1trp}

{bg, Z1pu, Z1trp}

Ø

t9

{bg, Z1pu, Z2pu, Z3pu, Z1trp}

{bg, Z1pu, Z2pu, Z3pu, Z1trp}

{bg, Z1pu, Z2pu, Z3pu, Z1trp}

{Z1pu, Z1trp}

{bg, Z1pu, Z1trp}

{bg, Z1pu, Z1trp}

{bg, Z1pu, Z1trp}

Ø

Ø

t10

{bg, Z1pu, Z2pu, Z3pu, Z1trp}

{bg, Z1pu, Z2pu, Z3pu, Z1trp}

{bg, Z1pu, Z2pu, Z3pu, Z1trp}

{Z1pu, Z1trp}

{bg, Z1pu, Z1trp}

{bg, Z1pu, Z1trp}

{bg, Z1pu, Z1trp}

Ø

Ø

Ø

t11

{Z1pu, Z2pu, Z3pu}

{Z1pu, Z2pu, Z3pu}

{Z1pu, Z2pu, Z3pu}

{bg, Z1pu}

{bg, Z1pu}

{bg, Z1pu}

{bg, Z1pu}

Ø

Ø

Ø

Ø

t12

{Z1pu, Z2pu, Z3pu}

{Z1pu, Z2pu, Z3pu}

{Z1pu, Z2pu, Z3pu}

{bg, Z1pu}

{bg, Z1pu}

{bg, Z1pu}

{bg, Z1pu}

Ø

Ø

Ø

Ø

Ø

t13

{Z1pu, Z2pu, Z3pu}

{Z1pu, Z2pu, Z3pu}

{Z1pu, Z2pu, Z3pu}

{bg, Z1pu}

{bg, Z1pu}

{bg, Z1pu}

{bg, Z1pu}

Ø

Ø

Ø

Ø

Ø

Ø

t14

{Z1pu, Z2pu, Z3pu}

{Z1pu, Z2pu, Z3pu}

{Z1pu, Z2pu, Z3pu}

{bg, Z1pu}

{bg, Z1pu}

{bg, Z1pu}

{bg, Z1pu}

Ø

Ø

Ø

Ø

Ø

Ø

Ø

t15

{Z1pu, Z2pu, Z3pu}

{Z1pu, Z2pu, Z3pu}

{Z1pu, Z2pu, Z3pu}

{bg, Z1pu}

{bg, Z1pu}

{bg, Z1pu}

{bg, Z1pu}

Ø

Ø

Ø

Ø

Ø

Ø

Ø

Ø

t16

{Z3pu}

{Z3pu}

{Z3pu}

{bg, Z2pu}

{bg, Z2pu}

{bg, Z2pu}

{bg, Z2pu}

Ø

Ø

Ø

Ø

Ø

Ø

Ø

Ø

Ø

t17

Ø

Ø

Ø

{bg, Z2pu, Z3pu}

{bg, Z2pu, Z3pu}

{bg, Z2pu, Z3pu}

{bg, Z2pu, Z3pu}

Ø

Ø

Ø

Ø

Ø

Ø

Ø

Ø

Ø

Ø

t18

Ø

Ø

Ø

{bg, Z2pu, Z3pu}

{bg, Z2pu, Z3pu}

{bg, Z2pu, Z3pu}

{bg, Z2pu, Z3pu}

Ø

Ø

Ø

Ø

Ø

Ø

Ø

Ø

Ø

Ø

Ø

t19

Ø

Ø

Ø

{bg, Z2pu, Z3pu}

{bg, Z2pu, Z3pu}

{bg, Z2pu, Z3pu}

{bg, Z2pu, Z3pu}

Ø

Ø

Ø

Ø

Ø

Ø

Ø

Ø

Ø

Ø

Ø

Ø

t20

Ø

Ø

Ø

Ø

Ø

Ø

Ø

{bg, Z1pu, Z2pu, Z3pu, Z1trp}

{bg, Z1pu, Z2pu, Z3pu, Z1trp}

{bg, Z1pu, Z2pu, Z3pu, Z1trp}

{Z1pu, Z2pu, Z3pu}

{Z1pu, Z2pu, Z3pu}

{Z1pu, Z2pu, Z3pu}

{Z1pu, Z2pu, Z3pu}

{Z1pu, Z2pu, Z3pu}

{Z3pu}

Ø

Ø

Ø

Ø

t21

Ø

Ø

Ø

Ø

Ø

Ø

Ø

{bg, Z1pu, Z2pu, Z3pu, Z1trp}

{bg, Z1pu, Z2pu, Z3pu, Z1trp}

{bg, Z1pu, Z2pu, Z3pu, Z1trp}

{Z1pu, Z2pu, Z3pu}

{Z1pu, Z2pu, Z3pu}

{Z1pu, Z2pu, Z3pu}

{Z1pu, Z2pu, Z3pu}

{Z1pu, Z2pu, Z3pu}

{Z3pu}

Ø

Ø

Ø

Ø

Ø

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.:

f_C(D )

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

= Z3pu × (Z1pu+Z1trp) (bg+Z1pu) (bg+Z2pu) × (bg+Z1pu) (bg+Z2pu) × (bg+Z1pu+Z2pu+Z3pu+Z1trp) × (Z1pu+Z2pu+Z3pu) × Z3pu → The final Conjunctive Normal Form (CNF)

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

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

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

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

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

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

→ The final Disjunctive Normal Form (DNF) of f_C(D)

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

Normalization in its final normal form, the last Boolean expression f_C(D) is recognized as Disjunctive Normal Form (DNF). DNF is analogous to Sum Of Product (SOP) boolean algebra in digital electronics logic. f_C(D) in DNF form is an alternative representation of the DS in which all its constituents are the D-reducts of C (i.e. RED_D(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 RED_D(C) of the above final f_C(D) reveals that either one of the D-reducts of C can be used alternatively to represent exactly the same equivalence relation U│IND(D) of the DS as that represented by the whole set of attributes C, i.e.,

$$\begin{array}{c}RE{D}_D\left(C\right)=\left\{bg,Z1pu,Z3pu\right\},\left\{bg,Z3pu,Z1trp\right\},\left\{bg,Z1pu,Z3pu\right\},\\ \left\{bg,Z1pu,Z3pu,Z1trp\right\},\left\{bg,Z1pu,Z2pu,Z3pu\right\},\left\{bg,Z2pu,Z3pu,Z1trp\right\},\\ \left\{Z1pu,Z2pu,Z3pu\right\},\left\{Z1pu,Z2pu,Z3pu,Z1trp\right\}\end{array}$$E21

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

• 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 CORE_D(C) = ∩RED_D(C) = Z3p, or

• Taking intersection of all D-reducts of C, i.e. CORE_D(C) = ∩RED_D(C) = Z3pu

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

CORE_D(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 U│C with respect to C) would have been,

$$COR{E}_{U|C}\left(C\right)=\left\{bg\right\}$$E22

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.

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

Cases of concurrence	Condition Attributes, ci∈C
Z1pu	Z2pu	Z3pu	Z4pu	Z1trp	Z2trp	Z3trp	Z4trp	Most significant attribute
Case 1	+	+	+		+				Z1pu
Case 1´	+	+	+			*			Z2pu
Case 2		+	+			*			Z2pu
Case 2´		+	+				*		Z3pu
Case 3			+				*		Z3pu
Case 4				+				*	Z4pu

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.

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.

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

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).

U│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, 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 U│IND(D). As before, similar consideration is taken in discerning events appearing only in different classes in D-space.

U

t1

t2

t3

t4

t5

t6

t7

t8

t9

t10

t11

t12

t13

t14

t15

t16

t17

t18

t19

t20

t21

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

{bg, Z1pu, Z1trp}

{bg, Z1pu, Z1trp}

{bg, Z1pu, Z1trp}

{Z1pu, Z1trp}

{bg, Z1pu, Z1trp}

{bg, Z1pu, Z1trp}

{bg, Z1pu, Z1trp}

Ø

t9

{bg, Z1pu, Z1trp}

{bg, Z1pu, Z1trp}

{bg, Z1pu, Z1trp}

{Z1pu, Z1trp}

{bg, Z1pu, Z1trp}

{bg, Z1pu, Z1trp}

{bg, Z1pu, Z1trp}

Ø

Ø

t10

{bg, Z1pu, Z1trp}

{bg, Z1pu, Z1trp}

{bg, Z1pu, Z1trp}

{Z1pu, Z1trp}

{bg, Z1pu, Z1trp}

{bg, Z1pu, Z1trp}

{bg, Z1pu, Z1trp}

Ø

Ø

Ø

t11

{Z1pu}

{Z1pu}

{Z1pu}

{bg, Z1pu}

{bg, Z1pu}

{bg, Z1pu}

{bg, Z1pu}

Ø

Ø

Ø

Ø

t12

{Z1pu}

{Z1pu}

{Z1pu}

{bg, Z1pu}

{bg, Z1pu}

{bg, Z1pu}

{bg, Z1pu}

Ø

Ø

Ø

Ø

Ø

t13

{Z1pu}

{Z1pu}

{Z1pu}

{bg, Z1pu}

{bg, Z1pu}

{bg, Z1pu}

{bg, Z1pu}

Ø

Ø

Ø

Ø

Ø

Ø

t14

{Z1pu}

{Z1pu}

{Z1pu}

{bg, Z1pu}

{bg, Z1pu}

{bg, Z1pu}

{bg, Z1pu}

Ø

Ø

Ø

Ø

Ø

Ø

Ø

t15

Z1pu}

Z1pu}

Z1pu}

{bg, Z1pu}

{bg, Z1pu}

{bg, Z1pu}

{bg, Z1pu}

Ø

Ø

Ø

Ø

Ø

Ø

Ø

Ø

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

t20

Ø

Ø

Ø

Ø

Ø

Ø

Ø

{bg, Z1pu, Z1trp}

{bg, Z1pu, Z1trp}

{bg, Z1pu, Z1trp}

{Z1pu}

{Z1pu}

{Z1pu}

{Z1pu}

{Z1pu}

Ø

Ø

Ø

Ø

Ø

t21

Ø

Ø

Ø

Ø

Ø

Ø

Ø

{bg, Z1pu, Z1trp}

{bg, Z1pu, Z1trp}

{bg, Z1pu, Z1trp}

{Z1pu}

{Z1pu}

{Z1pu}

{Z1pu}

{Z1pu}

Ø

Ø

Ø

Ø

Ø

Ø

Table 8.

D-discernibility matrix of modified C

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

f_C(D)

=(bg+Z1pu+Z1trp) (Z1pu) × (Z1pu+Z1trp) (bg+Z1pu) (bg) × (bg+Z1pu+Z1trp) (bg+Z1pu) (bg) × (bg+Z1pu+ Z1trp) × Z1pu

=(Z1pu) × (Z1pu+Z1trp) (bg) × (bg) × (bg+Z1pu+Z1trp) × Z1pu

=(Z1pu) (bg) → The final Disjunctive Normal Form (DNF) of f_C(D)

There is only one D-reduct of C, RED_D(C) = {bg, Z1pu}. As shown in Table 9, it can alternatively be used to represent exactly similar equivalence relation U│IND(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 CORE_D(C) = ∩RED_D(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 CORE_D(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.

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

Table 9.

Equivalent decision table with respect to RED_D(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),

$$k =\frac{card POS(C,D)}{card (CD-decision algorithm)}$$E23

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 → Trip_b

rule 9:ag₀ bg₁ cg₀ Z1pu₁ Z4pu₀ Z1trp₁ Z2trp₀ Z3trp₀ Z4trp₀ → Trip_b

rule 10:ag₀ bg₁ cg₀ Z1pu₁ Z4pu₀ Z1trp₁ Z2trp₀ Z3trp₀ Z4trp₀ → Trip_b

rule 11:ag₀ bg₀ cg₀ Z1pu₁ Z4pu₀ Z1trp₀ Z2trp₀ Z3trp₀ Z4trp₀ → Trip_b

rule 12:ag₀ bg₀ cg₀ Z1pu₁ Z4pu₀ Z1trp₀ Z2trp₀ Z3trp₀ Z4trp₀ → Trip_b

rule 13:ag₀ bg₀ cg₀ Z1pu₁ Z4pu₀ Z1trp₀ Z2trp₀ Z3trp₀ Z4trp₀ → Trip_b

rule 14:ag₀ bg₀ cg₀ Z1pu₁ Z4pu₀ Z1trp₀ Z2trp₀ Z3trp₀ Z4trp₀ → Trip_b

rule 15:ag₀ bg₀ cg₀ Z1pu₁ Z4pu₀ Z1trp₀ Z2trp₀ Z3trp₀ Z4trp₀ → Trip_b

rule 16:ag₀ bg₀ cg₀ Z1pu₀ Z4pu₀ Z1trp₀ Z2trp₀ Z3trp₀ Z4trp₀ → Trip_b

rule 17:ag₀ bg₀ cg₀ Z1pu₀ Z4pu₀ Z1trp₀ Z2trp₀ Z3trp₀ Z4trp₀ → Trip_b

rule 18:ag₀ bg₀ cg₀ Z1pu₀ Z4pu₀ Z1trp₀ Z2trp₀ Z3trp₀ Z4trp₀ → Trip_b

rule 19:ag₀ bg₀ cg₀ Z1pu₀ Z4pu₀ Z1trp₀ Z2trp₀ Z3trp₀ Z4trp₀ → Trip_b

rule 20:ag₀ bg₀ cg₀ Z1pu₀ Z4pu₀ Z1trp₀ Z2trp₀ Z3trp₀ Z4trp₀ → Trip₀

rule 21:ag₀ bg₀ cg₀ Z1pu₀ Z4pu₀ Z1trp₀ Z2trp₀ Z3trp₀ Z4trp₀ → Trip₀

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 pick-up (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 C′D-decision algorithm is independent and consistent, i.e. POS(C′,D) = POS(C,D). Therefore, the following terms are valid:

• C′D-decision algorithm is reduct of CD-decision algorithm.

• The set of all reducts of CD-decision algorithm is called RED(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 CORE_D(C) = ∩RED_D(C)).

The modified DS in Table 6 had found its only reduct of condition attributes, RED(C,D) = RED_D(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.,

rule 1:bg₀ Z1pu₀ → Trip₀

rule 2:bg₀ Z1pu₀ → Trip₀

rule 3:bg₀ Z1pu₀ → Trip₀

rule 4:bg₁ Z1pu₀ → Trip₀

rule 5:bg₂ Z1pu₀ → Trip₀

rule 6:bg₂ Z1pu₀ → Trip₀

rule 7:bg₂ Z1pu₀ → Trip₀

rule 8:bg₁ Z1pu₁ → Trip_b

rule 9:bg₁ Z1pu₁ → Trip_b

rule 10:bg₁ Z1pu₁ → Trip_b

rule 11:bg₀ Z1pu₁ → Trip_b

rule 12:bg₀ Z1pu₁ → Trip_b

rule 13:bg₀ Z1pu₁ → Trip_b

rule 14:bg₀ Z1pu₁ → Trip_b

rule 15:bg₀ Z1pu₁ → Trip_b

rule 16:bg₀ Z1pu₀ → Trip_b

rule 17:bg₀ Z1pu₀ → Trip_b

rule 18:bg₀ Z1pu₀ → Trip_b

rule 19:bg₀ Z1pu₀ → Trip_b

rule 20:bg₀ Z1pu₀ → Trip₀

rule 21:bg₀ Z1pu₀ → Trip₀

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):bg₀ Z1pu₀ → Trip₀

rule 2′ (4):bg₁ Z1pu₀ → Trip₀

rule 3′ (5, 6, 7):bg₂ Z1pu₀ → Trip₀

rule 4′ (8, 9, 10):bg₁ Z1pu₁ → Trip_b

rule 5′ (11, 12, 13, 14, 15):bg₀ Z1pu₁ → Trip_b

rule 6′ (16, 17, 18, 19):bg₀ Z1pu₀ → Trip_b

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:

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

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

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:

rule 1′ (1, 2, 3, 20, 21):Z1pu₀ → Trip₀

rule 2′ (4):Z1pu₀ → Trip₀

rule 3′ (5, 6, 7):Z1pu₀ → Trip₀

rule 4′ (8, 9, 10):Z1pu₁ → Trip_b

rule 5′ (11, 12, 13, 14, 15):Z1pu₁ → Trip_b

rule 6′ (16, 17, 18, 19):Z1pu₀ → Trip_b

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

rule 1′ (1, 2, 3, 20, 21):bg₀ → Trip₀

rule 2′ (4):bg₁ → Trip₀

rule 3′ (5, 6, 7):bg₂ → Trip₀

rule 4′ (8, 9, 10):bg₁ → Trip_b

rule 5′ (11, 12, 13, 14, 15):bg₀ → Trip_b

rule 6′ (16, 17, 18, 19):bg₀ → Trip_b

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.

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.

U	bg	Z1pu	Trip
1′ (1, 2, 3, 20, 21)	0	0	0
2′ (4)	1	0	0
3′ (5, 6, 7)	2	0	0
4′ (8, 9, 10)	1	1	a
5′ (11, 12, 13, 14, 15)	0	1	a
6′ (16,17,18,19)	0	0	a

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.

bg₀ Z1pu₀ ∨ bg₁ Z1pu₀ ∨ bg₂ → Trip₀

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

Z1pu₁ → Trip_b

bg₀ Z1pu₀ → Trip_b

i.e.

Z1pu₁ ∨ bg₀ Z1pu₀ → Trip_b

The combined form of the minimal CD-decision algorithm is

bg₀ Z1pu₀ ∨ bg₁ Z1pu₀ ∨ bg₂ → Trip₀

or,

Z1pu₀ (bg₀ ∨ bg₁) ∨ bg₂ → Trip₀

and

bg₀ Z1pu₀ ∨ Z1pu₁ → Trip_b

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

• The decision rule Z1pu₀ (bg₀ ∨ bg₁) ∨ bg₂ → Trip₀ 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:

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

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

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

• The decision rule bg₀ Z1pu₀ ∨ Z1pu₁ → Trip_b is interpreted as,

  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:

  1. when there is no more fault indication (bg = 0) and relay pick-up element has reset (Z1pu = 0), or

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

  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.

CD-decision algorithm		Removed; attribute		Resultant; rule to; check	At least one other rule in conflict	Core attribute
bg	Z1pu
rule 1′ (1, 2, 3, 20, 21): bg0 Z1pu0 → Trip0	{			Z1pu0 → Trip0	rule 6′: Z1pu0 → Tripb	} bg, Z1pu
		bg0 → Trip0	rule 5′: bg0 → Tripb
rule 2′ (4): bg1 Z1pu0 → Trip0	{			Z1pu0 → Trip0	rule 6′: Z1pu0 → Tripb	} bg, Z1pu
		bg1 → Trip0	rule 4′: bg1 → Tripb
rule 3′ (5, 6, 7): bg2 Z1pu0 → Trip0	{			Z1pu0 → Trip0	rule 6′: Z1pu0 → Tripb	} bg
		bg2 → Trip0	none
rule 4′ (8, 9, 10): bg1 Z1pu1 → Tripa	{			Z1pu1 → Tripb	none	} Z1pu
		bg1 → Tripb	rule 2′: bg1 → Trip0
rule 5′ (11, 12, 13, 14, 15): bg0 Z1pu1 → Tripb	{			Z1pu1 → Tripb	none	} Z1pu
		bg0 → Tripb	rule 1′: bg0 → Trip0
rule 6′ (16, 17, 18, 19): bg0 Z1pu0 → Tripb	{			Z1pu0 → Tripb	rule 1′: Z1pu0 →Trip0	} bg, Z1pu
		bg0 → Tripb	rule 1′: bg0 → Trip0

Table 11.

Eliminating unnecessary condition attribute in decision rules

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

Table 12.

Cores of decision rules

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

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. RED_D(C) = {bg, Z1pu}) has been discovered after formulating the attribute priority of the distance relay operation to trim the DS. RED_D(C) can alternatively be used to represent exactly the same equivalence relation U│IND(D) represented by the whole set of attributes C. Relying on the reduced number of condition attributes represented by RED_D(C), relay analysis that can be achieved at ease.

The D-core of C (i.e. CORE_D(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.

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.