CHAPTER 6Rule Base Design for Fuzzy Consequent

Variable for IDMT Relay

6.1 Introduction

6.2 Fuzzy Variable Design

6.3 Rule Base Development

6.4 Rule Base Inference

6.5 Approximation by Artificial Neural Network

6.5.1 Neuro - Fuzzy System

6.5.2 Neural Network Training

6.6 Approximation of Degree of Membership

6.6.1 Weight Assignment

6.6.2 Neural Network Training

6.7 Simulation Result

6.8 Discussion

The di ference between science and fuzzy subjects is that science requires reasoning .while those

other subjects merely requires scholarship."

Robert Heinlein

Time Enough for Love ,1973

Chapter 6.Rule Base Design of Fuzzy Consequent Variable for IDMT Relay

G.I IntroductionThe essential for designing a fuzzy systems are :

(a) Antecedent variables available at the input part o f the system.

(b) Consequent variables available at the output part o f the system.

(c) Implication rules

In the chapters 4 and 5 the design of antecedent variables for three rules and nine rules

respectively have been discussed in detail .These rules are developed after fuzzifiction of

characteristics curve o f IDMT relay .For every complex and non-linear systems whose input and

output parameters are known a fuzzy based system is recommended to design on the basis o f human

reasoning .This all will eliminate the complicated mathematical calculations while designing the

system.

In this chapter the consequent part o f the fuzzy system (Fuzzy relay sensor) has been discussed.

Here equation 6.1 instead o f IDMT curve is used to develop the rule base.

The antecedent and consequent variables available at the input and output part of a fuzzy

system are implicated using IF-THEN rules on the basis o f behavior characteristics of the system. The

antecedent variable is input variable while The consequent variable is the variable parameter which are

available at the output part of the fuzzy system after implication .

The relay in power system is a sensor, which senses abnormal signals in the power system and

trips the protective circuit. The inverse definite minimum time (IDMT) characteristics o f over current

and earth fault relay may be considered for developing fuzzy sensor. For standard IDMT relays, the

commanded expression for time current relationship is given by-

Here Ot represents Operating time o f relay and

Ipi represents Plug setting value of operating current.

The c h a ra c te r is t ic equation of overcurrent or earth fault IDMT relay given by equation 6.1 is

fuzzified and a number of fuzzy rules are developed for designing the consequent variable o f

proposed fuzzy sensor, and hence the static overcurrent shown in Fig. 4.3 can be replaced by a

proposed sch em e of overcurrent relay as shown in Fig. 4.4.

6.Z Fuzzy Variable DesignThe first design element of a consequent fuzzy variables is the word selection. Because o f the

semantic importance the word selection has more meaning than just assigning a symbol as is

frequently done in calculus [109]. The same consideration applies to the selection o f predicates (high,

medium, low). Semantically appropriate word selection between two layers is the essential key that

makes a fuzzy engine transparent in contrast to all other mathematical method such as neural network

which are expressed in pure symbolism [106] .The second design element o f a fuzzy variable is the

membership function .Designing membership functions is closely related to the overall solution, and

consideration beyond the domain depicted in Fig. 6.1 and Fig. 6.2 often become necessary [102]. The

first challenge includes finding the appropriate number of membership function and their locations on

the universe of the discourse. The second challenge is to determine shapes o f the membership

functions. As one can see, determining the number o f membership functions and their location

corresponds to the mapping o f input - output relationship o f the basic fuzzy inference mechanism.

Determining the shape of a membership function (i.e. Triangle, Gaussian curve, bell shape etc.) has

less impact on the overall solution. The third element o f variable design is the determination of the

universe of discourse. Universe o f discourse represents the extent o f validity o f the fuzzy inference

rules on each variable domain .Outside such a domain the performance o f a fuzzy inference engine

will be unpredictable or ill defined [110].

The linguistic interaction (i.e., taking values ) occurs between the first two layers at the top in

Fig. 6.1 and Fig. 6.2 .Therefore predicates are also called fuzzy values i.e. if “ operating time” is a

fuzzy variable, the predicate “Low” may be one o f the values it can take.

The overall structure o f fuzzy variable for IDMT relay is outlined in Fig 6.1 (antecedent fuzzy

variable) and Fig. 6.2 (consequent fuzzy variable).The antecedent parameter o f IDMT relay is plug

setting current level Ipi One can fuzzify this antecedent variable [52] by using linguistic fuzzy terms

i e. small level, medium level and high level as shown in Fig. 6.1.The different current ranges o f Ipi for

each of the linguistic variables has its own fuzzy universe o f discourse designated by min to max

values as below -

Universe of discourse for small = (Ip iS) mjn to (IpiS) max

Universe of discourse for medium = (I piM) min to ( IpiM) max

76

(Ipl ) min (Ipi )min(Ipl )m a x (Ipi )min(Ipl ) m a x (Ipl^)max

(Ipi) min Universe o f discourse (Ipl) max

Fig 6.1 Design of Antecedent variable for IDM T

The fuzzy universe of discourse for antecedent variable from rating o f IDMT relay sensor would be-

[(Ipl)min------------------------------------------ I pi, ------------------------------------------------(Ipl)max] . . . ( 6 .2 )

The consequent variable o f IDMT relay sensor from equation 6.1 would be -

[(OtW ------------------------------Oti-------------------------------------(Ot)min] ...(6 .3)

The linguistic design approach o f fuzzy variable design utilizes heuristic interpretation (or

expertise articulated in daily language) due to either the lack o f appropriate data or because o f nature

of problem . The linguistic design is from top to bottom referring to Fig. 6.1 and Fig. 6.2 . In this

approach to membership function development, an important concept called “decision boundary”

plays major role [109]. Decision boundaries indicate which regions in the input space correspond to

which regions in the output space in an input - output mapping process.

(Ot)min Universe o f discourse (Ot ) max

Fig. 6.2 Design of consequent variable for IDMT relay

Extracting decision boundaries from data set by membership function design, also known as

“partitioning can be accomplished in a sample manner when we know which output data points

correspond to which input data points. There are advanced mathematical methods in the literature to

accomplish partitioning. Among them, neural network and genetic algorithm have shown promising

results. In case of unknown decision boundaries, the expert interpretation o f data and C-means

clustering technique in partitioning is preferred [110]. The linguistic modifiers (linguistic hedges) are

also examined, which change the role o f a membership function in controlled manner .

The antecedent variables from the relay characteristics may be taken as fuzzy sets in fuzzy

universe of discourse. The IDMT relay curve has been plotted between plug setting current level (X-

axis) and operating time (Y-axis) exhibiting the inverse proportionality (See Fig. 4.5) .The plug setting

current parameters are indicated by fuzzy linguistic variables as IpiS ,IpiM ,IpiH for small, medium and

high predicates respectively taking four current values under each parameters (See equation 6.4) and

there corresponding membership functions is written as in equation 6.5 (See Fig 6.3 ) .

The corresponding universe o f discourse can also be obtained for operating time using

equation 6.1 and can be written as equation 6.6.

Plug setting current is fuzzified in universe o f discourse (2-20) for three predicates small,

medium, h igh and its corresponding operating time is calculated using equation 6.1 and is fuzzified in

universe o f discourse (2-8) as shown in Fig.6.3 and Fig.6.4 respectively.

Let us con sid er some training data (Four from each fuzzy variable) from universe o f discourse (2-20)

o f plug setting current as given in equation 6.4.

]p, = ( 2.5 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 15 , 17.5 , 2 0 ) ...(6 .4)

.......iPis — -» « — ipiM — -»«-— - v H------------ -»

This set is treated as training data set, their membership function from fuzzy universe of

discourse of plug setting current from Fig 6.3 would be -

m ) = ( 1, 0.98 , 0.88 , 0.67 , 0.22 , 0.90 , 1.0 , 0.99, 0.008 , 0.65 , 0.95 ,1 .0 ) ...(6.5)

.......M(pi )S-------- -----------M(Pi) M— ------------------------------M<Pi) H-“>

The operating time corresponding to the plug setting current (Ot ) values considered in

equation 6.4 for the IDMT relay can be calculated by the equation 6.1 as given in equation 6.6.

Ot = ( 7.57, 6 .3 , 4.98 ,4.28, 3 .8 4 ,3 .5 3 ,3 .3 ,3 .1 1 , 2 .9 ,2 .5 , 2.37,2.26) ...(6.6)

<-................ O tH-------------- <---------------O tM-------------- <-----------------O ts---------- -»

The membership function from fuzzy universe o f discourse o f operating time from Fig 6.4

would be -

mo,)=( 0.17, 0.75 , 0.56 , 0.16 , 0.93 , 0.66 , 0.4 , 0.13 , 0.8 , 0.66 , 0.49 , 0.33) ...(6.7)

<-....... H H (o.)------------- -» < ------- n M (oO------------------------ n s (o,)-------^

input variable 'PI-setling-currentM

Fig. 6.3 Fuzzification of plug setting (Antecedent) currents variable for IDMT relay

Low Medium High

8output variable ’Operaling-Time"

Fig. 6.4 Fuzzification of operating time (Consequent) for IDMT relay

6.3 Rule Base DevelopmentThe fuzzy IF-THEN rules in generalized form may be written as [52,106]:

IF Ipi is Py THEN Ot is Py

In equation 6.8 the symbols and notations are given as below:

• means IS

Pij means predicates for fuzzy antecedent variables Ipi .

—► means THEN

Pki means predicates for fuzzy consequent variable Ot.

In more specific form equation 6.8 may be written in the form o f FAM (Fuzzy Associated

Memories) rules

Predicates may be encoded by (Pij,Pki)

In general a FAM system F: Ipin-----^ Otp encodes and processes in parallel a FAM bank of

FAM rules [5,106]

Here Pjj is referred as antecedent term and Pki is referred as consequent term [5]. The

antecedent variables with referred to its membership function in Zadeh’s notation would be -

Ipi • Pij — ► O t • P k i ...(6.8)

ieN ...(6.9)

r Mi■A I j,=o 1

I / - t f r/=» ■'o/.

i e N

i e N

...(6.10)

...(6.11)

The subscripts S,M,H indicates sm all, medium and high predicates respectively [109].

The com plete antecedent variable may be written in generalized form

A A &j Ss ’ j M ’ t H

\ pl i P'i Pli )i e N

The corresponding consequent variable using equation 6.1 can be written as given below -

* u H °,H - JsNy=0 H \0.02

u pli ’

o,L= I 0.14j e N

...(6.12)

...(6.13)

...(6.14)

...(6.15)

...(6.16)j =0 L̂ 0.02 ,

1 Pl j ‘

The subscripts L,M,H indicate low, medium and high predicates respectively. The complete

consequent variable thus designed may be written in Zadeh’s notation [71]. From equation 6.14,6.15

and 6.16.

0 ,= (0 ,H,0 t M,0 t L)

■ u H " u M " u l0 = f V 1 V 11 '■Z-i 0.14 ’ Z - ( 0.14 ’ 0.14

7=0 (1 plj H )002-l '=° (lpljM )002 -1 J=° (/p, / f 01-l

Using equation 6.12 and 6.17 the fuzzy associated memory (FAM) rules is developed for the IDMT

relay as given in Table 6.1 [102,109].

...(6.17)

...(6.18)

Table 6.1 FAM rule Table

\ 0 t

Ipl \

H M L

s 1

M R2 BH

i v ^ i

The FAM rules developed above (Table 6.1) can be written in linguistic form using fuzzy reasoning as given below [8,12].

to IHi

p'i

M

If Ipi is THEN Ot is V - A -~ T S 1 Z - l _____0.14

7=0 (Ipl j H)002- l

l f , p'is S f V t h e n Ot is Z - %'■° pi, 7=0 (Ipl. M)0u2-1

lf Ipl is ™ E N o , is y ^ “ L~ T H 0.14to IPh 7=0 Up'/ ) 001-1

i j e N

MATLAB output of linguistic form o f FAM rule is shown in Fig 6.5 as given below -

j . If (F l-setting-current is S m all) t h e n IU D eratin g-T im e is H iq h l (1i £ 'settin9 'current is MediumJ then (Operating-Time is Medium) (1) 3. If (Pl-setting-current is High) then (Operating-Time is Low) (1)

Fig. 6.5 Rule Base for IDMT relay

6.4 Rule Base InferenceR u le b a se developed and discussed above can be used to infer operating time for different plug

setting current as shown in Fig. 6.6 (a) and (b). In Fig 6.6 (a) two rules are firing for plug setting

current =6.66 Amp and giving the operating time = 4.92 seconds .This value o f operating time is

alm ost similar to the value obtained from equation 6.1.In the same manner Fig. 6.6 (b) is showing

operating time (2.76 sec.) corresponding to its plug setting current (13 Amp.).

Pl-se ting-current = 6.66Opera£ng-T*ne = 4.92

2 8

(a)

Pl-seting-current = 13

‘X

/ V2 20

OperaSng-Time =2.76

_

£(b)

Fig 6.6 (a), (b) Rule base inferencing process for different plug setting current

Fig. 6.7 shows the simulated characteristics curve o f IDMT relay. This curve satisfies the

inverse proptional rule of IDMT relay .The curve is rough due to less number o f rules (Only three

rules). Smooth curve can be obtained by increasing the number o f rules. But it is clear from the above

simulated curve that the working behavior o f Fuzzy IDMT relay is very much similar to the

conventional IDMT relay.

Fig 6.7 Simulated characteristics curve of IDMT relay

B.5 Approximation by Artificial Neural NetworkThe linguistic systems and the fuzzy systems are known to be easy to design and develop [14].

A finite number of combinations is needed to completely describe a linguistic system and the

corresponding fuzzy systems [109]. On the other hand, the neural networks have a proven ability in

learning, moreover good learning algorithms are available for them therefore it is practically

interesting to combine fuzzy systems and neural networks [14,64]. A Neuro-Fuzzy system comprises a

system having either a neural network structure but using fuzzy neurons instead o f a classic neuron,

or systems composed o f both fuzzy systems and neurons, or neural networks, each o f these subsystems

preserving the classic characteristics o f its class in the hybrid configuration. Fuzzy systems allow

modeling of the imprecise but rational behavior o f human experts, where as neural network can be

trained to model empirical behavior o f human experts or to model complex nonlinear systems[14].

6.5.1 Neuro - Fuzzy SystemThere are six main classes of Neuro - Fuzzy (hybrid) systems namely class A, class B, class C,

class D class E and class F . Here class B includes adaptive fuzzy systems that are trained by a neural

network, is considered. Either the rules or the membership functions can be modified during the

adaptation process by the neural network [106]. A neural network can have any number o f layers. A

multilayer neural network with one hidden layer is shown in Fig. 6.8 for training the consequent

(output) variables of IDMT relay using EBPA. In this figure ‘f is the characteristic function of

hidden layer and output layer. In hidden layer function f is sigmoidal function and in outer layer it is

pure linear function for better performance.

‘Wji’ is the weight of interconnected neuron of input and hidden layer

‘W kj ’ is the weight o f interconnected neuron o f hidden and output layer

The most widely used activation function is sigmoidal function which is given below

1S=

l + e~

Where X is total input in a neuron .In terms o f Operating time (Ot) and plug setting current ( Ipi) it can

be written as -

1Ot

l + e

Input layer (ith layer) Hidden layer (j* layer) Output layer (kth layer)

► O f

► o M

► OtL

(Ot)n

(Ot)n

Fig. 6.8 Multilayer Neural Network for training the consequent fuzzy variable of IDMT

relay

The features of the models based on neural network are not “ rational but analytical, in the

sense that in their description one does not explain by some logic rules the way one determines the

output values as for fuzzy systems [14] .The weights in a neural network are determined by successive

adaptation steps. So a neuron and a neural network can stand for a representation o f the empirical

formula, and of empirically determined (Learned , trained ) controls and decision - making processes.

They cannot stand for rational rule-based processes .

Fig. 6.9 shows a proposed Neuro- Fuzzy system for IDMT relay as fuzzy sensor .This system

is a series of fuzzy system and neural network architecture represented in a general form [106]. The

equation is - Ot = f (Zn , Zn ,... Zn )

Where Zn' ,Zn2,...Znr represent the first second ...rth values o f the sampled output membership

function of the fuzzy system at time moment n.The fuzzy system is not provided with a defuzzifier .

One simple (but not only) interpretation o f this structure is as follows [64]. One has to build a non

linear system, and some raw basic knowledge exists about the desired behavior. This knowledge is

incorporated into the fuzzy system. However a tuning o f the obtained characteristic function is

necessary (may be for the purpose o f adaptability of robustness). The tuning is performed by the

neural network that is trained to perform the best defuzzyfying - tuning mixed operation [14,64].

Fig. 6.9 Neuro - Fuzzy System for IDMT relay as fuzzy sensor

0.5.2 Neural Network TrainingNeural network architecture given in Fig. 6.8 is trained in supervised manner using EBPA and

the simulated results are shown in Table 6.3.This table contains consequent variable for its desired and

simulated values .The error is in acceptable range. It is tried to develop and approximate the

consequent variables o f IDMT relay as fuzzy sensor. The performance goal was meet after 308

epochs. The performance curve for training o f neural network is shown in Fig. 6.10.

io a

Fig 6.10 Performance of neural network on an epoch by epoch basis as training proceeds

G.E Approximation of Degree of MembershipThe neural network architecture shown in Fig 4.14 of chapter 4 is again considered here for

approximation of degree of membership for antecedent and consequent variables of IDMT relay based

on equation 6.1. The data given in equation 6.4 to 6.7 are considered for training.

G.6.1 Weight AssignmentTable 6.2 The initial weights assigned to different paths of Neural Network shown in Fig. 4.14

Input to hidden layer weights Hidden to output layer weights

Weights Assigned values Weights Assigned values

M-(Ipi) H<O0

W,(l) 0.95 W /2) 0.20

W2(,) 0.95 w 2(2) 0.80

w3(l) 0.90 w 3(2) 0.60

w4(,) 0.65 w 4(2) 0.20

w5(,) 0.25 w 5(2) 0.95

w6(1> 0.92 w 6(2) 0.65

w7(,) 0.98 w 7(2) 0.42

w 8<’> 0.95 w 8(2) 0.20

w9(,) 0.01 w 9(2) 0.85

W,o(1) 0.70 w 10(2) 0.65

W„(,) 0.96 W „(2) 0.50

W12(l) 0.98 W,2(2) 0.35

Weight assignment is very important part of training the neural network. Weight should be

assigned to neural network in expertise manner in order to minimize the learning time. Here initial

weights are considered from the equation 6.5 and 6.7 for neural network, after slight modification as

shown in T a b le 6.2 for input to hidden layer and for hidden to output layer.

G.G. 2 Neural Network TrainingThe neural network shown in Fig. 4.14 is trained in supervised manner with initial weights

given in Table 6.2 and by presenting input pattern with one element each o f plug setting current and

operating time to obtain its corresponding output (Degree o f membership). Performance goal was meet

after 500 epochs. The training performance curve is shown in Fig. 6.11.

Fig. 6.11 Training performance curve of neural network

E.7 Simulation ResultAfter training the neural network results are shown in Table 6.4. This table contains

membership of antecedent variable for its desired and simulated values. In the same manner Table 6.5

contains membership of consequent variable for its desired and simulated values. The error calculated

and tabulated is in acceptable range. It is tried to develop and approximate the consequent variables o f

IDMT relay as fuzzy sensor for power system protection.

The test impulses of fault currents of transient nature has been developed corresponding to the

amplitudes of different plug setting currents ‘Ipi ‘. The consequent variables ‘Ot ‘ referring to equation

6.1 are observed to be a fuzzy set of variables around its corresponding values with its membership

functions shown with horizontal line segments as illustrated in Fig 6.12 i.e. an input test impulse o f

amplitude. This figure depicts the conceptual pictorial view o f , showing that how the amplitude of

transient current affects the characteristics curve o f IDMT relay.

B.8 DiscussionThere is a stabilized relationship between plug setting currents and operating time o f IDMT

relay given by the equation 6.1.Some relations have been derived in fuzzy universe o f discourse for

fuzzified values of plug setting current and operating time in small, medium, high and high, medium

and low respectively. The consequent variables Ot for IDMT relay have been determined to develop

the structure of fuzzy variables or linguistic variables and also helped to generate FAM rules for

consequent fuzzy variables.

The behavior of inverse definite minimum time relay has been fuzzified and a fuzzy sensor has

been developed to have its performance same as IDMT relay .It is tried to develop the fuzzy sensor,

and its simulation and testing have been performed to test its characteristics whether it is coping with

that of conventional relay.

Table 6.3 Comparison of desired and simulated Operating Time (Consequent variable) for

corresponding plug setting current (Antecedent variable)

I noT Antecedent

variable (I pi )

Desired Consequent

Variable (Ot)

Simulated consequent

Variable (Ot)

Error

i 2.5 7.57 7.5713 -0.0013

2 3.0 6.3 6.3048 -0.0048

3 4.0 4.98 4.9755 0.0045

4 5.0 4.28 4.2798 0.0002

5 6.0 3.84 3.8142 0.0258

6 7.0 3.53 3.5351 -0.0051

7 8.0 3.3 3.2985 0.0015

8 9.0 3.11 3.11 0.000

9 10 2.9 2.9005 -0.005

10 15 2.5 2.5023 -0.0023

11 17.5 2.37 2.3696 0.0004

12 20 2.26 2.2601 -0.0001

Table 6.4 Comparison of desired and simulated membership function of antecedent variables

S.NO. Antecedent

variables (Ipit) in

seconds

Desired Antecedent

Membership

function

Simulated

Antecedent

membership function

Error

1 2.5 1.0 1.0000 0.000

2 3.0 0.98 0.97836 0.00164

3 4.0 0.88 0.88195 -0.00195

4 5.0 0.67 0.67007 -0.00007

5 6.0 0.22 0.22101 -0.00101

6 7.0 0.9 0.89866 0.00134

7 8.0 1.0 1.0000 0.0000

8 9.0 0.99 0.9889 0.00119 10 0.08 0.0092051 0.070794910 15 0.65 0.64919 0.0008111 17.5 0.95 0.95803 -0.0080312 20 1.0 0.99278 0.00722

Table 6.5 Comparison of desired and simulated membership function of consequent variables

S.NO. Consequent

variable Ot in Sec.

Desired Consequent

Membership

Function

Simulated consequent

membership function Error

1 7.57 0.17 0.17432 0.00432

2 6.3 0.75 0.7496 0.0004

3 4.98 0.56 0.56091 -0.00091

4 4.28 0.16 0.15891 0.00109

5 3.84 0.93 0.93016 -0.00016

6 3.53 0.66 0.65749 0.00251

7 3.33 0.4 0.40411 -0.00411

8 3.11 0.13 0.12852 0.00148

9 2.9 0.8 0.80089 -0.00089

10 2.5 0.66 0.65989 0.00011

11 2.37 0.49 0.49027 -0.00027

12 2.26 0.33 0.329^3 -0.00011

7.5

7

6.5

6.0

5.5

5.0

4.5

4.0

3.5

3.0

2.5

2

0

a ?2 3 H 52.3̂ 3 ft <-*-

5* 5’*T3C

3T3C

B5N*<3oao

U ) Ul| O'! 0C| s d

Ipi (Plug setting current) in Amp.

►—1 tOri ©U)

?

7 s

?

Rela’Fig .6.12 Conceptual pictorial view of approximate consequent variable Relay92