Power System Voltage Stability: Indications, Allocations and Voltage Collapse Predictions

ISSN (Print) : 2320 – 3765
ISSN (Online): 2278 – 8875
International Journal of Advanced Research in Electrical, Electronics and Instrumentation Engineering

Vol. 2, Issue 7, July 2013
Power System Voltage Stability: Indications,

Allocations and Voltage Collapse Predictions
F.A. Althowibi 1,M.W. Mustafa2
PhD Student, Dept. of EPE, UniversitiTeknologi Malaysia, Skudai, Johor, Malaysia 1
Professor, Dept. of EPE, UniversitiTeknologi Malaysia,Skudai, Johor, Malaysia 2
ABSTRACT:Voltage collapse has been recognized as a series threat in power system stability and operation. Fast and
accurate indications and allocations of voltage stability in power systems are a challenging task to accomplish. Voltage
violations and undesirable line outages might be inevitable when power systems operated close to its transmission
capacity limits. Unexpected load increases or insufficient reactive power supply may contribute to partial or total
voltage collapse threatening system security. The ability to draw a clear and complete picture of system voltage
stability with accurate indications and precise voltage collapse allocations allow operators to take the necessary action
to prevent such incidents. A successful avoidance of such system collapse is based on method’s accuracy, speed of
indication, and very low computation time.
This paper presents a new approach of studying voltage stability in power systems at which voltage stability in
transmission lines and system buses are carefully analysed based on their V-Q and V-P relationships. Four indices are
proposed; two for voltage stability analysis at system buses designated as VPIbus and VQIbus studying the dynamics of
loads and generators while VPILine and VQILine are for line voltage stability analysis studying transmission lines stress
and outages. Voltage collapse is precisely predicted by the proposed indices for the system as a whole and for every
bus and line. These developed indices are simple, fast, and accurate proving a clear and complete picture of power
flow dynamics indicating maximum active and reactive power transfers through transmission systems. The proposed
approach was demonstrated on the IEEE 14-bus and 118-bus systems and compared with existing methods to show its
effectiveness and efficiency.
Keywords: Voltage collapse, Line voltage stability index, Voltage stability analysis
I. INTRODUCTION
The growth of power systems have been witnessed worldwide by engineers, utilities and customers. As
energy demands is increasing rapidly, power system is expanding to accommodate the rapid load growths by
constructing new power plants, transmission lines, substations, and control devices. The shape of electric power
industry also has been changed and persistently pressured by government agencies, large industries, and investors to
privatize, restructure, and deregulate [1]. This continues expansion and persistent pressure make power system more
complex or may be vulnerable and difficult to maintain its stability and ensure its security.
As a result, several blackouts directly related to voltage collapse have been occurred costing lots of million
dollars, and still a threat to power system stability and security. Some well-known incidents of blackout recorded in
Germany in 2006 and Russia in 2005[2-4], in Greece 2004 [5, 6], Italy in 2003[2] and in the same year, blackouts
occurred in USA and Canada[7], Sweden-East Denmark[2], London, UK[8] and Croatia and Bosnia Herzegovina[9].
Voltage collapse is most likely to occur when a power system is operated near its capacity limits. Some
electric utilities are forced at such level due to the difficulty of constructing a new transmission line because of the new
regulations and policy, or to reduce their operational costs and maximize their profits, but doing so raises risks that
make it challenge for operators to contain or control.
Operating within operational design limits makes power system more protected and secure, yet operating
beyond those limits can lead to the absence of generator synchronism, transmission outages or might result in partial or
total system voltage collapse. Hence, maintaining voltage stability in power networks plays a significant role in
preventing voltage collapse.
Voltage stability is well-defined and classified in [10] while addressing an appropriate analysis for voltage
stability phenomena among engineering and researches are still debatable. Voltage stability has been studied using two
main approaches: static and dynamic analysis, where voltage instability as fact is considered as dynamic phenomenon.
Although the dynamic analysis is preferable by most utilities, the static voltage stability approach is commonly used in
research and on-line applications providing an insight into stability problems with high speed analysis.
Copyright to IJAREEIE www.SeminarsTopics.com 3138
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Several methods have been used for static voltage stability analysis to measure voltage stability proximity
estimating the point of voltage collapse. A number of methods proposed in the literature use the singularity of power
flow Jacobian matrix as base, sing or indicator of voltage collapse. Several methods are developed using eigenvalue or
Jacobian matrix singularity monitoring the smallest eigenvalue[11-14], are based on a reducing Jacobian
determinants[15, 16], identifying the critical buses using a tangent vector [17], or computing eigenvalues and
eigenvectors of a reduced Jacobian matrix as introduced in[11, 18] and named as modal analysis. Other methods took
another approach determining maximum loadability at line [19-23] while others are specifying system stability margins
at bus attempting to determine the weaken bus [24-27].
Recently, voltage collapse prediction index (VCPI) has been introduced in [28] to evaluate voltage stability
and predict voltage collapse while an improved voltage stability index designated as Lij has been described in [29]
taken into account the influence of the load model.
A combination of voltage stability static and dynamic approaches was introduced in [30] named as PTSI and
used to study power system dynamic and behaviour and to predict voltage collapse point. Similar approach was
developed in [31] evaluating voltage stability when a small disturbance is applied.
P-V curves introduced in [32] using only one power flow solution instead of multiple solutions to estimate
voltage collapse point while another indices based on voltage stability margins presented in[33, 34]. Similarly, two
sensitive indices with deviation approach developed in [35] used to monitor the available voltage stability margin to the
collapse point terms as TRGGP and TRGGQ.
Other researchers took a different approach to study voltage stability problem using a non-iterative technique
based on Taylor’s expansion [36], using an equivalent local network model in[37], or considering a distribution
network with high/medium [38] to evaluate voltage stability and determine system voltage collapse point.
Method diversity in applications, conditions, purposes, and usability is vulnerability in voltage stability
analysis. Some of these methods are robust or accurate, but might consume more time particularly in large power
systems while others tend to fail if any power system element such control devices is involved. Some of these models
are also just applicable for specified applications or certain conditions which might be inappropriate to be used in
general. Other methods such on-line monitoring methods dispread to have real-time data to determine the proper
corrective strategies in insecure conditions while other approaches such load modelling are not practical when every
load should be modelled accurately.
Because of above these shortcomings, research in this area is needed. Readability and clear indications,
speediness and lower computation consumption, result accuracy, correct indications and precise allocations of voltage
stability, and simplicity rather than complexity are the keys of preventing voltage instability and avoiding voltage
collapse event. If such indices are developed, system blackouts can be avoidable.
This paper presents a new approach of conducting voltage stability analysis, where transmission lines and
power system buses are both analysed based on V-P and V-Q associations providing a clear and complete picture of
system dynamics. Four indices are proposed; two for voltage stability analysis at system buses designated as VPIbus and
VQIbus studying the load and generator dynamics while VPILine and VQILine are for line voltage stability analysis
studying transmission lines stress and outages. VPIbus and VQIbusgenerate indications of voltage stability for each power
system bus while VPILine and VQILineproduce voltage stability indications at each individual line. VPIbus ,VQIbus, VPILine
and VQILineine predict accurately the point of voltage collapse for the system as a whole and also for every bus and line
indicating powerful tools in conducting voltage stability analysis.
The proposed indices are simple, accurate and fast generating voltage stability indications with low
computation time needed to avoid such voltage collapse events. The introduced indices can be applied in real-time
application producing quick voltage stability indications, allowing controls to take necessary action to prevent such
incidents. The proposed indices might fulfil the needs of electric sectors and meet application’s requirements to prevent
future blackout incidents. This method has been demonstrated on the IEEE 14-bus and 118-bus Test Systems to show
its effectiveness and efficiency.
II. THE PROPOSED INDICES
The paper proposed two indices of conducting voltage stability analysis based on bus and line power system.
The first index is named as VPI where voltage profiles and maximum power transfer in power systems is related while
the relationship between voltages and load reactive powers are established in the second index termed as VQI. Both
indices,VPIandVQI, are able to conduct voltage stability based on bus system designated as VPIbusandVQIbus studying
the dynamics of load-generation powers at bus while VPILineandVQILine curry out line voltage stability analysis studying
line stress conditions and outages.

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A. Voltage Stability Analysis at system buses:
The system shown in, Fig.1 is representative of general power system and the transmission line is connecting a
generator with load. The network is presented by an equivalent generator modelled in steady state and it is assumed that
the generator voltage, E, is in normal condition and equal to the voltage at generation bus Vi. Vi will be kept constant
using generator excitation systems.
Vi∟δi Vj∟δj
(G +jB)ij
Pi +jQi ILine
Bus i Bus j
Pj+jQj
Fig. 1 Simple power system-Bus System

where:
Vi , V j sending and receiving voltages at system buses
δi, δj. sending and receiving voltages angle at systembus i and j
Pi, Pj sending and receiving real powers at buses
Qi , Q j sending and receiving reactive powers at buses
Y bus (G+jB) line admittance between bus i and j
This general system can be extended to an n-bus power system used to establish a relationship between the
sending voltage,Vi, and the receiving real powers, Pj at bus. This gives an accurate estimate of voltage stability margin
with real power at the specified or critical bus indicating the maximum MW that can be transferred via transmission
systems.
Let’s assume bus i is as a reference bus, then, the line current, I, is calculated by:
I  Vi  V j  Ybus (1)
By using the receiving powers, the current i also is determined by
 Sj  P  jQ j  (2)
*
I    j
V  Vj   j
 j 
Rearranging equation (1) and (2) yields:
Pj  jQ j  V j ViYbus    j   V j
2
 Ybus  (3)
The real and imaginary parts can be separated from equation (3) as:
2
Re : Pj  V j ViYbus  cos(   j )  V j  Ybus  cos( ) (4)
2
Im : Q j   V j Vi Ybus  sin(   j )  V j  Ybus  sin( ) (5)
By substituting equation (5) into equation (3) to establish Vi-Pjrelationship, yields:
Pj  j   V j Vi Ybus  sin(   j )  V j  Ybus  sin( )  (6)
2
 
 V j Vi Ybus    j   V j
2
 Ybus 
With a few algebra manipulations, the new equation is expressed as
2 V j Vi  cos(   j ) Pj
Vj    0 (7)
cos( ) Ybus  cos( )
Let’s assume the load at receiving terminal is constant and consequently the power factor of demands
powers remains constant. δj is relatively small and can be assumed to be ignored due to its limited contribution on
Eq.(7) as a whole, then the whole term of (cos(θ- δj)/cos(θ) ) is eliminated. Beside, since Gbusis a representation of Y
bus .cos (θ), the new equation is given by
Pj
V j2  V j Vi   0 (8)
Gbus
The solution ofVj in equation (8) is:

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4  Pj
Vi  Vi 2 
Gbus (9)
Vj 
2
Equation (9) gives one distinct real root or two equal roots of Vj if (∆=b2 – 4ac) is discriminated to zero. The
discriminated real roots of Vj can be expressed as
 4 Pj
Vi  0
2
……  Gbus (10)

 4 Pj 1
G V 2
 bus i
The Vj varies from zero to one producing several indications due to the real root limitation. This limitation can
be usedas voltage stability indicator, where real root voltage is greater than zero and lower than one. If not, the voltage
stability is compromised. This proves that the developed equation determines voltage stability at bussystem and be
termed as
4 Pj
VPIbus   1.0 (11.1)
diag( Gbus )  Vi
2
4 Pj  diag( Rbus )
0r  2
 1.0 (11.2).
Vi
Fig.1 is also used here to establish a relationship between the sending-end voltage,Vi, and the receiving-
endreactive powers, Qj. It perhaps may give more accurate voltage stability analysis. Reactive power is closely
associated with system voltage security giving an accurate estimate of voltage stability margin with reactive power at
the specified or critical bus. The reactive power margin is the MVAr distance starting from the normal operation point
and ending by the point of voltage collapse.
By substituting equation (4) into equation (3), the V-Q relationship is establishedas
V j Vi Ybus  cos(   j )  V j2 Ybus  cos( )  jQ j

(12)
 V j Vi Ybus    j   V j2  Ybus 
With a few algebra manipulations, the new equation is expressed as
sin(    Qj
V j2  V j Vi 
 0 (13)
j
sin( )
Ybus  sin( )
By using the same assumptions and following the same procedures, the discriminated Vj root in relationship with load
reactive powers is expressed as
4Q j
VQI   1.0
Bbus  Vi 2
As Bbus has dominant diagonal elements; the VQI can be expressed as
4Q j
VQIbus   1.0 (14.1)
diag ( Bbus )  Vi 2
4Q j  diag ( X bus )
0r   1.0 (14.2)
Vi 2
VPIbus and VQIbus vary from zero to one indicating system stability boundaries. When the values of VPIbus and
VQIbus close or near to the unity, the voltage stability reaches stability limits. Voltage collapse occurs when both VPIbus
and VQIbusareexceeding their stability limits.
B. Line Voltage Stability Analysis:
The number lines in power systems are usually higher than the number of buses as to assure power delivery to
main load centres and to improve system reliability and stability ensuring its security. Due to heavy loaded or stressed
system lines, transmission line is more likely to be collapsed or outage and mayeventually contribute to partial or total
system collapse. Studying the maximum transfer of real and active powers that each transmission line can handle
before its outage or collapse is essential.
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Because transmission lines are connecting buses in power systems, the same assumption should be applied in
this developed theory, which is the generator voltage, E, is equal to the voltage at generation bus Vi.Vi will be kept
constant using generator excitation systems.
The system shown in, Fig.2 is a simple line power system connecting the sending bus by the receiving bus
through transmission line. This simple line system can be extended to an n-line power system.
Bus k Bus m
Pk +jQk Pm +jQm
ILine (G +jB)km
Vk∟δk Vm∟δ
m
Fig.2 Simple line power system
where,
Vk , V m sending and receiving voltages at system buses
δk, δm. sending and receiving voltages angle at system busk and m
Pk, Pm sending and receiving real powers at buses
Qk , Q m sending and receiving reactive powers at buses
Ykm (G+jB) line admittance between bus k and m
Here, the sending-end voltage profiles, Vk,,and receiving-end powers,Pm, relationship at system line is
established. Let’stake bus k as a slack bus, then, the line current, I, is:
I  Vk  Vm Ykm (15)
By using the receiving-end apparent powers, the current between sending-end bus and receiving-end
bus,I,obtainedby:
P  jQm  (16)
*
S 
I   m   m
 Vm  Vm    m
Rearranging equation (15) and (16) yields:
Pm  jQm  VmVk Ykm    m   Vm  Ykm  (17)
2
By separating the real and imaginary of equation (17) yields

2
Re : Pm  VmVk Ykm  cos(   m )  Vm  Ykm  cos( ) (18)
2
Im : Qm   VmVk Ykm  sin(   m )  Vm  Ykm  sin( ) (19)
By substituting equation (19) into equation (17) to establish Vk-Pmconnection, yields
Pm  j   VmVk Ykm  sin(   m )  Vm  Ykm  sin( )  (20)
2
 
 VmVk Ykm    m   Vm  Ykm 
2
With a few algebra manipulations, the new equation is formed as

Pm VmVk  cos(   m ) 2 (21)
  Vm  0
Ykm  cos( ) cos( )
Let’s assume the load at receiving-end terminal is constant and consequently the power factor of demand powers
remains constant. Beside, some of receiving-end powers are generators acted as negative sources and their voltage
profiles are already assumed to be reserved constant. Thus,δm is relatively small and can be ignored due to its limited
contribution on Eq.(21) . Then, the new equation is expressed as
P
V 2  V V  m  0 (22) m m k
Gkm
The, the root of Vm is expressed as:
4  Pm
Vk  Vk2 
Gkm (23)
Vm 
2
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If Vm is discriminated to zero, the real roots of Vmisobtained by

 2 4P
…… Vk  G  0 (24)
m
 km

 4 Pm 1
 Gkm  Vk 2

The Vm varies from zero to the unity and can be fit for line voltage stability indicator expressed as
4  Pm
VPI Line   1.0 (25.1)
G km  V k
2
4  Pm  R km
0r 2
  1.0 (25.2)
Vk
The sending-end voltages, Vk, and receiving-end reactive powers, Qm, are also can be connected in line power
system. By using Fig.2 and the same assumptions mentioned earlier, the connection between Vk-Qm can be derived
from equations (17) to (19) by substituting equation (18) into equation (17) andyields,
VmVkYkm  cos(   m )  Vm2Ykm  cos( )  Qm
(26)
 VmVkYkm    m   Vm2  Ykm 
By arraigning equation (26) with a few algebra manipulations, the new equation is expressed as
Vm2  VmVk
sin(   m   Qm
0 (27)
sin( )  Ykm  sin( ) 
By using the same assumptions of neglectingδmand following the same procedures, the discriminated Vm root in
relationship with load reactive powers is expressed as
4  Qm
VQILine   1.0 (28.1)
Bkm  Vk2
4  Qm  X km
or   1.0 (28.2)
Vk2
VPILineandVQILinevary from zero to one presenting line voltage stability margins. Once the value of
VPILineandVQILineapproach unity, the voltage stability reaches stability limits. The occurrence of voltage collapseis
taken place once VPILineandVQILineexceedingtheir stability limits. VPILineandVQILinedetermine how far the transmission
line is from its instability or collapse point.
III. REVIEW OF VOLTAGE STABILITY METHODS:
Three voltage stability methods arebriefly discussed in this sectionwhich are: Modal analysis introduced in
[77, 90, 91], voltage collapse prediction index at bus (VCPI) [17], andLine stability index (Lmn) [13].
A Modal Analysis
Modal analysis is analytical tools used to determine system stability margins and network limits. It
computesthe eigenvalues and eigenvectors of a reduced power flow Jacobian matrix, where the eigenvalue defines
stability modes and themagnitude of eigenvectors approximate system voltage instability. The equations of power
floware expressed in matrix format as
P   J P J PV   
Q   J 
J QV  V 
(29)
   Q
where
∆P and ∆Q the changes in the real and reactive powers
∆V and∆δ the deviations in bus voltage magnitude and angle
Once assumed ∆P=0in Eq.(29), the V-Q sensitivity is established and expressed as
V
 J r1 (30)
Q
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
where, J r  J QV  J Q  J P1  J PV . 
Then, the right and left eigenvector of Jrmatrix can be expressed as
J r1    1 (31)
where
ζ and η the left and the right eigenvectors
Λ the diagonal eigenvector matrix of Jrmatrix
The V-Q sensitivity is then expressed as

V    1  Q (32)
Once the eigenvectors are normalized as it did in practice,  i  i  1 where i  1,2,.... n then the equation
expressed as
  V  1  Q (33)
Since η = η-1, the final ∆V and ∆Q relationship is expressed as
V  1  Q (34)
1
where  can be expressed in matrix form as
1 0........ . 0 
 0  ..... . 0 
_ 1   2 
. . . 0
 
 0 0..... 0 n 
Hence, V
bus
 i1  Q , i  1,2,.... n . (35)
For any i, if λ > 0, then the variation of Vi and Qiare in the same direction representing system stability
whileinstable system occurred when λ < 0 for any i. Once assumed ∆Q=0in Eq.(29), the V-P sensitivity is established
and expressed as
V
 J r1 (36)
P

where, J r  J PV  J P  J Q1  J QV . 
The right and left eigenvector of the Jr matrix is given by
J r1    1 (37)
By doing the same procedure, the V-P sensitivity is then expressed as
V bus  i1  P , i  1,2,.... n . (38)
For any i, if λ > 0, then the variation of Vi and Qiare in the same direction and the system is stable while the
system is considered unstable when λ < 0 for any i.
B VCPI
VCPI is voltage stability index used for voltage stability evaluation able to define system stability margins. The
index is based on bus system and derived from the basic power flow equations usingsystem apparent power, Si. Sican be
expressed as:
 Vk 2  Vk  cos k  j Vk  sin  k  
  (39)
S k    N

  Y
 kk
    Vm  cos m  j Vm  sin  m 
 
  mm 1k 
 
whereV’m is given by:
 Y
Vm  km  Vm , and Y   Ykm
Y k 1, j  k
Based on equation (38), the introduced index is expressed as

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V 
m 1, m  k
m
VCPI bus  1 
(40)
Vk
VCPI varies from zero to one and its limitation specifies voltage stability margins for the entire system. Once
the value of VCPIis nearthe unity or exceeding it, the system voltage stability is comprised and the voltage collapse is
more likely to occur.
C Lmn
Lmn is a new voltage stability approach used to calculate line voltage stability attempting to detect the stress and
condition of power system lines and predict line voltage collapse. Using a single power system line model as shown in
Fig.2, the line quadratic voltage is calculated by
Vk  sin(   )  Vk  sin(   m )2  4 XQ4
Vm  (41)
2
where the real root of receiving voltage generates more values between zero and one and its boundaries representing
line system stability, used as an index named as Lmn and expressed as
4 XQ4 (42)
Lmn Line 
Vk  sin(   ) 2
IV. RESULTS AND DISCUSSION
This section is a demonstration of the proposed indices,VPIbus, VQIbus,VPILine andVQILine, on IEEE 14-bus and
118-bus systems to carry out line and bus voltage stability analysis. The proposed bus indices were compared first with
receiving-end voltages to check V-P and V-Q associations, and then compared to VCPI and Modal analysis (dV/dP and
dV/dQ) at bus system to measureVPIbus’s andVQILine’s relative effectiveness and efficiency. The proposed line indices
wereafterward compared with Lmn first to verify their accuracy in conducting line voltage stability analysis (because
they have similar characteristics) and then to VCPI and Modal analysis to validate their accuracy of voltage collapse
predictions.
VCPI and Modal analysis ( dV/dP anddV/dQ) are used for bus voltage collapse predictions; where VCPI is
suggested for online voltage stability evaluation and Modal analysisisproposed to compute the eigenvalues and
eigenvectors of power flow Jacobian matrix to estimate voltage dynamic collapse. These mentioned methods are
different and have different characteristics, yet they all share voltage stability margins estimating one system collapse
event or point. Few loading scenarios were considered in this study where each loading scenario was increased
gradually till the system collapsed.
TABEL.1
VPIbus and VQIbus on IEEE 14-Bus system at Initial Steady- State
bus VPIbus VQIbus bus VPIbus VQIbus

No. No.
1 0 0 8 0 0
2 0.0031 0.0151 9 0.0222 0.027
3 0.0875 0.0758 10 0.0145 0.0157
4 0.0203 0.0039 11 0.0083 0.0084
5 0.0028 0.0017 12 0.027 0.0117
6 0.0099 0.0166 13 0.0337 0.0218
7 0 0 14 0.0554 0.0387
Total 0.2847 0.2286

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A. IEEE 14-bus system:
1. Initial state scenario at bus:

Table.1 showed theVPIbusandVQIbus performances on IEEE 14-bus system, where both conducted bus voltage
stability analysis at system steady-state condition. The results presented that VPIbusandVQIbus successfully indicated
voltage stability at each individual system bus recoding 0.2847 and 0.2286 at zero loading rate kfor the summation of
VPIbus and VQIbus respectively. There was no voltage stability indication at a reference bus and nor voltage stability
results when there were no records of load powers as indicated at bus 7 and 8.
Voltage Collapse Voltage Collapse

Unstable Unstabl Unstable
Stable System e Stable System Voltage Collapse Stability Margin
Stability Margin Stability Margin
VQIbus VQIbus
Voltage Collapse
VPIbus VPIbus Stability Margin
Vbus 9 Vbus
10
(a) At bus (9) (b)At bus (10) (a) VPIbus (b)dV/dP
Voltage Collapse
VQIbus
VQIbus Voltage Collapse
VPIbus
VPIbus Stability Margin
Vbus
Vbus
13
14
Load Factor k Load Factor k Load Factor k Load Factor k

(c) At bus (13) (d)At bus (14) (c) VQIbus (d)dV/dQ
Fig.3 1st scenario on IEEE 14-bus system, VPIbus,VQIbus and Fig.4 2nd scenario on IEEE 14-bus system, VPIbus,VQIbusand Modal
Voltage at bus vs. load factor k Method (dV/dP, dV/dQ) vs. load factor k
2. Loading scenario at bus:

In this application, the loads were remained constant and increased in small steps, 0.01 units, until the IEEE
14-bus system collapsed. At this point, the voltage collapse is predicted and voltage stability margin is calculated. Two
load scenarios were considered in this study. There were two loading scenarios: load components were increasedat all
buses once with identical k until the system collapsed while the real load components were increasedsimultaneously at
all bus with identical k until reached to a point, where the system voltage collapse occurred. The power factors were
kept constantpresumably.
Figure 3 illustrates the first loading scenario where the loads were incrementally increased at all buses until the
IEEE14-bus system collapsed. Here, four buses were selected randomly (9, 10, 13 and 14) in which the proposed bus
indices were compared against the receiving-end voltage,Vbus. The results showed that both VPIbus andVQIbusindices
predicted the point of voltage collapse at loading rate k = 2.75 intercepting with the receiving-end voltage,Vbus
indicating an accurate projection of voltage collapse at that particular buses. At the point of collapse, the both VPIbus
andVQIbus passed their unity exceeding their stability while Vbus dropped sharply approaching zero. From subfigure
(3.a), as loading rate k increased at bus (9), VPIbus andVQIbusincreased until reached to a point, where both indices went
sharply to zero passing its stability limits and Vbus rose sharply going to infinity. Maximum real and active power
transfers were also estimated by VPIbus andVQIbusfor each individual bus starting by the system initial state and ending
by the point of voltage collapse.
Figure 4 illustrates the second loading scenario where the load real powers only were simultaneously increased
at all buses until the 14-bus system collapsed. VPIbus and VQIbus compared to modal analysis (dV/dP and dV/dQ) to
verify their accuracy of estimating voltage collapse point. In each of these subfigures, VPIbus and VQIbus are shown in
sub-figures (a) and (c), while modal analysis-dV/dP and dV/dQ are shown in subfigures (b) and (d) respectively.

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All indices started together in normal system condition and collapsed at loading rate k= 2.96 showing that they
had an identical voltage collapse point as they had the same system stability margin. At loading rate k= 2.96,VPIbus
Voltage Collapse Voltage Collapse
Unstable Unstabl Unstable
Stable System e Stable System
Stability Margin Stability Margin

Sensitivity
VQILine Sensitivity Margin
Margin
VPILine VQILine
VPILine System Collapse System Collapse
Lm
Vbus
n 2 Lm
Vnbus
14
(a) At Line (9) (b)At Line (20) (a) VPILine (b)dV/dP
Line outages
VQILine VQILine
VPILine VPILine
Lm Lm
Vbus
n 5 Vnbus
14

(c) At Line (5) (d)At Line (17) (c) VQILine (d)dV/dQ
Fig.5 2nd scenario on IEEE 14-bus system, VPILineandVQILine, Fig.6 1st scenario on IEEE 14-bus system, VPILineandVQILineand
andreceiving-end Voltage vs. load factor k Modal Method (dV/dP, dV/dQ) vs. load factor k
andVQIbus went sharply passing their unity and exceeding their stability limits while dV/dP and dV/dQ dropped
gradually to zero, where their matrixes approached singularity.
As seen in subfigures (4.a) and (4.c), VPIbus andVQIbusindicated voltage stability for the system as a whole and
for every bus providing maximum real and active powers that every buswas able to deliver while dV/dP and dV/dQ
gave a clear voltage collapse prediction as shown in subfigures (4.b) and (4.d) respectively.
3. Initial state scenario at line system:
Table.2 showed the VPILineand VQILine performances on IEEE (14-bus, 20 lines) system, where both were
carried out line voltage stability analysis at system normal condition. The result outcomes effectively indicated voltage
stability at 20 power system lines, recording 0.895 k and 0.918 at zero loading rate k for the summation of VPILine and
VQILine respectively. There was a difference of only 0.0595 loading rate k between the total summations of two
methods confirming their similarity results. Our results also showed there no voltage stability indications were
recorded by VPILine when the line resistance is or close to zero as indicated in lines 8, 9, 10, 14, and 15.
4. Loading scenario at line system:

Two load scenarios were illustrated in figures (5) and (6) in which the load components were amplified in the
system at all buses simultaneously with identical rate k until the system collapsed as a first, while the real load powers
were increased in the system a second.
Figure 5 illustrated the comparison among VPILine, VQILine,,Lmnandthe receiving-end voltage,Vbus, , where the
load real powers were incrementally increased at all buses until the IEEE14-bus system collapsed. Here, voltage
stability analysis conducted on the selected lines 5, 9, 17 and 20 to measure VPILine’sandVQILine’s relative effectiveness
and efficiency.
When the system subjected to load increase, VPILine, Lmnand VQILine increased gradually as Vbus at selected
lines decreased slowly until all indices reached to loading rate k = 3.96, whereVbuseitherdropped to zero or sharply went
to infinity, and VPILine,Lmn, andVQILine passed their unity indicating system collapse point. Between system steady state
and the point of voltage collapse,system voltage stability margin was estimated by all indices having very
similarresults.

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Here, VPILine showed its capability to estimate the maximum real power delivery for every line while, Lmnand
VQILine explained how much reactive powers can be delivered at each line. Vbusis no longer be taken for granted as a
voltage collapse indicator, line may be outage or appeared as a perfect steady voltage as shown in subfigure (5.a).
As it can be seen from subfigures (5.a), (5.b), and (5.d), all performed indices predicted the point of voltage
collapse at loading rate k =3.96 while in subfigure (5.c) Lmn predicted only the point of voltage collapse earlier by
loading rate difference k = 0.11 than other indices indicating an error in its projection. At subfigure (5.c), VPILine
andVQILine, projected the point of voltage collapse where they both intercepted with the receiving-end
voltage,Vbusindicating their accuracy measures in producing voltage stability indications along with load increase.
TABEL.2
VPILine and VQILine on IEEE 14-Bus System at Initial Seady State
Line From to VPILine VQILine

No. k m
1 1 2 0.1085 0.089
2 1 5 0.1449 0.008
3 2 3 0.1234 0.066
4 2 4 0.1177 0.017
5 2 5 0.0851 0.024
6 3 4 0.0616 0.021
7 4 5 0.032 0.015
8 4 7 0 0.031
9 4 9 0 0.075
10 5 6 0 0.044
11 6 11 0.0245 0.035
12 6 12 0.0365 0.031
13 6 13 0.0444 0.046
14 7 8 0 0.193
15 7 9 0 0.098
16 9 10 0.0074 0.015
17 9 14 0.0495 0.046
18 10 11 0.0103 0.017
19 12 13 0.0135 0.014
20 13 14 0.0359 0.033
Total ----- ----- 0.895 0.918
Figure 6 illustrated the first loading scenario where the load powers were increased steady at all buses with
identical loading rate k until the IEEE 14-bus system collapsed. The results showed that VPILine, VQILine,dV/dP and
dV/dQ indices had an identical system voltage collapse prediction recording loading rate k =2.75.
At this point, all indices shared equal voltage stability margin. VPILine, VQILine indicated maximum real and
reactive powers that can be transferred through the transmission lines for the system as a whole and also for every line
while dV/dP and dV/dQ had voltage stability indications with limited information assigned to each line.
VQILineindicated the outage of three lines due to their maximum reactive power limits at loading rate k = 2.44, 2.58and
2.67 respectively, while VPILine,dV/dP and dV/dQshowed no such outage indications. These line outages led to cascade
voltage collapse.
B. IEEE 118-bus system:

This section exhibits an implementation of the proposed method, VPIbus,VQIbus, VPILine and VQILine on an IEEE
118-bus system to conduct line and bus voltage stability analysis. The proposed indices were compared with
alternatives to verify their accuracy in predicting voltage collapse for large power systems. In this application, loads
were remained constant and subjected to a gradual increase with a step of 0.01 units until the system collapsed. The
first load scenario was considered here and power factor also remains constant.
1. Loading scenario at bus system:

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Figures (7) showed the performance of the proposed bus indices on IEEE 118-bus system and compared to
VCPI and Modal analysis-dV/dP methods. In each of these subfigures, VPIbus and VQIbus are shown in sub-figures (a)
and (c), while VCPI and dV/dP are shown in subfigures (b)and (d) respectively.Here, the first loading scenario was
applied where the loads were increased simultaneously at all buses with identical loading rate k until the IEEE118-bus
system collapsed.
The results showed that VPI bus, VQI bus, VCPI and dV/dP had very similar system voltage collapse prediction
recorded at loading rate k =0.86. At this point, VPIbus, VQIbusand VCPI had exceeded their stability limits passing their
unity while some of dV/dQcomponents went to zero and others approached infinity indicating its matrix reached
singularity. All indices started with system initial condition at zero loading factor k and ended at the point of voltage
collapse sharing equal system voltage stability margins.
System Collapse Sensitivity Margin Sensitivity Margin
Sensitivity Margin Sensitivity Margin
Line outages System Collapse
System Collapse
System Collapse
(a) VQI bus (b) VCPI (a) VQI bus (b) VCPI
Sensitivity Margin
Sensitivity Margin
Sensitivity Margin
Line
System Collapse
System Collapse outag
es

(c) VPI bus (d)dV/dP (c) VPI bus (d)dV/dP
Fig.7 First scenario on IEEE 118-bus system, VPIbusandVQIbusand Fig.8 1st scenario on IEEE 118-bus system, VPILine,VQILine,, VCPI and
Modal MethoddV/dP vs. load factor k Modal MethoddV/dP vs. load factor k
2. Loading scenario at line system:

Figures (8) show the performance of VPILine and VQILine on IEEE118-bus system and compared to VCPI and
Modal analysis-dV/dP methods to measure their relative effectiveness and efficiency in predicting voltage collapse at
large power system. In each of these subfigures, VPILine and VQILine are shown in sub-figures (a) and (c), while VCPI
and dV/dP are shown in subfigures (b)and (d) respectively.
The procedure described above was repeated whereIEEE118-bus system was subjected to a gradual load
increase at once until the system collapsed. The results showed the occurrence of voltage collapse at loading rate k
=0.86, at whichVPILine, VQILineandVCPIindices passed their voltage stability limits while dV/dPwent to zero or
approached to infinity.
Although the point of system collapse accurately projected by VCPI and dV/dQ methods, they showed no
signs of line outages as VPILine and VQILine did. VQILine projected approximately 15 lines to be outage passing their
maximum reactive power transfer limits dragging the entire system to collapse, when the cascade voltage collapse
occurred. Three lines collapsed indicated by VPILine recorded in 0.44, 0.5 and 0.73 loading rate k respectively before
system total collapse; indicating their maximum real power transfers reached to their limits.
Thus, VPILine and VQILine are able to provide maximum system loadability in both active and reactive powers
not only for the system as a whole, but also for every line. With their simplicity, speeds of readability and knowing
system loadability limits, operators may act fast to preserve system reliability not to prevent voltage instability.
3. Indices Computation time:

The time consuming during voltage stability analysis is rather important particularly when power systems are
operating daily or even hourly under stressed. As a result, the computation time of the proposed indices were
demonstrate on IEEE 118-bus system and compared with alternative methods to show their effectiveness of executing
voltage stability analysis.
Table 3 showed that the proposed indices had the lowest solution time to calculate and allocate voltage
stability at each individual bus and/or line, recording 1.1166sec, which is probably the fast voltage stability analysis.
The results showed that VQILine was the fastest index among the proposed indices recoding 1.1166sec while VQIbus was
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slowest in computation time estimated at 1.1166sec. Our results also indicated that VPIbus ,VQIbus,VPILine, andVQILine
were still faster in producing voltage stability indications comparing to the alternative methods. Modal indices, for
example, recorded the highest computation time estimated at 1.4635sec and 1.5154sec for dV/dQ and dV/dP
respectively.
TABEL. 3
A Computation Time Comparison AmongMethods Implemented in IEEE 118-Bus System
Methods Time/Sec N. Iterations

VQILine 1.1166 10
VPIbus 1.1960 10
VPILine 1.2134 10
VQIbus 1.2461 10
VCPI 1.2584 10
Lmn 1.3363 10
dV/dQ 1.4635 10
dV/dP 1.5154 10
C. Overall Results:
Overall, VPIbus, VQIbus VPILine and VQILine have very similar stability margins providing almost identical
voltage collapse point for power system as a whole while they have the same voltage stability results comparing to
modal analysis, Lmn,VCPI, and receiving-end voltage Vbus. The results also exhibit high voltage stability analysis with
low computation time using VPIbus, VQIbus VPILine and VQILineindices generating accurate voltage stability indications
and allocations with clear readability for each individual line and bus and also for entire power system.
VCPI and modal analysis are powerful approach conducting voltage stability analysis in bus system while line
voltage stability is effectively conducted by Lmnmethod. However, modal indices are complex and consuming much
calculation times to conduct voltage stability analysis. Their stability indications are based on Jacobian sensitivities
which are not easy to extract voltage stability at each individual bus or system line. For a system with several thousand
buses it is impractical to calculate all of eigenvalues even calculating the minim eigenvalues of Jris not sufficient;
because there is more than one weak mode associated withdifferent part of the system. Although VCPI has advantages
of lowering analysis computation time with good voltage stability indications for ever bus, its information is based on
voltage ratios with no sing of remedial actions.
VPIbus and VQIbus generate indications of voltage stability for each power system bus while VPILine and VQILine
produce voltage stability indications at each individual line. Voltage collapse are accurately predicted by VPIbus ,
VQIbus, VPILine and VQILine comparing to alternative methods estimating voltage stability margins for each power
system bus and line; and indicating maximum active and reactive power transfers through transmission systems.
VPIbusandVQIbus study the load and generation dynamics while VPILineandVQILine study transmission lines
stress and outage. The synchronism of generators and loads might be predicted or detected using these indices while
other methods such VCPIshow no signs of line outages during load increase.
The proposed indices produce more voltage stability indicationsonce a power system subjected to an
unexpected increase of load components. Limited power transfers and shortage in supplying reactive power demands
may cause a partial or total system breakdown. Our results also show that the occurrences of voltage collapse take
faster rates when a system subjected to an increase of active and reactive load components. Although those indices are
valid for the linearized model as dynamic analysis is preferred by some utilities, the proposed indices can be applied at
points in timein a dynamic simulation and can be also implemented easily in on-line voltage stability applications.
Thus, the proposed indices,VPIbus ,VQIbus, VPILine and VQILineare superior in their simplicity, accuracy, speed
calculations and direct VQ and VP associations indicating a powerful tool to conduct a static voltage stability analysis
and predicts precisely the point of voltage collapse. They also are capable of identifying the weaken buses and which
area are involved with. With this simplicity, speeds of readability and knowing system maximizing power transfers,
operators may act faster than before particularly when the system subjected to a sudden disturbance; not only to prevent
voltage instability, but also to improve system stability ensuring its security.
V. CONCLUSION:
This paper presented a new approach of analysing voltage stability in power systems. Voltage stability in
transmission lines and power system buses are carefully analysed as to prevent line outages and/or to avoid partial or
total system voltage collapse. Four indices are proposed; VPIbus and VQIbus for voltage stability analysis at system buses
studying the dynamics of loads and generatorswhile VPILine and VQILineare for line voltage stability analysisstudying
transmission lines stress and outages.
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VPIbus and VQIbus generate indications of voltage stability for each power system bus while VPILine and VQILine
produce voltage stability indications at each individual line. Voltage collapse are accurately predicted by VPIbus ,
VQIbus, VPILine and VQILine comparing to the alternative methods estimating voltage stability margins for each power
system bus and line; and indicating maximum active and reactive power transfers through transmission systems. In
other words, the proposed indices indicate how far the bus and/or line are from their severe loading condition or
outages.
The accuracy ofVPIbus ,VQIbus, VPILine and VQILine in conducting voltage stability analysis and theirestimation
of voltage collapse were verified extensively, indicating that they have very similar system stability margins and
voltage collapse detectioncomparing to alternative methods. Line outages are predicted by VPILine and VQILinecaused by
either limited real power transfers or insufficient reactive power to support the required demand.
The results also show the proposed indices are superior to theirsimplicity, accuracy, speed calculations and
direct VQ and VP associations indicating a powerful tool to conduct a static voltage stability analysis and predict
precisely the point of voltage collapse. With this simplicity, speeds of readability and knowing system maximizing
power transfers, operators may act faster than before particularly when the system subjected to a sudden disturbance;
not only to prevent voltage instability, but also to improve system stability ensuring its security.
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BIOGRAPHY
F. A. Althowibi received the B.S. degree in electrical engineering from College of Technology, Saudi Arabia in 2000
and the degree of M.S. from University of Queensland, Australia in 2006. He is currently pursuing PhD degree at
UniversitiTeknologi Malaysia, Malaysia.
M. W. Mustafa received his B.Eng degree (1988), M.Sc (1993) and PhD (1997) from University of Strathclyde,
Glasgow. His research interest includes power system stability, deregulated power system, FACTS, power quality and
power system distribution automation. He is currently deputy dean graduate studies and research, Faculty of Electrical
Engineering at UniversitiTeknologi Malaysia.

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International Journal of Advanced Research in Electrical, Electronics and Instrumentation Engineering