A

Ya

b

a

ARRAA

KEPSV

1

ouaassmoeoiow

osstvon

(

0d

Electric Power Systems Research 79 (2009) 265–271

Contents lists available at ScienceDirect

Electric Power Systems Research

journa l homepage: www.e lsev ier .com/ locate /epsr

new node voltage stability index based on local voltage phasors

ang Wanga, Wenyuan Lib,∗, Jiping Lua

The State Key Laboratory of Power Transmission Equipment & System Security and New Technology, Chongqing University, Chongqing, 400030, ChinaBritish Columbia Transmission Corporation, Suite 1100, Four Bentall Center, 1055 Dunsmuir Street, P.O. Box 49260, Vancouver, BC V7X 1V5, Canada

r t i c l e i n f o

rticle history:eceived 19 September 2007eceived in revised form 17 June 2008

a b s t r a c t

The paper proposes an equivalent system model (ESM), which includes effects of both local network andsystem outside the local network. A new node voltage stability index called the equivalent node voltagecollapse index (ENVCI), which is based on ESM and uses only local voltage phasors, is presented. The

ccepted 18 June 2008vailable online 26 July 2008

eywords:nergy management system (EMS)hasor measurement unit (PMU)

main advantages of ENVCI are accuracy in modeling and calculations, and ease in real time or on-lineapplications. The simulation results show that the ENVCI can identify not only the weakest node (bus)causing system instability but also the system voltage collapse point when it is near zero. This featureenables us to set an index threshold to monitor and predict system stability on-line so that a proper actioncan be taken to prevent the system from collapse.

Crown Copyright © 2008 Published by Elsevier B.V. All rights reserved.

scmddow[TvadTstaiea

ystem collapseoltage stability index

. Introduction

Considerable voltage instability-related outage events haveccurred around the world and resulted in major system fail-res (blackouts) in recent years. Voltage stability has becomen important concern for utilities. Static voltage stability can bessessed using continuation power flow calculations [1–3]. Manytatic voltage assessment methods have been proposed so far,uch as the minimum singularity value method, mode analysisethod and sensitivity method [4–6]. The main disadvantages

f the continuation power flow-based methods include consid-rable computational efforts making implementation difficult inn-line applications, possible premature divergence due to numer-cal instability in power flow calculations, inconsistency betweenff-line model and real life situation, and incapability of identifyingeak lines or buses that cause system collapse.

It is well known that unlike angle instability, voltage instabilityften starts in a local network and gradually extends to the wholeystem. This feature makes the process of system losing voltagetability much slower (a few seconds or even longer) compared to

hat of losing angle stability, and also enables us to predict staticoltage stability using local measurements. There are two typesf local evaluation techniques for voltage stability: line-based andode (bus)-based techniques. Conceptually, if a line or a node in the

∗ Corresponding author. Tel.: +1 604 699 7379.E-mail addresses: wangyanghh82@126.com, wangyanghh@hotmail.com

Y. Wang), wen.yuan.li@bctc.com (W. Li), lujiping@cqu.edu.cn (J. Lu).

ef

•

•

378-7796/$ – see front matter. Crown Copyright © 2008 Published by Elsevier B.V. All rioi:10.1016/j.epsr.2008.06.010

ystem is critically voltage-instable, the whole system approaches aollapse point [7–14]. Refs. [7,8] presented voltage stability assess-ent techniques using local bus phasors whereas Refs. [9–11]

erived different line-based voltage stability indices. A commonemerit of the existing line-based indices is the fact that impactsf the rest of system outside the line have been ignored and thisill lead to inaccurate or even incorrect results in some cases. Ref.

12] proposed an internal and external impedance method usinghevenin theorem whereas Ref. [13] introduced the limit of nodeoltage into the method. Ref. [14] proved the concept of the internalnd external impedance method using Tellegen’s theory. The mainisadvantage of this method is the assumption that the equivalenthevenin voltage and impedance are constant in the two or moreystem states. If the two system states are far apart, this assump-ion is obviously invalid but if they are too close, it may result in

large calculation error in the estimate of equivalent Theveninmpedance since the estimation process may be associated with anxtremely small value in the denominator. This causes inaccuracynd difficulties in the actual implementation.

This paper proposes a new voltage stability index, which is calledquivalent node voltage collapse index (ENVCI). It has the followingeatures:

Compared to the existing line-based voltage indices, the effect of

the rest of a system outside the local network is included throughan equivalent model of the system. This assures accuracy of theindex in modeling.Compared to the internal and external impedance method, theequivalent system impedance that needs to be estimated using

ghts reserved.

2 ystem

•

•

•

mla

2

2

adtpNippalrPatlg

n

�I

w

Y

wbpin

S

Ls

S

�IFwiapa

2

Ea

66 Y. Wang et al. / Electric Power S

two system states is only a small part of the total impedancewhereas the impedances of local network (branches that aredirectly connected to the considered node) are known and neednot be estimated using two system states.Compared to the conventional method based on continuationpower flows, the computation of ENVCI is much faster since itis associated with very simple calculations and no system-widepower flow is required.The calculation of ENVCI only requires the information of localvoltage phasors, which can either be obtained via synchronizedphasor measurement units (PMU) or through the state estima-tion of energy management system (EMS) at control centers ofutilities. This enables ENVCI to be easily applied in on-line (EMS)or real time (PMU) environment.The presented method can identify the weakest node(s) of thesystem in light of the ENVCI values of all monitored nodes. TheENVCI at the weakest node will be very near zero when the sys-tem approaches its voltage collapse point. Therefore, a thresholdof ENVCI can be easily set up to trigger an emergency remedialaction scheme to protect the system from voltage collapse.

The rest of the paper is organized as follows. The equivalentodel and formulation of ENVCI are presented in Section 2. Simu-

ation results are given in Section 3 to demonstrate the feasibilitynd effectiveness of ENVCI, followed by conclusions in Section 4.

. Equivalent network model and formulation of ENVCI

.1. Equivalent local network model (ELNM)

As shown in Fig. 1(a), a local network for any node (bus) N intransmission system is defined as all the components that are

irectly connected to it. The local network is divided into two por-ions. The first portion is represented by an equivalent outgoingower flow Pon + jQon. It is the total power flowing out of the node, which is the sum of line power flows on all the lines with outgo-

ng flows, plus the load, generator power and compensated reactiveower at the node N. Note that all the individual power flow com-onents within the dashed-line frame at the right side of the nodere not shown in the figure. The second portion includes all theines with power flows entering the node N, each of which is rep-esented by a �-circuit at its left side. The Pn + jQn in Fig. 1(b) is

on + jQon plus the reactive charging powers at the receiving end ofll the lines in the second portion. The reactive charging powers athe sending end of all the lines are included in the system equiva-ent impedance that is introduced in the equivalent system modeliven in Section 2.2. It is obvious that the outgoing current at the

swptw

Fig. 1. (a) Original local network model; (

s Research 79 (2009) 265–271

ode N can be expressed as

n = S∗n

�V∗n

=M∑

i=1

Yni( �Vi − �Vn) =M∑

i=1

Yni�Vi −

M∑i=1

Yni�Vn

=M∑

i=1

Yni�Vi − Yeq �Vn (1)

here,

ni = 1Zni

, Yeq =M∑

i=1

Yni

here, Yni and Zni are the admittance and impedance of linesetween the nodes i and n respectively; �Vi and �Vn are the voltagehasors at the node i and n; the superscript * represents conjugate

n the paper and thus �V∗n is the conjugate phasor of �Vn; and M is the

umber of lines with power flows entering the node N.Multiplying �V∗

n at the both side of Eq. (1) yields

∗n = �V∗

n�In = �V∗

n

M∑i=1

Yni�Vi − �V∗

n Yeq �Vn (2)

etting �Veq = Veq∠�eq =∑M

i=1Yni�Vi/Yeq and �Vn = Vn∠�n, and sub-

tituting them into Eq. (2), we have

∗n = ( �V∗

n�Veq − V2

n )Yeq (3)

n = ( �Veq − �Vn)Yeq (4)

rom Eqs. (3) and (4), an equivalent single line model (i.e., ELNM)hich represents the second portion of the local network contain-

ng the lines with power flows entering the node N is obtainednd shown in Fig. 1(b). In this model, only voltage phasors and linearameters (impedances) of the second portion of the local networkre needed. Note that Zeq in the figure is the reciprocal of Yeq.

.2. Equivalent system model (ESM)

A dummy voltage source �Ek with impedance Zkm is added to theLNM to include the effect of the system outside the local network,s shown in Fig. 2. Note that all the grounding branches repre-

enting reactive charging powers at the sending ends of the linesith power flows entering the node N have been assumed to beart of Zkm, which can be estimated later. The �Ek and Zkm will havehe exactly same effect as the whole system outside the local net-ork as long as they can assure the identical voltage phasors and

b) Equivalent local network model.

Y. Wang et al. / Electric Power Systems Research 79 (2009) 265–271 267

lent sy

pe

(

wlreE

L

K

L

a

E

IZI

E

E

H

K

BE(

Z

wlk

ictpeaveftIeE

2

sf

P

IleZi

Ealatawtto

Fig. 2. Equiva

ower flows for the equivalent line. To make this hold, the followingquation has to be satisfied:

Pkm + jQkm)∗ = �V∗eq ·

�Ek − �Veq

Zkm= �V∗

eq ·�Veq − �Vn

Zeq(5)

here, Pkm and Qkm are the real and reactive powers flowing into theocal network, which corresponds to

∑Mi=1(Pi + jQi) in Fig. 1. The Zkm

epresents the equivalent system impedance that the power flowncountered in the system before it reaches the local network. Fromq. (5), the following equations can be derived:

Veq∠ − �eq · Ek∠�k − Veq∠�eq

Zkm= Veq∠ − �eq · Veq∠�eq − Vn∠�n

ZeqEk∠(�k − �eq) − Veq

Zkm= Veq − Vn∠(�n − �eq)

ZeqZkm

Zeq= Ek∠(�k − �eq) − Veq

Veq − Vn∠(�n − �eq)

etting Zkn = Zkm + Zeq and introducing a complex coefficient K yields

= Zkm + Zeq

Zeq= Zkn

Zeq= Ek∠(�k − �eq) − Vn∠(�n − �eq)

Veq − Vn∠(�n − �eq)(6)

etting �E′k = E′

k∠�′

k= Ek∠(�k − �eq), �V ′

n = V ′n∠�′

n = Vn∠(�n − �eq)

nd �V ′eq = V ′eq∠0 = Veq in Eq. (6), we have the following expression:

�′k = K · V ′

eq + (1 − K) �V ′n (7)

t is assumed that the equivalent voltage source �E′k and impedance

km are constant between two adjacent system equilibrium states.t follows from Eq. (7) that:

�′k = K · V ′

eq 1 + (1 − K) �V ′n1 (8)

�′k = K · V ′

eq2 + (1 − K) �V ′n2 (9)

ere, subscripts 1 and 2 represent the system states 1 and 2.Solving Eqs. (8) and (9) yields:

= 1

1 − (V ′eq1 − V ′

eq2)/( �V ′n1 − �V ′

n2)(10)

y substituting K into Eq. (8) or (9), �E′k can be obtained and then

k ∠ �k can be calculated. With K, it is easy to calculate Zkn from Eq.6):

= K · Zeq (11)

kn

In the above derivation process, a single line ESM is obtained, inhich effects of both the local network and the system outside the

ocal network have been established. By using this model, the twoey quantities that are needed to calculate the new ENVCI, which

msloa

stem model.

s derived in the next subsection, can be either measured or cal-ulated from the node voltage phasors at two ends of the lines inhe second portion of the local network. Apparently, the voltagehasor Vn ∠ �n at the node N can be directly measured whereas thequivalent source voltage Ek ∠ �k can be estimated from the volt-ge phasors and line parameters through intermediate equivalentoltage phasor Veq ∠ �eq and equivalent parameter Yeq. It should bemphasized that the ESM derived here is not a single equivalenceor the whole system but an equivalence representing the effects ofhe whole system on a single node (bus) in a specific system state.n other words, each node in a system state corresponds to a differ-nt ESM. These ESMs of individual nodes are used to calculate theirNVCI indices.

.3. Formulation of ENVCI

With the single line equivalent system model for the node N ashown in Fig. 2, the outgoing power at this node has to satisfy theollowing simple power flow equations:

n + jQn = �Vn ·( �Ek − �Vn

Zkn

)∗(12)

f the voltage phasors at the two nodes of the equivalent singleine model are expressed in the rectangular coordinates, i.e., �Ek =k + jfk and �Vn = en + jfn, and the line impedance is expressed bykn = Rkn + jXkn, Eq. (12) can be separated into a real part and anmaginary part as follows:

PnRkn + QnXkn = en(ek − en) + fn(fk − fn)PnXkn − QnRkn = fken − ekfn

(13)

q. (13) is essentially the power flow equation for solving the volt-ge phasor at the receiving node (i.e., the node N) of the equivalentine that represents the effects of whole system (both local networknd system outside the local network) when the voltage phasor athe sending node is given. If each ESM for all equivalent lines insystem state has a mathematical solution for its receiving node,hich means that voltages at all nodes in the system state exist,

hen the system should have an overall power flow solution andhe system voltage stability holds. Conversely, if an ESM for at leastne receiving node does not have a mathematical solution, which

eans that the operational voltage at the node does not exist, the

ystem cannot have a system-wide power flow solution and willose voltage stability. In other words, the system stability dependsn the solvability of Eqs. (13) for all nodes in the system. The solv-bility of Eqs. (13) can be judged by singularity of its Jacobian

268 Y. Wang et al. / Electric Power Systems Research 79 (2009) 265–271

Table 1Simulation results of IEEE 14-bus system

Load Node 14 Node 13 Node 9 Node 4

� MW ENVCI V(pu) ENVCI V(pu) ENVCI V(pu) ENVCI V(pu)

1.0000 62.7 N/A 1.0315 N/A 1.0495 N/A 1.0497 N/A 1.02951.319 82.7 1.0066 1.0053 1.0871 1.043 1.0839 1.0426 1.0468 1.02521.638 102.7 0.94368 0.97646 1.0717 1.0358 1.0656 1.0345 1.036 1.02051.9569 122.7 0.87291 0.9445 1.0544 1.0278 1.0439 1.0251 1.024 1.01522.2759 142.7 0.79173 0.90837 1.0348 1.0187 1.0183 1.014 1.0098 1.00932.5949 162.7 0.69599 0.8664 1.0117 1.0081 0.98714 1.0005 0.99381 1.00252.9139 182.7 0.57751 0.81523 0.98358 0.99502 0.94734 0.98353 0.97383 0.994453.2329 202.7 0.41529 0.74607 0.94553 0.97734 0.91843 0.95973 0.96394 0.984033.3126 207.7 0.35957 0.72296 0.93274 0.97144 0.88387 0.95163 0.94931 0.98073.3923 212.7 0.29281 0.69486 0.91752 0.96426 0.86128 0.9417 0.94009 0.976783.4721 217.7 0.20261 0.65635 0.89646 0.95446 0.83048 0.92801 0.92855 0.971653.4912 218.9 0.17126 0.64299 0.88921 0.95107 0.81943 0.92325 0.92466 0.969943.512 220.2 0.13076 0.6258 0.88008 0.94671 0.80731 0.91711 0.9175 0.9677933

N CI wa

m

J

d

wm

t

E

Ad

E

w

�

pttsTmtmsaat

3

TS

L

�

11111112222223333333333

.5311 221.4 0.06032 0.59528 0.86352

.5343 221.6 0.048278 0.5895 0.86013

ote: V(pu) denotes node voltage in per unit in Tables 1–4. N/A means that the ENV

atrix, i.e.,

=[

ek − 2en fk − 2fnfk −ek

](14)

et(J) = 2(eken + fkfn) − (e2k + f 2

k ) = 0 (15)

here the symbol det denotes the determinant of the Jacobianatrix J.Eq. (15) provides a new voltage stability index, which is called

he equivalent node voltage collapse index:

NVCI = 2(eken + fkfn) − (e2k + f 2

k ) (16)

lso, it is easy to derive the expression of ENVCI in the polar coor-inates, which is given by

NVCI = 2EkVn cos �kn − E2k (17)

here,

kn = �k − �n

toL

able 2imulation results of IEEE 30-bus system

oad Node 30

MW ENVCI V(pu)

.0000 13.0 N/A 0.99188

.1538 15.0 0.95871 0.98355

.3077 17.0 0.93966 0.97495

.4615 19.0 0.91785 0.96543

.6154 21.0 0.89532 0.95546

.7692 23.0 0.87191 0.94508

.9231 25.0 0.8489 0.93426

.0769 27.0 0.81911 0.9226

.2308 29.0 0.793 0.91032

.3846 31.0 0.76378 0.89737

.5385 33.0 0.73097 0.88366

.6923 35.0 0.6975 0.86906

.8462 37.0 0.65892 0.85343

.0000 39.0 0.62141 0.83654

.1538 41.0 0.5776 0.8181

.3077 43.0 0.52221 0.79702

.4615 45.0 0.45878 0.77216

.6154 47.0 0.38631 0.74254

.7692 49.0 0.24071 0.69435

.8077 49.5 0.1806 0.67565

.8462 50.0 0.12116 0.6485

.8538 50.1 0.095389 0.64003

.8615 50.2 0.065992 0.62853

0.939 0.78202 0.9062 0.90932 0.964130.93755 0.78019 0.90414 0.90834 0.96347

s not calculated in the first state since two system states are required.

It can be seen that the calculation of ENVCI only needs voltagehasors at the two ends of the ESM. Each node has an ESM. Whenhe ENVCI of at least one node is close to zero in a system state,he system approaches the voltage collapse point and the corre-ponding node is the weakest node that causes system instability.he ENVCI can be easily used in a real time or on-line environ-ent since it can be calculated very fast using the voltage phasors

hat are obtained from either PMU measurements or the state esti-ator of EMS. It is worthy to note that in general, not all nodes but

elected nodes need to be monitored in actual applications as oper-tors know the fact that the system will never lose voltage stabilityt many nodes. Conceptually, however, it is not difficult to calculatehe ENVCI for all nodes if necessary.

. Simulation results

The following simulations on the four IEEE standard test sys-ems [15] are performed using the voltage stability module VSATf the commercial software package DSA developed by Powertechabs Inc. in Canada. In the simulations, node loads are gradually

Node 29 Node 27

ENVCI V(pu) ENVCI V(pu)

N/A 1.0034 N/A 1.02320.98782 0.99533 1.0327 1.01940.97024 0.98705 1.0236 1.01550.95023 0.97788 1.0119 1.01080.92957 0.96827 1.0001 1.00570.90805 0.95827 0.98788 1.00050.88532 0.94785 0.98851 0.994950.86056 0.93662 0.97161 0.988790.83447 0.9248 0.95729 0.982240.80686 0.91235 0.94224 0.975290.77755 0.89917 0.9261 0.967860.74618 0.88515 0.90852 0.95990.71234 0.87014 0.88938 0.951290.67553 0.85396 0.86861 0.941940.63427 0.83631 0.84553 0.931640.5859 0.81615 0.81655 0.91960.52431 0.79242 0.78243 0.90510.45659 0.76421 0.74184 0.887690.31901 0.71843 0.6585 0.857720.25724 0.70074 0.62559 0.845920.1906 0.67515 0.58338 0.828770.1676 0.66719 0.56581 0.82340.13547 0.65639 0.55156 0.81613

Y. Wang et al. / Electric Power System

Table 3Simulation results of IEEE 57-bus system

Load Node 33 Node 32

� MW ENVCI V(pu) ENVCI V(pu)

1.0000 3.8 N/A 0.94759 N/A 0.949881.5263 5.8 0.8597 0.9287 0.85353 0.932282.0526 7.8 0.82008 0.90855 0.81229 0.913462.5789 9.8 0.77711 0.88682 0.79445 0.893153.1053 11.8 0.72976 0.86325 0.73902 0.871073.6316 13.8 0.6783 0.8373 0.68846 0.846734.1579 15.8 0.61865 0.80822 0.63089 0.819414.6842 17.8 0.54845 0.77469 0.56352 0.787855.2105 19.8 0.46446 0.73409 0.48017 0.749535.7368 21.8 0.34531 0.67993 0.36723 0.698295.8684 22.3 0.30505 0.66211 0.32695 0.681396.0000 22.8 0.24962 0.64061 0.28164 0.660996.1316 23.3 0.17726 0.61194 0.21945 0.633756.1579 23.4 0.17312 0.60441 0.20136 0.626596.1842 23.5 0.14033 0.59576 0.1832 0.6183666

abaGat

isiincdwTatnTt

(

(

(

(

can be used to judge the system voltage stability by identifying

TS

L

�

12333333333333

.2105 23.6 0.1177 0.58497 0.15859 0.60808

.2368 23.7 0.061598 0.56956 0.12497 0.5934

nd proportionally increased to stress the system with a factor �y assuming a constant power factor. In other words, node loadsre increased by multiplying the factor � in each stressing step.enerator power outputs are correspondingly increased with anverage scale coefficient and the unmatched power is absorbed byhe swing bus.

Tables 1–4 present the results showing variations of ENVCIndices and voltages of the key nodes (including the weakest voltagetability node and those directly connected to it) with load stress-ng in the four different IEEE test systems. Note that the numbern the “load” column of the tables is the total MW value at theodes whose loads are increased, and the last line in each tableorresponds to the system collapse point where the power flowiverges. Fig. 3 displays the variation of ENVCI of the weakest nodeith an increase of � in the four IEEE systems. It can be seen from

ables 1–4 that the ENVCI indices of all key nodes decrease gradu-lly and monotonically as � increases. However, only the ENVCI ofhe weakest node approaches zero but the ENVCI indices of otherodes still keep a distance from zero when the system collapses.he following observations and analyses can be made for each ofhe four IEEE systems.

1) For the IEEE 14-bus system: P and Q loads are increased at thenodes 4 (25%) and 14 (75%) without reactive limits at generators.

able 4imulation results of IEEE 118-bus system

oad Node 44

MW ENVCI V(pu)

.0000 87.0 N/A 0.98501

.1494 187.0 0.79512 0.90165

.2989 287.0 0.53348 0.78269

.5862 312.0 0.37715 0.71589

.6575 318.2 0.32214 0.69287

.7299 324.5 0.26211 0.66477

.8011 330.7 0.17932 0.6256

.8195 332.3 0.15527 0.61152

.8379 333.9 0.11446 0.59228

.8425 334.3 0.11117 0.58554

.8471 334.7 0.084267 0.57672

.8483 334.8 0.096304 0.57389

.8494 334.9 0.070963 0.57044

.8506 335.0 0.070616 0.56595

s Research 79 (2009) 265–271 269

Although loads are increased at the node 4 and node 14 simul-taneously, the node 14 is the weakest node causing systemvoltage instability. This depends on several factors including theamount of load increase, distance from PV nodes and droppedvoltages at each node. As shown in Table 1, the ENCVI canexactly identify the weakest node 14 since its ENCVI basicallyreaches zero while others do not when the power flow diverges.

2) For the IEEE 30-bus system: P and Q loads are increased atthe nodes 29 (50%) and 30 (50%) with reactive power limitsat generators.

Similarly, the weakest node 30 causing system voltage insta-bility is identified by its ENCVI that almost reaches zero whenthe power flow diverges. It can be seen from Table 2 that theENVCI of the node 29 is also small. This is due to the fact thatthe loads at these two remote nodes are increased with thesame proportion and their locations are very close electrically.

3) For the IEEE 57-bus system: P and Q loads are increased at thenode 33 with reactive power limits at generators.

The ENVCI indices of both nodes 33 and 32 are very close tozero. By analyzing the single line diagram of this system [15], itwas found that both the nodes are located at a remote place thatis far away from generators and the node 33 is the end point of avery short radial line from the node 32. Therefore when load isincreased at the node 33, both the nodes 32 and 33 are stressedand a heavy power flow travels over a long distance before itarrives at the node 32. The ESM of the node 33 includes the effectof the whole network before the node 32. This can be viewedas a case in which both nodes 32 and 33 are the weakest nodescausing system collapse with the node 33 a little bit weakerthan the node 32.

4) For the IEEE 118-bus system: P and Q loads are increased atthe nodes 43 (25%), 44 (50%) and 45 (25%) with reactive powerlimits at generators.

When the power flow diverges, the ENVCI at the node 44basically reaches zero while the ENVCI at the nodes 43 and45 still have somewhat distance from zero although they aresecondly close to zero. This example indicates that the ENVCIcould distinguish severity degrees contributing to system volt-age instability of adjacent nodes whose loads are increasedsimultaneously.

The results of simulations indicate that the proposed ENVCI

the weakest node(s). Whenever the ENVCI of at least one nodeis close to zero, it indicates that system voltage collapse willhappen. The ENVCI enables us to set an index threshold to pre-dict voltage instability so that a proper action can be taken to

Node 45 Node 43

ENVCI V(pu) ENVCI V(pu)

N/A 0.98665 N/A 0.978480.85289 0.93343 0.84617 0.924510.6862 0.85741 0.68512 0.84370.52594 0.80756 0.56777 0.799890.4787 0.79026 0.53512 0.785080.42989 0.7692 0.49593 0.767170.36297 0.74 0.44212 0.742540.34025 0.72956 0.42306 0.733810.30876 0.71535 0.39769 0.721970.29857 0.71039 0.38824 0.717860.28439 0.70391 0.37756 0.712510.28069 0.70183 0.37488 0.71080.27602 0.6993 0.36882 0.708720.26746 0.69602 0.36389 0.70603

270 Y. Wang et al. / Electric Power Systems Research 79 (2009) 265–271

F � in I4 essing

4

a(tEonfiaitjnemsmtAooa

A

R

R

[

ig. 3. (a) ENVCI of Node 14 versus � in IEEE 14 system; (b) ENVCI of Node 30 versus4 versus � in IEEE 118 system (Note: X-axis � is the multiple of load increase in str

prevent the system from collapse. The ENVCI should also workfor a contingency case in which a component fails since it can becalculated within very short time (generally less than one sec-ond) while the process of system losing voltage stability willtake a few seconds or even longer.

. Conclusions

The paper presents the equivalent local network model (ELNM)nd ESM for calculating the voltage stability index of each nodebus) in a transmission system. The local network is defined as allhe components that are directly connected to a considered node.ach node has an ESM in a system state. The ESM includes the effectsf both local network and system outside the local network on theode voltage phasor and power flows. A new index ENVCI for identi-

ying system voltage instability via recognition of the weakest nodes derived from the ESM. One feature of ENVCI is its relatively highccuracy in both modeling and calculations not only because thempacts of grid system are included in the model but also becausehe system equivalent impedance that needs to be estimated isust a small part of the total equivalent impedance. The ENVCI onlyeeds voltage phasor information of local network, which can beasily obtained from synchronized PMUs or the existing state esti-ator of EMS at control centers, and can be calculated in a very

hort time so that it can be applied in a real time or on-line environ-ent. The simulation results of four IEEE test systems demonstrate

he feasibility and effectiveness of the proposed index and method.system will lose voltage stability whenever the ENVCI of at least

ne node is near zero. This feature enables us to set an index thresh-ld to monitor and predict voltage stability on-line so that a properction can be taken to prevent the system from collapse.

[

[

EEE 30 system; (c) ENVCI of Node 33 versus � in IEEE 57 system; (d) ENVCI of Nodethe system).

cknowledgment

The authors are grateful for the partial support from the Nationalesearch and Development Fund of China (2004CB217908).

eferences

[1] C.W. Taylor, Power System Voltage Stability, McGraw-Hill, Inc., New York, Amer-ica, 1994.

[2] P.W. Sauer, M.A. Pai, Power system steady-state stability and the load-flowJacobian, IEEE Trans. Power Syst. 5 (November (4)) (1990) 1374–1383.

[3] N. Yorino, H. Sasaki, et al., An investigation of voltage instability problems, IEEETrans. Power Syst. 7 (May (2)) (1992) 600–611.

[4] P.-A. löf, G. Andersson, D.J. Hill, Voltage stability indices for stressed powersystems, IEEE Trans. Power Syst. 8 (February (1)) (1993) 326–335.

[5] G.K. Morison, B. Gao, P. Kundur, Voltage stability analysis using staticand dynamic approaches, IEEE Trans. Power Syst. 8 (August (3)) (1993)1159–1171.

[6] B. Lee, V. Ajjparapu, Invariant subspace parametric sensitivity (ISPS) ofstructure-preserving power system models, IEEE Trans. Power Syst. 11 (May(2)) (1996) 845–850.

[7] G. Verbic, F. Gubina, A new concept of protection against voltage collapse basedon local phasors, IEEE Trans. Power Deliv. 19 (April (2)) (2004) 576–581.

[8] F. Gubina, B. Strmcnik, Voltage collapse proximity index determination usingvoltage phasors approach, IEEE Trans. Power Syst. 10 (May (2)) (1995)788–794.

[9] B. Venkatesh, R. Ranjan, H.B. Gooi, Optimal reconfiguration of radial distribu-tion systems to maximize loadability, IEEE Trans. Power Syst. 19 (February (1))(2004) 260–266.

10] M. Moghavemmi, F.M. Omar, Technique for contingency monitoring and voltagecollapse prediction, IEE Proc. Gener. Transm. Distrib. 145 (November (6)) (1998)634–640.

11] V. Balamourougan, T.S. Sidhu, M.S. Sachdev, Technique for online predictionof voltage collapse, IEE Proc. Gener. Transm. Distrib. 151 (July (4)) (2004)453–460.

12] K. Vu, M.M. Begovic, et al., Use of local measurements to estimatevoltage-stability margin, IEEE Trans. Power Syst. 14 (August (3)) (1999)1029–1035.

ystem

[

[

[

YeiUa

DTUpmCanada and a recipient of Significant Reviewer Award from IEEE Power Engineering

Y. Wang et al. / Electric Power S

13] M. Begovic, B. Milosevic, D. Novosel, A novel method for voltage instability pro-tection, in: Proceedings of the 35th Hawaii International Conference on SystemSciences, 2002.

14] I. Smon, G. Verbic, F. Gubina, Local voltage-stability index using Tellegen’s the-orem, IEEE Trans. Power Syst. 21 (August (3)) (2006) 1267–1275.

15] IEEE test systems, available at: http://www.ee.washington.edu/research/pstca/.

ang Wang was born in Jiangyou, China on 17 May 1977. He received the B.S. degree inlectrical engineering from Sichuan University, Sichuan, China, in 2002. Currently, hes working towards to a Ph.D. degree in College of Electrical Engineering, Chongqingniversity, China. His research interests include power system voltage stability, SPSnd risk analysis in power systems.

S

DUp

s Research 79 (2009) 265–271 271

r. Wenyuan Li (IEEE fellow) is currently a principal engineer at British Columbiaransmission Corporation (BCTC) in Canada and an advisory professor of Chongqingniversity in China. He is the author of four books and considerable technicalapers in power system operation, planning, optimization and reliability assess-ent. Dr. Li was the winner of the 1996 “Outstanding Engineer” awarded by the IEEE

ociety.

r. Jiping Lu is a professor at the College of Electrical Engineering of Chongqingniversity, China. His research interests include power system automation, relayrotection and probability application in power systems.