A Novel Fuzzy application to power system

A novel fuzzy index for steady state voltage stability analysis andidentification of critical busbars

P.K. Satpathy, D. Das *, P.B. Dutta Gupta

Department of Electrical Engineering, Indian Institute of Technology, Kharagpur 721 302, India

Received 28 March 2001; received in revised form 4 March 2002; accepted 10 April 2002

Abstract

In this paper, a novel fuzzy index is proposed for the prediction of steady state voltage stability conditions in transmission

networks. The uncertainties in the input parameters are efficiently modeled in terms of fuzzy sets by assigning trapezoidal and

triangular membership functions. The results include fuzzy load flow solutions for the base case and critical conditions with and

without contingencies. The proposed fuzzy voltage stability index clearly indicates the location and status of critical busbars. Case

studies have been conducted on standard test systems (IEEE 14-bus, 30-bus, and 57-bus) with proper validation of the results.

# 2002 Elsevier Science B.V. All rights reserved.

Keywords: Voltage stability index; Critical busbars; Fuzzy set theory; Membership functions

1. Introduction

The history of fuzzy set theory (FST) dates back to

the year 1965 when it was first introduced by Zadeh [1]

with the fundamental concept of representing uncertain-

ties. The advent of FST rendered a mathematical

platform for representing the imprecise notions and

concepts of human interpretation by the help of

membership functions (MF). Consequently the fact

that FST application is gaining popularity in many

spheres, researchers are now on the run to explore even

better means and applicability of its principles to handle

uncertainties in power systems. References [2/5] indi-cate some power systems areas where FST has success-

fully been applied (viz. load forecasting, load flows,

operation and control of PSS, optimal VAR planning,

transient rotor stability evaluation, unit commitment,

fault diagnosis in transformers and transmission lines).

A review of the literature also reveals that no

significant research has been carried out on fuzzy

voltage stability analysis. Although a number of re-

search contributions is available highlighting the appli-

cations of fuzzy-expert control approach for voltage

stability monitoring and enhancement [6/8], the objec-tives of this paper are quite different from them.

Reference [6] shows a control model based on FST for

voltage stability enhancement by using the Newton/Raphson load flow technique. In Ref. [7], a fuzzy-expert

rulebase is reported that formulates voltage stability

control strategies by monitoring the eigenvalue of the

load flow Jacobian with the help of modal analysis.

Reference [8] also reports another expert fuzzy control

approach for voltage stability enhancement by monitor-

ing the L-index calculated from load flows. Although,

Refs. [6/8] have used sets of fuzzy rules for theenhancement and control of voltage stability, its mon-

itoring is done by use of traditional load flow techni-

ques, but not the fuzzy power flow (FPF).

The most remarkable difference between the tradi-

tional load flows and the FPF is that the former uses

crisp values for the input parameters (bus injections),

where as the later one makes use of FST techniques to

model the related uncertainties associated with them [9/12]. The results of FPF as reported in [9], are mainly the

possibility distributions of line power flows and bus

voltages, which may provide some immediate and

interesting conclusions in the form of what may happen

corresponding to a different degree of possibility or

credibility (strictly in a human sense). In Ref. [10] the

* Corresponding author. Tel.: /91-3222-78053; fax: /91-3222-78707

E-mail address: [email protected] (D. Das).

Electric Power Systems Research 63 (2002) 127/140

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authors used interval arithmetic to model load uncer-

tainties in formulating the FPF problem. Another FPF

for the base case is reported in [11], which could not

justify fuzzy distribution for all states.In view of these facts, the authors of this paper

strongly feel that FST applications to voltage stability

analysis must be paid attention in order to find out the

effect of parameter uncertainties on the system states

and limiting conditions, if any. With this motivation, the

authors of this paper have tried to justify the possibility

and benefits of applying FST approach to voltage

stability studies within the framework of steady stateanalysis. This objective is accomplished in two steps: (i)

developing an efficient FPF algorithm by assigning

suitable MFs for each input parameter; and (ii) regular-

izing the FPF algorithm by incorporating the continua-

tion technique. In the past, the continuation technique

has been applied to traditional load flows for computa-

tional benefits [13,14]. In view of these advantages, we

propose a new fuzzy continuation power flow (FCPF),which has been obtained by extending the FPF algo-

rithm to support the continuation technique.

The main advantage of FCPF over the FPF and the

traditional load flows is that it remains capable of

withstanding numerical ill conditioning effects resulting

from Jacobian singularity at higher loading conditions.

Therefore, the results around the base case, and up to

the steady state voltage stability threshold may beobtained by the proposed FCPF technique. With these

modifications we observe that the simulation results for

all states (both at the base case and voltage stability

threshold conditions) have been regularized and all of

them show fuzzy possibility distributions.

Major findings of voltage stability analysis on the

basis of crisp parameter formulation include steady state

voltage conditions [15/19,28,29] and identification ofcritical busbars [20/23]. In this paper, we present someinteresting results of steady state voltage stability

analysis in view of a fuzzy parameter formulation. In

addition, a new fuzzy voltage stability index (FVSI) is

also proposed. The proposed FVSI serves as a good

indicator for identification of critical busbars both in

normal and contingency conditions. The authors claim

that the results obtained from this novel approachwould provide better insight to planners and operators

in the field of power engineering to handle the uncer-

tainties effectively.

The organization of the paper is as follows. Section 2

highlights the basic facts about FST. Section 3 deals

with the fuzzy modeling of input parameters considering

trapezoidal and triangular MFs. In Section 4, the

general procedure is outlined to obtain base casesolutions by FPF. The necessary modifications for the

development of the proposed FCPF algorithm are

highlighted in Section 5. In Section 6, the procedure to

obtain the proposed FVSI is presented. Section 7

outlines the simulation results obtained from the case

studies conducted on standard IEEE 14-bus, 30-bus,

and 57-bus test systems with proper justification.

2. Basic facts about fuzzy set theory (FST)

The term fuzzy implies something that is imprecise,

unclear, and above all not well defined. Therefore, fuzzysets do not have any sharp boundary like the crisp sets

[26]. In order to make the understanding of fuzzy sets

simpler, some discussion on crisp sets is presented here.

A crisp set A may be imagined as a cluster of subsets

x having one-to-one correspondence as defined by the

characteristic function /mA/ of Eq. (1). The implication of

Eq. (1) is that the elements x may have either 100%

correspondence (as implied by mA(x)//1) or null corre-spondence (as implied by mA(x)//0) with the parent setA . Such a correspondence is crisp in nature. However, in

practical life, the elements often exhibit intermediate

values of correspondence with their parent sets ranging

between 0 and 1. Such a correspondence leads to

uncertain or fuzzy events.

mA(x)1; if x A0; if xQA

(1)

Zadeh [24] reported that the imprecise knowledge and

perception of human beings could be modeled in a morenatural way to generate meaningful probability distri-

butions with the application of fuzzy sets. In view of

this, a fuzzy set /A/ in the universe of discourse U , may

be imagined as a cluster of subsets x whose correspon-

dence with the parent set may be represented as a MF

given by mA(x); such that

mA(x) [0; 1] (2)

This membership indicates the degree or extent that x

belongs to A: The value of mA(x) can be any wherebetween 0 and 1, and this range is what makes it

different from a crisp set. The closer the value of mA(x) is

to 1, the more x belongs to A: Elements of fuzzy sets areordered pairs comprising of the set element x and the

corresponding membership grade mA(x): This relation-ship is expressed mathematically in Eq. (3).

Af(x; mA(x))jx Ug (3)

While solving practical problems, it is important to

adopt appropriate fuzzy operators to satisfy the desired

fuzzy reasoning. This may be taken care of on the basis

of the following guidelines. Firstly, the problem to be

solved must be stated mathematically or linguistically.

Secondly, the upper and lower threshold boundaries (i.e.

the highest and the lowest degree of satisfaction) for thevariables should be well imagined. Thirdly, proper

forms of MFs or possibility distributions reflecting the

changes in the degree of satisfaction with those of the

P.K. Satpathy et al. / Electric Power Systems Research 63 (2002) 127/140128

changes in the variables should be constructed. Finally,the required fuzzy operations are to be so selected that

the results thus obtained suitably match with the

expected ones.

3. Fuzzy modeling of input parameters

A common practice followed in several simulation

studies is the use of crisp numbers for the specified

voltages and scheduled power injections, which hardly

maintain specific values in practice. Power systems beinglarge, complex and geographically widely distributed,

are highly influenced by unexpected events and uncer-

tainties. Therefore, a lot of uncertainties may be

associated with the input parameters for implementation

in any analytical method. These facts make it difficult in

dealing with power system problems through strict

mathematical formulations alone. Fuzzy logic on the

other hand, is a natural choice and seems to bepromising in modeling these uncertainties with the

help of FST [24,27]. In references [9,11,12], uncertainty

modeling for loads and generations only, has been

considered for the FPF simulation. However, the

authors of this paper feel that the specified voltages at

the PV buses may be another valid candidate for

uncertainty modeling, as this is also practically affected

by changes in loading and network configurations.Therefore, in this paper, we have considered three input

parameters (i.e. voltages at the PV buses, loads and

generations) and modeled them as L/R fuzzy distribu-

tions by applying trapezoidal and triangular MFs. A

fuzzy number is defined to be of the L/R type [25], ifthere are L/R shape functions in association withpositive scalars a (left spread) and b (right spread).

A trapezoidal L/R fuzzy MF as shown in Fig. 1(a) isgenerally expressed by the characteristic points a1, a2, a3and a4 such that the fuzzy number under study can

assume any value between a1 and a4, but values within

the range a2 and a3 are most likely to take place. All the

values within the range a2 and a3 have a membership

value mx /1, that indicates complete membership forthe event. However, values within the ranges (a1/a2)and (a3/a4) have memberships 05/mx 5/1, which in-dicates partial membership values. Any value outside

the range of a1 and a4 has membership values mx / 0,indicating non-membership for the parameter. A spe-

cific relationship between the element x and its degree of

membership mx satisfying the trapezoidal MF can be

mathematically stated in the form of Eq. (4), where L (x )

and R (x ) refer to the L/R functions of the fuzzydistribution.

mA(x)

L(x) for a15x5a2;1 for a25x5a3;R(x) for a35x5a4;0 otherwise:

8>>>:

(4)

where,

L(x)x (a2 a)

a ja]0

(5)

and

R(x)(a3 b) x

b jb]0

(6)

The mathematical implications of Eq. (4) may be

stated as a simple linguistic declaration to make the idea

clear. For instance, let us assume a practical example of

a particular power system node, where the peak demand

excursions for the real power at the i th bus are not likelyto go below 5 MW or above 20 MW, but most likely to

occur within 10/15 MW. Such a linguistic declarationcan be easily translated into a trapezoidal fuzzy dis-

tribution as shown in Fig. 1(a) by setting the character-

istic points with a1/5 MW, a2/10 MW, a3/15 MWand a4/20 MW. On the other hand, a triangular MF asshown in Fig. 1(b) can be derived from the trapezoidal

MF of Fig. 1(a) with the assumption that the character-istic points a2 and a3 coincide with each other.

4. Fuzzy power flow (FPF)

The mathematical formulation of the traditional load

flow problem results in a system of algebraic equations

(Eq. (7)), describing the power systems.

Fig. 1. (a) Trapezoidal L/R fuzzy MF. (b) Triangular L/R fuzzy MF.

P.K. Satpathy et al. / Electric Power Systems Research 63 (2002) 127/140 129

f (d; V )0 (7)

The solution of these equations is based on an

iterative technique because of their non-linearity [30].

The real and reactive powers at the i th bus in terms of

voltage magnitudes and angles are shown in Eqs. (8) and(9). Given an initial set of bus voltage magnitudes and

angles, the real and reactive powers are calculated from

these equations.

Pi Xnj1

[ViVjYij ] cos(uijdidj) (8)

QiXnj1

[ViVjYij] sin(uijdidj) (9)

The uncertainty modeling of the input parameters

results in a set of fuzzy equations representing the power

systems, which may be written in its compact form asshown in Eq. (10). The fuzzy power mismatches, which

show the difference between the specified and calculated

powers are expressed in Eqs. (11) and (12).

f (d; V )0 (10)

DPi [Pi(specified)Pi] (11)

DQi [Qi(specified)Qi] (12)

where,

Pi(specified)(PGiPLi) (13)

Qi(specified)(QGiQLi) (14)

The voltage magnitude and angle updates (Dd; DV )/are found iteratively from the FPF equation as shown in

Eq. (15), and the process is repeated until the power

mismatches fall within a specified tolerance. The new set

of values at the end of each iteration, for voltage

magnitudes and angles are found by adding the updates

to their corresponding old values.

DdiDV i

H

J j NL1DPi

DQi

(15)

where the various elements of the Jacobian sub-matrices

are given by

Hij @Pi=@djji"j; Nij @Pi=@Vjji"jJij @Qi=@djji"j; Lij @Qi=@Vjji"jHii@Pi=@di; Nii @Pi=@ViJii@Qi=@di; Lii@Qi=@Vi

g (16)

Using these modified load flow equations the New-

ton/Raphson load flow program is run iteratively so asto generate the base case solutions in a fuzzy environ-ment. However, results beyond the base case are

obtained through FCPF algorithm as described in the

next section.

5. Fuzzy continuation power flow (FCPF)

In electric power systems the steady state voltagestability conditions are often analyzed through PV/QV

diagrams. In order to get a complete trace of a PV curve,

one shown in Fig. 2(a), it is desirable that the static

power flow solutions be obtained at various loading

conditions between the base case point and the critical

point. The near critical solutions often show conver-

gence difficulties due to singularity of the power flow

Jacobian. Continuation techniques [13,14] have beenused to steer out this numerical shortcoming. In this

section, an FCPF technique has been proposed that

inherits the fundamental attributes of the continuation

power flow technique and is powered to run in a fuzzy

environment. While applying continuation method to

work in a fuzzy environment, the compact form of load

flow equations as shown in Eq. (10) are reformulated

and the modified form of this is shown in Eq. (17).

f (d; V ; l)0 (17)

The variable (l) introduced in Eq. (17) simulates the

load-changing scenario of the network. In this paper, we

have assumed a uniform load growth all over the

network. With this assumption, we present the load

growth expressions as a function of their respective base

values, such that Pl/Pbase(1/l) and Ql/Qbase(1/l ). The base loads have been simulated with the helpof the static polynomial ZIP model. In this paper, we

have assumed uniform ZIP coefficients for all buses in

the network (i.e. Z/0.2 p.u., I/0.3 p.u. and P/0.5

Fig. 2. (a) P/V curve at i th bus. (b) l /V curve at i th bus.


p.u.). Now we define a suitable range for the load

parameter l, for which we are interested in finding thesolutions. The lower limit of this range (lb) refers to thebase case loading, and the upper limit (lc) is referred tothe critical loading. Having solved for the base case, l isincremented in small steps and the continuation power

flow is run every time until the critical point is reached.

Continuation power flow is based on a locally

parameterized continuation technique that employs a

predictor/corrector algorithm. The predictor algorithmhelps to predict the solutions along a tangential direc-

tion. The tangent predictor is obtained from the partial

differentiation of Eq. (17), such that;

@[f (d; V ; l)]0 or fd[@(d)]fV [@(V )]fl[@(l)]0

or

[fd fV fl][@(d) @(V ) @(l)]T 0 (18)

A solution of Eq. (18) gives the desired tangent

vectors

Table 1

Line outage simulation for 14-bus

Line outage case

no.

Code no. of outaged lines

(Line data for these lines is available in Tables A1,

A2, A3 and A4 of Appendix A)

1 Nil

2 19

3 20

4 4

5 6

6 11

7 5

8 3

9 2

10 12,19

11 3,17

12 5,11

13 2,18

14 4,13,16

15 5,15,17

Table 2


Line outage

case no.

Code no. of outaged lines



1 Nil

2 6

3 3

4 41

5 10

6 37

7 18

8 6,18

9 3,41

10 10,28

11 6,17,27

12 12,27,37

13 3,20,28,41

14 6,18,28,37,40

15 3,12,17, 27,40,41

Table 3


Line outage

case no.

Code No. of outaged lines



1 Nil

2 79

3 65

4 74

5 7,65

6 5,54

7 5,10,14

8 35,54,79

9 5,22,74,79

10 5,10,25,35,65

11 8,11,14,19,74,78

12 8,10,11,13,19,54,65

13 5,7,14,25,35,74,78,79

14 6,7,10, 11,19,25,35,54,78

15 5,8,9,13,14,22,25,65,74,79

Fig. 3. Base voltage (Vb) distribution.

Fig. 4. Critical voltage (Vc) distribution.


[@(d) @(V ) @(l)]T

and then, a suitable step size s is used to predict the

length of these tangent vectors as in Eq. (19). Using the

tangent vectors, the new set of predicted solutions are

obtained from Eq. (20). The next step is to correct the

predicted solutions through a corrector algorithm.

[t(d) t(V ) t(l)]T s[@(d) @(V ) @(l)]T (19)

[d V l]Tnew [d V l]Told [t(d) t(V ) t(l)]

T (20)

The corrector algorithm is based on a locally para-

meterization technique that employs the traditional load

flow program in a slightly modified form. The mod-

ification used is to specify any one out of d; V or l as acontinuation parameter. The process is repeated until

the critical point is reached. By monitoring the magni-

tude and sign of the tangent vector @(l), correspondingto the load parameter l , the critical point can be sensed.

The value of @(l) is positive before the critical point,which turns zero at critical point and negative beyond it.

It is to be noted that the value of the load parameter for

the base case (and the critical point) may be referred as

Fig. 6. Fuzzy distribution of FVSI.

Table 4

FCPF (critical) results for 14-bus

Case nos. (Ref. Table 1) Critical results

Corresponding to MF1.0 l

Critical bus lc (p.u.) Critical rank

1 8 4.0749 15

2 14 3.8428 14

3 8 3.8031 13

4 8 3.6099 11

5 8 3.4955 9

6 9 3.4942 8

7 8 3.2915 7

8 8 3.0861 5

9 7 1.5078 2

10 14 3.5514 10

11 8 3.0146 4

12 9 2.9901 3

13 7 1.5071 1

14 11 3.6346 12

15 14 3.2425 6

Table 5


Case nos. (Ref. Table 2) Critical results corresponding to MF1.0 l


1 30 3.1406 15

2 8 2.6020 8

3 30 2.8666 12

4 30 3.0413 14

5 30 2.6134 9

6 29 2.1843 4

7 30 2.9066 13

8 30 2.5786 5

9 30 2.8602 11

10 30 2.5879 7

11 30 2.5821 6

12 29 2.0699 3

13 30 2.8308 10

14 29 1.2434 2

15 30 0.5370 1

Table 6


Case nos. (Ref. Table 3) Critical results corresponding to MF1.0 l


1 57 3.7342 15

2 31 3.5694 14

3 51 2.5967 7

4 57 3.3151 13

5 51 2.5844 6

6 57 3.2037 12

7 57 3.1463 11

8 31 2.9921 8

9 57 3.0593 9

10 51 2.0042 2

11 57 3.0992 10

12 51 2.2925 4

13 31 2.3959 5

14 42 2.1226 3

15 51 1.7110 1


lb (and lc), which are assigned values lbjl0 andlcjlc0 and @(l)0 respectively.

6. Fuzzy voltage stability index (FVSI)

A new fuzzy index is proposed in this paper for

identification of critical busbars in a transmission net-

work. A critical busbar (i.e. the weakest among all the

buses in a network) may be defined as a bus, which is

almost in the verge of experiencing voltage collapse.Otherwise stated, the weakest bus would be the one,

whose operating point is the closest to the steady state

stability margin or the nose of the P /V curve or P /lcurve. Examining the status of the busbars it is possible

to derive this required information through static

analysis for formulation of voltage stability indices

(VSIs). A performance index should be numerically

stable and also be capable of measuring the amount ofload increase that the system can tolerate before the

voltage collapses in the network.

The FVSI proposed here is based on continuation

power flow technique and hence is claimed to be

numerically stable. Starting from a power flow solution

around the base case corresponding to lbjl0; the nexthigher solutions are obtained through FCPF until

finally the critical point corresponding tolcjlc0 and @(l)0 is reached, as shown in Fig. 2(b).The percentage drop in fuzzy voltage magnitude is then

calculated for each busbar corresponding to the load

increase from lb to lc, which serves as the fuzzy voltage

stability index. Thus the proposed FVSI for any

arbitrary system bus (i th bus) is expressed as

(FVSI)i %DV i

V ijlb jl0 V ijlc jlc0 and @(l)0V ijlb jl0

100 (21)

Using Eq. (21) the FVSI values are obtained for all

the busbars considering specific MFs and desired

operating conditions. A comparison of these values

identifies the most critical busbar in the network on

the basis that the busbar having the largest FVSI is

considered most critical in the context of voltage

collapse. The numerical results obtained in support of

this are presented in the next section.

7. Case study and numerical results

Case studies on IEEE 14-bus, IEEE 30-bus and IEEE

57-bus standard test systems are conducted and results

thus obtained, are presented in this section. In thispaper, we have used three types of input database off

which the first one is the standard line data for the three

test systems. Although such database is available in the

literature, they are presented here in Tables A1, A2, A3

and A4 of Appendix A for a quick reference. It is to be

noted that, bus no.1 in all the three test systems is

treated as slack bus, bus nos. 2/5 in the 14-bus system,bus nos. 2/6 in the 30-bus system and bus nos. 2/7 inthe 57-bus system are treated as generator (PV) buses.

All other buses in the respective systems are treated as

load (PQ) buses.

Table 7

Base case and critical results for 14-bus

MF Results for critical busbar (bus-7)

(Ref. Case No. 13 of Tables 1 and 4)

Vb Vc lc (FVSI)c

0.0 (L) 1.0621 0.8799 1.9391 23.364

0.3 (L) 1.0523 0.8655 1.8000 22.981

0.6 (L) 1.0424 0.8533 1.6693 22.380

1.0 (L) 1.0291 0.8364 1.5071 21.635

1.0 (R) 1.0005 0.8040 1.1814 19.690

0.6 (R) 0.9862 0.7917 1.0580 18.342

0.3 (R) 0.9753 0.7816 0.9716 17.390

0.0 (R) 0.9643 0.7709 0.8902 16.496

Table 8




Vb Vc lc (FVSI)c

0.0 (L) 0.9649 0.4194 3.2561 53.024

0.3 (L) 0.9523 0.4095 2.8200 52.228

0.6 (L) 0.9395 0.4044 2.4605 50.905

1.0 (L) 0.9221 0.3998 2.0699 48.852

1.0 (R) 0.8661 0.3741 1.3992 43.416

0.6 (R) 0.8469 0.3729 1.1962 40.836

0.3 (R) 0.8321 0.3714 1.0642 38.576

0.0 (R) 0.8168 0.3706 0.9466 36.307

Table 9




Vb Vc lc (FVSI)c

0.0 (L) 1.0077 0.3782 2.1036 63.238

0.3 (L) 0.9999 0.3712 1.9755 62.866

0.6 (L) 0.9921 0.3698 1.8568 61.935

1.0 (L) 0.9815 0.3654 1.7110 60.921

1.0 (R) 0.9364 0.3545 1.2762 55.782

0.6 (R) 0.9250 0.3504 1.1797 54.610

0.3 (R) 0.9163 0.3447 1.1115 54.010

0.0 (R) 0.9075 0.3414 1.0470 53.130


Table 10

FPF (base case) results for 14-bus (for Case No. 13 of Table 1)

Item(s) MF distribution and corresponding base case results

0.0 (L) 0.3 (L) 0.6 (L) 1.0 (L) 1.0 (R) 0.6 (R) 0.3 (R) 0.0 (R)

V6 1.0785 1.0693 1.0599 1.0473 1.0189 1.0056 0.9955 0.9852

V7 1.0621 1.0523 1.0424 1.0291 1.0005 0.9862 0.9753 0.9643

V8 1.0658 1.0565 1.0471 1.0345 1.0079 0.9943 0.9839 0.9733

V9 1.0806 1.0694 1.0582 1.0431 1.0098 0.9939 0.9818 0.9695

V10 1.0828 1.0711 1.0593 1.0435 1.0093 0.9928 0.9802 0.9674

V11 1.0955 1.0847 1.0738 1.0593 1.0302 1.0152 1.0039 0.9926

V12 1.1065 1.0960 1.0855 1.0715 1.0451 1.0310 1.0204 1.0098

V13 1.0948 1.0838 1.0727 1.0580 1.0297 1.0147 1.0034 0.9920

V14 1.0765 1.0638 1.0511 1.0340 0.9967 0.9789 0.9654 0.9517

d6 0.203 0.219 0.236 0.259 0.317 0.343 0.363 0.383d7 0.180 0.191 0.203 0.219 0.260 0.277 0.291 0.305d8 0.145 0.155 0.165 0.179 0.214 0.229 0.240 0.252d9 0.215 0.234 0.254 0.281 0.348 0.378 0.402 0.426d10 0.213 0.233 0.254 0.282 0.354 0.386 0.410 0.435d11 0.207 0.227 0.248 0.277 0.351 0.383 0.408 0.433d12 0.207 0.228 0.250 0.280 0.361 0.394 0.420 0.446d13 0.208 0.230 0.252 0.283 0.363 0.397 0.424 0.451d14 0.221 0.243 0.266 0.298 0.376 0.411 0.438 0.466Pg1 1.8062 1.9297 2.0528 2.2150 2.6250 2.7883 2.9109 3.0339

Qg1 0.963 0.784 0.604 0.363 0.0458 0.2957 0.4850 0.6758Qg2 0.7114 0.5566 0.4066 0.2130 0.010 0.176 0.294 0.406Qg3 0.2649 0.2983 0.3325 0.3785 0.4636 0.5155 0.5557 0.5971

Qg4 0.2216 0.2432 0.2660 0.2973 0.4295 0.4701 0.5025 0.5366

Qg5 0.2637 0.2830 0.3027 0.3292 0.3746 0.4040 0.4266 0.4498

Ploss 0.1564 0.1697 0.1852 0.2091 0.2827 0.3204 0.3518 0.3862

Table 11

FCPF (critical) results for 14-bus (for Case No. 13 of Table 1)

Item(s) MF distribution and corresponding critical results

0.0 (L) 0.3 (L) 0.6 (L) 1.0 (L) 1.0 (R) 0.6 (R) 0.3 (R) 0.0 (R)

V6 0.9779 0.9650 0.9532 0.9373 0.9038 0.8907 0.8806 0.8703

V7 0.8799 0.8655 0.8533 0.8364 0.8040 0.7917 0.7816 0.7709

V8 0.9102 0.8949 0.8815 0.8633 0.8283 0.8148 0.8041 0.7928

V9 0.9609 0.9451 0.9305 0.9112 0.8709 0.8551 0.8431 0.8310

V10 0.9779 0.9609 0.9452 0.9246 0.8828 0.8659 0.8532 0.8406

V11 1.0402 1.0260 1.0125 0.9949 0.9604 0.9452 0.9338 0.9224

V12 1.0992 1.0865 1.0741 1.0581 1.0268 1.0123 1.0016 0.9911

V13 1.0530 1.0394 1.0264 1.0094 0.9778 0.9627 0.9516 0.9406

V14 0.9695 0.9514 0.9346 0.9127 0.8668 0.8489 0.8357 0.8226

d6 0.844 0.876 0.902 0.935 0.999 1.016 1.030 1.045d7 0.776 0.794 0.806 0.823 0.851 0.856 0.860 0.866d8 0.607 0.623 0.635 0.650 0.678 0.683 0.687 0.692d9 0.878 0.917 0.949 0.991 1.073 1.097 1.116 1.135d10 0.869 0.911 0.948 0.994 1.086 1.114 1.135 1.157d11 0.847 0.891 0.929 0.977 1.079 1.108 1.130 1.153d12 0.844 0.891 0.932 0.983 1.100 1.131 1.155 1.180d13 0.849 0.898 0.940 0.993 1.106 1.139 1.164 1.190d14 0.894 0.942 0.984 1.037 1.142 1.175 1.200 1.225Pg1 7.6838 7.7586 7.7853 7.8084 7.8253 7.8366 7.8780 7.9240

Qg1 0.527 0.284 0.055 0.2508 0.7843 1.0559 1.2629 1.4731Qg2 4.3184 4.2413 4.1212 3.9664 3.7619 3.5127 3.3405 3.1814

Qg3 3.8560 3.7998 3.7178 3.6341 3.4070 3.3051 3.2483 3.2067

Qg4 1.4558 1.5518 1.6317 1.7341 2.0093 2.0587 2.0957 2.1332

Qg5 0.9034 0.9419 0.9732 1.0152 1.0794 1.1026 1.1211 1.1408

Ploss 3.0642 3.0628 3.0594 3.0513 3.0172 2.9623 2.9369 2.9246

Qloss 9.3983 9.5633 9.6329 9.7672 9.9534 9.9799 10.001 10.048


The second database refers to the formulation of

characteristic values for use in the uncertainty modeling

of the input parameters. In this paper, we have formed

this complete database for the characteristic values (/

a1; a2; a3; a4; a and b); corresponding to all the threetest systems, which are presented in Tables A5, A6, A7

and A8 of Appendix A.

The third database has been prepared by us to

simulate various line outage contingency cases. This is

necessary to observe the uncertainty based critical

performance of the system under various operating

conditions. The contingencies simulated in this paper

include single and multiple line outage cases. While

forming the database for multiple contingencies, the

maximum number of lines taken for outage has been

limited to 15% of the total lines existing in the original

network. Tables 1/3 indicate the database for selectedcontingencies of the 14-bus, 30-bus and 57-bus systems

respectively.

The results highlighted in this section may be split

into four groups to justify the objectives of this paper

efficiently. We now precisely present the results obtained

for all the three test systems, justifying these objectives.

Grouping of results are as follows:

. Group 1: Verification of FPF (base case) results. Theobjective is to justify fuzzy distribution of all output

states corresponding to base case conditions, in

agreement with the fuzzy distribution of input para-meters.

. Group 2: Verification of FCPF (critical) results. Theobjective is to justify fuzzy distribution of all output

Fig. 7. Load bus voltage distribution (14-bus).

Fig. 8. Load bus angle distribution (14-bus).

Fig. 9. Generation and loss distribution (14-bus).

Table A1

Line data for 14-bus system

Line code no. Buses at both ends Line impedances (p.u.)

R X B

1 1/2 0.01938 0.05917 0.02642 2/3 0.04699 0.19797 0.02193 2/7 0.05811 0.17632 0.01874 1/8 0.05403 0.22304 0.02465 2/8 0.05695 0.17388 0.01706 3/7 0.06701 0.17103 0.01737 8/4 0.00000 0.25202 0.00008 6/7 0.00000 0.20912 0.00009 6/5 0.00000 0.17615 0.0000

10 7/9 0.00000 0.55618 0.000011 6/9 0.00000 0.11001 0.000012 9/10 0.03181 0.08450 0.000013 4/11 0.09498 0.19890 0.000014 4/12 0.12291 0.25581 0.000015 4/13 0.06615 0.13027 0.000016 9/14 0.12711 0.27038 0.000017 10/11 0.08205 0.19207 0.000018 12/13 0.22092 0.19988 0.000019 13/14 0.17093 0.34802 0.000020 7/8 0.01335 0.04211 0.0064


states corresponding to critical conditions, in agree-

ment with the fuzzy distribution of input parameters.

. Group 3: To analyze the effect of various contingen-cies on maximum power transferring capability of the

network. The objective is to rank the contingencies

according to their severity.

. Group 4: To identify the critical busbars at the steadystate voltage stability threshold. The objective is to

monitor its status subject to parameter uncertainties.

Results for Group 1 and Group 2 have been obtained

for 14-bus, 30-bus and 57-bus test systems, for all

contingency cases listed in Tables 1/3. Group-1 results

show the possibility distributions of various items at the

base case (i.e. per unit voltage magnitudes/angles at all

load buses, per unit active power generation at the slack

bus, per unit reactive power generations at slack bus and

all PV buses, and per unit power losses of the system)

corresponding to parameter uncertainties. It is observed

from these results that all items satisfactorily followed

similar membership patterns as considered for the input

parameters. In order to justify this claim, we presented

in Tables 10 and 11 the detail numerical values showing

the possibility distributions obtained for the 14-bus base

case and critical results subject to a contingency opera-

tion, i.e. a multiple line outage case as in Case 13 of

Table 1.

Table A2

Line data for 30-bus system


R X B

1 1/2 0.0192 0.0575 0.02642 1/8 0.0452 0.1852 0.02043 2/11 0.0570 0.1737 0.01844 8/11 0.0132 0.0379 0.00425 2/5 0.0472 0.1983 0.02096 2/13 0.0581 0.1763 0.01877 11/13 0.0119 0.0414 0.00458 5/7 0.0460 0.1160 0.01029 13/7 0.0267 0.0820 0.0085

10 13/3 0.0120 0.0420 0.004511 13/9 0.0000 0.2080 0.000012 13/10 0.0000 0.5560 0.000013 9/4 0.0000 0.2080 0.000014 9/10 0.0000 0.1100 0.000015 11/12 0.0000 0.2560 0.000016 12/6 0.0000 0.1400 0.000017 12/14 0.1231 0.2559 0.000018 12/15 0.0662 0.1304 0.000019 12/16 0.0945 0.1987 0.000020 14/15 0.2210 0.1997 0.000021 16/17 0.0824 0.1923 0.000022 15/18 0.1070 0.2185 0.000023 18/19 0.0639 0.1292 0.000024 19/20 0.0340 0.0680 0.000025 10/20 0.0936 0.2090 0.000026 10/17 0.0324 0.0845 0.000027 10/21 0.0348 0.0749 0.000028 10/22 0.0727 0.1499 0.000029 21/22 0.0116 0.0236 0.000030 15/23 0.1000 0.2020 0.000031 22/24 0.1150 0.1790 0.000032 23/24 0.1320 0.2700 0.000033 24/25 0.1885 0.3292 0.000034 25/26 0.2544 0.3800 0.000035 25/27 0.1093 0.2087 0.000036 27/28 0.0000 0.3960 0.000037 27/29 0.2198 0.4153 0.000038 27/30 0.3202 0.6027 0.000039 29/30 0.2399 0.4533 0.000040 3/28 0.0636 0.2000 0.021441 13/28 0.0169 0.0065 0.0599

Table A3

Line data for 57-bus system (for lines 1/40)


R X B

1 1/2 0.0083 0.0280 0.06452 2/3 0.0298 0.0850 0.04093 3/8 0.0112 0.0366 0.01904 8/9 0.0625 0.1320 0.01295 8/6 0.0430 0.1480 0.01746 6/12 0.0200 0.1020 0.01387 6/4 0.0339 0.1730 0.02358 4/5 0.0099 0.0505 0.02749 5/10 0.0369 0.1679 0.0220

10 5/11 0.0258 0.0848 0.010911 5/7 0.0648 0.2950 0.038612 5/13 0.0481 0.1580 0.020313 13/14 0.0132 0.0434 0.005514 3/15 0.0269 0.0869 0.011515 1/15 0.0178 0.0910 0.049416 1/16 0.0454 0.2060 0.027317 1/17 0.0238 0.1080 0.014318 3/15 0.0162 0.0530 0.027219 8/18 0.0000 0.5550 0.000020 8/18 0.0000 0.4300 0.000021 9/6 0.0302 0.0641 0.006222 12/4 0.0139 0.0712 0.009723 10/7 0.0277 0.1262 0.016424 11/13 0.0223 0.0732 0.009425 7/13 0.0178 0.0580 0.030226 7/16 0.0180 0.0813 0.010827 7/17 0.0397 0.1790 0.023828 14/15 0.0171 0.0547 0.007429 18/19 0.4610 0.6850 0.000030 19/20 0.2830 0.4340 0.000031 21/20 0.0000 0.7767 0.000032 21/22 0.0736 0.1170 0.000033 22/23 0.0099 0.0152 0.000034 23/24 0.1660 0.2560 0.004235 24/25 0.0000 1.1820 0.000036 24/25 0.0000 1.2300 0.000037 24/26 0.0000 0.0473 0.000038 26/27 0.1650 0.2540 0.000039 27/28 0.0618 0.0954 0.000040 28/29 0.0418 0.0587 0.0000


The graphical interpretation of results shown in

Tables 10 and 11 is presented in Figs. 7/9, markedseparately as base case and critical. Fig. 7 confirms the

trapezoidal distribution of all load bus voltage magni-

tudes (V6/V14); Fig. 8 confirms the trapezoidal distribu-tion of all load bus voltage angles (d6/d14); and Fig. 9confirms the trapezoidal distribution of active/reactive

power generations at the slack bus (Pg1, Qg1), reactive

power generations at the PV buses (Qg2/Qg5), andsystem losses (Ploss, Qloss).

Due to large number of items involved in Group 1

results, and the space constraint, we limit our complete

presentation of base case and critical results to 14-bus

only. However, limited FPF (base case) results for few

voltage magnitudes have been presented for 14-bus, 30-

bus and 57-bus systems as shown in Tables 7/9 underthe nomenclature Vb. In Table 7 we presented the

distribution of Vb for bus-7 of 14-bus system subject

to line outage (Case-13 of Table 1). These values areeventually same as item V7 of Table 10 that holds all

base case items of 14-bus system. In Table 8 we

presented the distribution of Vb for bus-29 of 30-bus

system subject to line outage (Case-12 of Table 2). In

Table 9 we presented the distribution of Vb for bus-51 of

57-bus system subject to line outage (Case-15 of Table

3). Graphical interpretation for the distribution of Vbpresent in Tables 7/9 has been shown in Fig. 3, whichjustifies a clear trapezoidal distribution for 14-bus, 30-

bus and 57-bus systems.

In addition to the above, Tables 7/9 also figurenumerical distributions of few other FCPF (critical)

result items coming under Group 2. These items are the

critical voltage Vc, critical loading level lc and the

proposed FVSI for 14-bus, 30-bus and 57-bus systems

pertaining to those busbars only, which are critical forthe particular outage condition considered. Although all

the listed contingency cases have been successfully tried

for the analysis, results have been presented here

selectively (as in Tables 7/9), with a clear mention ofthe contingency case and critical busbar in each table.

Graphical interpretation for the distribution of Vcpresent in Tables 7/9 has been shown in Fig. 4, thatjustifies a clear trapezoidal distribution for 14-bus, 30-bus and 57-bus systems. Similarly, graphical interpreta-

tion for the distributions of lc and FVSI present inTables 7/9 has been shown in Figs. 5 and 6 respectively,justifying clearly trapezoidal distributions for 14-bus,

30-bus and 57-bus systems.

Group 3 and Group 4 results obtained for all

contingency cases reported in Tables 1/3, have beenpresented in Tables 4/6 for 14-bus, 30-bus and 57-bussystems respectively. The severity of contingency cases

has been assessed through the evaluation of critical

loading level lc. From the results of Tables 4/6, weobserved that the critical loading level lc is different for

different contingency conditions. It is the highest for

normal healthy operating conditions without any con-

tingency and the least for the most severe contingencies.

A lower value of lc indicates a reduction in themaximum power transfer capability that implies a

higher degree of severity for the said contingency. In

view of this, we monitored the value of lc to scale the

contingencies according to their severity ranking and

presented them in the name of critical rank, as shown in

Tables 4/6. Critical busbars have been identifiedthrough the evaluation of the proposed FVSI. Regard-

ing the identification of critical busbars we observedthat it is a function of network loading and type of

contingency. Monitoring the proposed FVSI values for

all network buses and comparing them for the highest

FVSI, critical busbars have been identified and pre-

Table A4

Line data for 57-bus system (for lines 41/80)


R X B

41 12/29 0.0000 0.0648 0.000042 25/30 0.1350 0.2020 0.000043 30/31 0.3260 0.4970 0.000044 31/32 0.5070 0.7550 0.000045 32/33 0.0392 0.0360 0.000046 34/32 0.0000 0.9530 0.000047 34/35 0.0520 0.0780 0.001648 35/36 0.0430 0.0537 0.000849 36/37 0.0290 0.0366 0.000050 37/38 0.0651 0.1009 0.001051 37/39 0.0239 0.0379 0.000052 36/40 0.0300 0.0466 0.000053 22/38 0.0192 0.0295 0.000054 11/41 0.0000 0.7490 0.000055 41/42 0.2070 0.3520 0.000056 41/43 0.0000 0.4120 0.000057 38/44 0.0289 0.0585 0.001058 15/45 0.0000 0.1042 0.000059 14/46 0.0000 0.0735 0.000060 46/47 0.0230 0.0680 0.001661 47/48 0.0182 0.0233 0.000062 48/49 0.0834 0.1290 0.002463 49/50 0.0801 0.1280 0.000064 50/51 0.1386 0.2200 0.000065 10/51 0.0000 0.0712 0.000066 13/49 0.0000 0.1910 0.000067 29/52 0.1442 0.1870 0.000068 52/53 0.0762 0.0984 0.000069 53/54 0.1878 0.2320 0.000070 54/55 0.1732 0.2265 0.000071 11/43 0.0000 0.1530 0.000072 44/45 0.0624 0.1242 0.002073 40/56 0.0000 1.1950 0.000074 56/41 0.5530 0.5490 0.000075 56/42 0.2125 0.3540 0.000076 39/57 0.0000 1.3550 0.000077 57/56 0.1740 0.2600 0.000078 38/49 0.1150 0.1770 0.003079 38/48 0.0312 0.0482 0.000080 5/55 0.0000 0.1205 0.0000


sented for all contingency cases, as shown in Tables 4/6.From these results it is clear that critical busbar location

is different for different contingency cases.

Although the computation of results under Group 3

and Group-4 has been carried out for various possible

uncertainty levels (i.e. MF/0.0, 0.3, 0.6, and 1.0 oneither side slopes of the trapezoid), it could not be

possible to present them all, due to want of space. It is to

be noted that the results reported in Tables 4/6correspond to a MF value 1, taken on the left side

slope of the trapezoidal distribution function.

In the course of analyzing these output results plotted

in Figs. 3/9, it is observed that the trend of trapezoidaldistribution is retained. This finding justifies the most

demanding objective of the FST application, as implied

in [9]. According to the author of Ref. [9], the results of

FPF are mainly the possibility distributions of line

power flows and bus voltages, which are also extensively

verified in this paper. Rather, in view of the FCPF

findings, we would like to add that the results of the

proposed FCPF might be considered as the possibility

distributions of critical voltages, critical loading levels

and critical FVSIs, from which a lot of information may

be derived to improve the operator intelligence in

handling power system uncertainties.

Table A5

Fuzzy modeling of input parameters (14-bus)

Item(s)* Characteristic values for trapezoidal MF

a1 a2 a a3 a4 b

V2 1.00 1.03 0.03 1.05 1.08 0.03

V3 0.98 1.00 0.02 1.02 1.04 0.02

V4 1.03 1.06 0.03 1.08 1.11 0.03

V5 1.06 1.08 0.02 1.10 1.12 0.02

PG2 0.30 0.40 0.10 0.50 0.60 0.10

QG2 0.30 0.35 0.05 0.45 0.50 0.05

QG3 0.20 0.25 0.05 0.30 0.35 0.05

QG4 0.05 0.10 0.05 0.15 0.20 0.05

QG5 0.10 0.15 0.05 0.20 0.25 0.05

PL2 0.15 0.20 0.05 0.25 0.30 0.05

PL3 0.85 0.90 0.05 0.95 1.00 0.05

PL4 0.05 0.10 0.05 0.15 0.20 0.05

PL7 0.40 0.45 0.05 0.50 0.55 0.05

PL8 0.01 0.06 0.05 0.10 0.15 0.05

PL9 0.20 0.25 0.05 0.30 0.35 0.05

PL10 0.02 0.07 0.05 0.11 0.16 0.05

PL11 0.01 0.02 0.01 0.04 0.05 0.01

PL12 0.03 0.05 0.02 0.08 0.10 0.02

PL13 0.05 0.10 0.05 0.15 0.20 0.05

PL14 0.08 0.13 0.05 0.17 0.22 0.05

QL2 0.05 0.10 0.05 0.15 0.20 0.05

QL3 0.10 0.15 0.05 0.20 0.25 0.05

QL4 0.02 0.05 0.03 0.09 0.12 0.03

QL7 0.01 0.03 0.02 0.05 0.07 0.02

QL8 0.00 0.01 0.01 0.02 0.03 0.01

QL9 0.10 0.15 0.05 0.20 0.25 0.05

QL10 0.03 0.05 0.02 0.07 0.09 0.02

QL11 0.00 0.01 0.01 0.02 0.03 0.01

QL12 0.00 0.01 0.01 0.02 0.03 0.01

QL13 0.04 0.05 0.01 0.06 0.07 0.01

QL14 0.03 0.04 0.01 0.06 0.07 0.01

* Missing parameters may be assumed as zero.

Table A6



a1 a2 a a3 a4 b

V2 1.035 1.040 0.005 1.050 1.055 0.005

V3 1.000 1.005 0.005 1.015 1.020 0.005

V4 1.000 1.005 0.005 1.015 1.020 0.005

V5 1.075 1.080 0.005 1.084 1.089 0.005

V6 1.065 1.070 0.005 1.072 1.077 0.005

PG2 0.200 0.300 0.100 0.500 0.600 0.100

QG2 0.150 0.250 0.100 0.350 0.450 0.100

QG3 0.250 0.300 0.050 0.400 0.450 0.050

QG4 0.150 0.200 0.050 0.300 0.350 0.050

QG5 0.150 0.250 0.100 0.350 0.450 0.100

QG6 0.100 0.150 0.050 0.250 0.300 0.050

PL2 0.160 0.210 0.050 0.220 0.270 0.050

PL3 0.250 0.290 0.040 0.310 0.350 0.040

PL5 0.800 0.900 0.100 1.000 1.100 0.100

PL7 0.175 0.200 0.025 0.250 0.275 0.025

PL8 0.015 0.020 0.005 0.030 0.035 0.005

PL10 0.050 0.055 0.005 0.065 0.070 0.005

PL11 0.050 0.070 0.020 0.080 0.100 0.020

PL12 0.050 0.100 0.050 0.120 0.170 0.050

PL14 0.055 0.060 0.005 0.065 0.070 0.005

PL15 0.075 0.080 0.005 0.085 0.090 0.005

PL16 0.025 0.030 0.005 0.040 0.045 0.005

PL17 0.075 0.085 0.010 0.095 0.105 0.010

PL18 0.025 0.030 0.005 0.035 0.040 0.005

PL19 0.075 0.090 0.015 0.100 0.115 0.015

PL20 0.015 0.020 0.005 0.025 0.030 0.005

PL21 0.100 0.150 0.050 0.200 0.250 0.050

PL23 0.025 0.030 0.005 0.035 0.040 0.005

PL24 0.050 0.085 0.035 0.090 0.125 0.035

PL26 0.010 0.025 0.015 0.045 0.060 0.015

PL29 0.015 0.020 0.005 0.030 0.035 0.005

PL30 0.075 0.100 0.025 0.110 0.135 0.025

QL2 0.100 0.120 0.020 0.135 0.155 0.020

QL3 0.200 0.275 0.075 0.325 0.400 0.075

QL5 0.150 0.180 0.030 0.200 0.230 0.030

QL7 0.080 0.105 0.025 0.115 0.140 0.025

QL8 0.005 0.010 0.005 0.015 0.020 0.005

QL10 0.005 0.015 0.010 0.025 0.035 0.010

QL11 0.005 0.010 0.005 0.020 0.025 0.005

QL12 0.050 0.070 0.020 0.080 0.100 0.020

QL14 0.005 0.010 0.005 0.020 0.025 0.005

QL15 0.005 0.015 0.010 0.035 0.045 0.010

QL16 0.005 0.010 0.005 0.020 0.025 0.005

QL17 0.050 0.055 0.005 0.060 0.065 0.005

QL18 0.000 0.005 0.005 0.015 0.020 0.005

QL19 0.020 0.030 0.010 0.040 0.050 0.010

QL20 0.001 0.004 0.003 0.010 0.013 0.003

QL21 0.050 0.100 0.050 0.120 0.170 0.050

QL23 0.010 0.015 0.005 0.020 0.025 0.005

QL24 0.050 0.060 0.010 0.075 0.085 0.010

QL26 0.015 0.020 0.005 0.025 0.030 0.005

QL29 0.000 0.005 0.005 0.015 0.020 0.005

QL30 0.010 0.015 0.005 0.025 0.030 0.005



8. Conclusions

In the majority of cases, the imprecision and uncer-

tainties in loads and generations are generally over-

looked for simplistic reasons. However, to obtain more

viable, realistic solutions and assure proper operations

of the power systems the merits of FST and other

emerging technologies must be explored. In this paper,

the principles of FST have been exploited and a

judicious choice is made for fuzzy modeling of these

parameters by using trapezoidal and triangular MFs for

the purpose of steady state voltage stability analysis. In

Section 7, FPF and FCPF results on the basis of

trapezoidal distribution of input uncertainties have

been presented. However, similar observations have

been verified for triangular membership distribution of

input parameters too. Fuzzy reasoning offers a realistic

way to understand these problems and also allows

Table A7



a1 a2 a a3 a4 b

V2 1.035 1.040 0.005 1.050 1.055 0.005

V3 1.000 1.005 0.005 1.015 1.020 0.005

V4 1.075 1.080 0.005 1.084 1.089 0.005

V5 1.000 1.005 0.005 1.015 1.020 0.005

V6 1.065 1.070 0.005 1.072 1.077 0.005

V7 1.005 1.010 0.005 1.020 1.025 0.005

PG2 0.100 0.110 0.010 0.125 0.135 0.010

PL2 0.100 0.110 0.010 0.125 0.135 0.010

PL3 0.120 0.140 0.020 0.160 0.180 0.020

PL5 0.350 0.430 0.080 0.454 0.534 0.080

PL7 0.200 0.250 0.050 0.290 0.340 0.050

PL9 0.070 0.080 0.010 0.100 0.110 0.010

PL10 0.010 0.015 0.005 0.025 0.030 0.005

PL13 0.090 0.100 0.010 0.120 0.130 0.010

PL14 0.070 0.075 0.005 0.085 0.090 0.005

PL15 0.100 0.110 0.010 0.130 0.140 0.010

PL16 0.100 0.115 0.015 0.145 0.160 0.015

PL17 0.100 0.110 0.010 0.130 0.140 0.010

PL18 0.140 0.160 0.020 0.184 0.204 0.020

PL19 0.005 0.010 0.005 0.016 0.021 0.005

PL20 0.005 0.010 0.005 0.016 0.021 0.005

PL23 0.025 0.030 0.005 0.036 0.041 0.005

PL25 0.025 0.030 0.005 0.036 0.041 0.005

PL27 0.035 0.040 0.005 0.046 0.051 0.005

PL28 0.015 0.020 0.005 0.032 0.037 0.005

PL29 0.090 0.100 0.010 0.120 0.130 0.010

PL30 0.010 0.015 0.005 0.025 0.030 0.005

PL31 0.020 0.025 0.005 0.031 0.036 0.005

PL32 0.002 0.004 0.002 0.008 0.010 0.002

PL33 0.020 0.025 0.005 0.031 0.036 0.005

PL35 0.010 0.015 0.005 0.025 0.030 0.005

PL38 0.090 0.100 0.010 0.120 0.130 0.010

PL41 0.005 0.010 0.005 0.020 0.025 0.005

PL42 0.060 0.070 0.010 0.072 0.082 0.010

PL43 0.010 0.015 0.005 0.025 0.030 0.005

PL44 0.090 0.100 0.010 0.120 0.130 0.010

PL47 0.145 0.155 0.010 0.160 0.170 0.010

PL49 0.100 0.110 0.010 0.130 0.140 0.010

PL50 0.120 0.130 0.010 0.150 0.160 0.010

PL51 0.130 0.140 0.010 0.160 0.170 0.010

PL52 0.040 0.045 0.005 0.055 0.060 0.005

PL53 0.080 0.090 0.010 0.110 0.120 0.010

PL54 0.030 0.037 0.007 0.045 0.052 0.007

PL55 0.060 0.065 0.005 0.070 0.075 0.005

PL56 0.060 0.070 0.010 0.082 0.092 0.010

PL57 0.050 0.060 0.010 0.075 0.085 0.010

* Remaining part continued in Table A8.

Table A8

Fuzzy modeling of input parameters (57-bus), continued from Table

A7


a1 a2 a a3 a4 b

QG2 0.120 0.140 0.020 0.160 0.180 0.020

QG3 0.350 0.430 0.080 0.454 0.534 0.080

QG4 0.200 0.250 0.050 0.290 0.340 0.050

QG5 0.070 0.080 0.010 0.100 0.110 0.010

QG6 0.010 0.015 0.005 0.025 0.030 0.005

QG7 0.090 0.100 0.010 0.120 0.130 0.010

QL2 0.080 0.090 0.010 0.105 0.115 0.010

QL3 0.080 0.090 0.010 0.110 0.120 0.010

QL5 0.080 0.090 0.010 0.110 0.120 0.010

QL7 0.120 0.130 0.010 0.150 0.160 0.010

QL9 0.010 0.015 0.005 0.025 0.030 0.005

QL10 0.005 0.008 0.003 0.012 0.015 0.003

QL13 0.005 0.010 0.005 0.016 0.021 0.005

QL14 0.015 0.020 0.005 0.026 0.031 0.005

QL15 0.010 0.015 0.005 0.025 0.030 0.005

QL16 0.005 0.008 0.003 0.012 0.015 0.003

QL17 0.020 0.025 0.005 0.035 0.040 0.005

QL18 0.035 0.040 0.005 0.055 0.060 0.005

QL19 0.001 0.002 0.001 0.004 0.005 0.001

QL20 0.003 0.005 0.002 0.011 0.013 0.002

QL23 0.005 0.008 0.003 0.014 0.017 0.003

QL25 0.005 0.008 0.003 0.016 0.019 0.003

QL27 0.001 0.002 0.001 0.004 0.005 0.001

QL28 0.008 0.011 0.003 0.015 0.018 0.003

QL29 0.010 0.013 0.003 0.019 0.022 0.003

QL30 0.007 0.009 0.002 0.013 0.015 0.002

QL31 0.010 0.015 0.005 0.023 0.028 0.005

QL32 0.001 0.002 0.001 0.004 0.005 0.001

QL33 0.008 0.010 0.002 0.012 0.014 0.002

QL35 0.005 0.008 0.003 0.012 0.015 0.003

QL38 0.015 0.018 0.003 0.022 0.025 0.003

QL41 0.005 0.008 0.003 0.012 0.015 0.003

QL42 0.035 0.040 0.005 0.050 0.055 0.005

QL43 0.004 0.007 0.003 0.013 0.016 0.003

QL44 0.005 0.008 0.003 0.014 0.017 0.003

QL47 0.105 0.110 0.005 0.122 0.127 0.005

QL49 0.035 0.040 0.005 0.050 0.055 0.005

QL50 0.090 0.100 0.010 0.110 0.120 0.010

QL51 0.020 0.026 0.006 0.040 0.046 0.006

QL52 0.010 0.016 0.006 0.028 0.034 0.006

QL53 0.005 0.008 0.003 0.012 0.015 0.003

QL54 0.010 0.012 0.002 0.016 0.018 0.002

QL55 0.025 0.030 0.005 0.040 0.045 0.005

QL56 0.018 0.020 0.002 0.024 0.026 0.002

QL57 0.015 0.017 0.002 0.023 0.025 0.002



incorporating the operators own intuition, intelligence

and knowledge acquired from past experiences in

solving them. The basic objective of this paper is to

observe whether the output results follow some kind offuzzy membership or possibility distribution. This

objective has been validated through case studies on

IEEE 14-bus, 30-bus and 57-bus test systems in view of

the fuzzy results at the base case and critical point under

various system configurations. Analysis of the results

reveals that the voltage magnitudes/angles, power gen-

erations, and power losses in the system follow similar

type of fuzzy memberships. Also the paper introduces anovel fuzzy voltage stability index for identifying critical

busbars subject to normal and contingency mode of

operations. The critical results reflect the maximum

loading margin in the system before the voltage may

collapse. The proposed fuzzy approach enables power

system operators and planners to operate the system

more realistically for a given range of loads and

generations. The above findings make it evident thatthe FST can be a very useful supplement to the

traditional mathematical tools in solving power system

problems. Thus it is concluded that for voltage stability

analysis the fuzzy reasoning can be successfully applied

in modeling the imprecision and uncertainty offered by

the system parameters and variables to understand the

system behavior during normal and contingency condi-

tions as well.

Appendix A

Tables A1, A2, A3, A4, A5, A6, A7 and A8

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A novel fuzzy index for steady state voltage stability analysis and identification of critical busbarsIntroductionBasic facts about fuzzy set theory (FST)Fuzzy modeling of input parametersFuzzy power flow (FPF)Fuzzy continuation power flow (FCPF)Fuzzy voltage stability index (FVSI)Case study and numerical resultsConclusionsAppendicesAppendix A

References

A Novel Fuzzy application to power system

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