
Review Article


Power System Voltage Stability Assessment Using Network EquivalentsA Review 

P. Nagendra,
T. Datta,
S. Halder
and
S. Paul



ABSTRACT

The voltage instability is a growing problem in the modern power systems and is associated with rapid voltage drop due to heavy load demand. Fast computing techniques have been tried to properly analyze the voltage stability for predicting the onset of voltage collapse so as to avoid any unwanted events of the system. Several approaches are available in the open literature for assessing the voltage stability of a power system. A fast and easier way to assess the voltage stability of power system is through its equivalent network and widely explored in current literature. This study presents a review on the voltage stability assessment methodologies particularly based on the network equivalent techniques.





Received: February 24, 2010;
Accepted: April 11, 2010;
Published: July 14, 2010


INTRODUCTION
The voltage drop along a transmission line mostly depends upon the type of
load at receiving end. The consumer demanding more reactive power is responsible
for higher voltage drops as voltage in power transmission system is strongly
coupled with reactive power status of the system (Kundur,
1994; Taylor, 1994; Van Custem,
2000). If the reactive power demand goes on increasing then system suffer
from a gradual voltage drop and at a certain load point it reaches the critical
point beyond which no stable operating point is possible even with any type
of corrective measures e.g. installation of SVC, use of FACTS controllers (Hingorani
and Gyugyi, 1999; Thurkaram and Lomi, 2000; Song
et al., 2004; Abido, 2009) or strategic load
shedding. Efforts have been made by the researchers to analyze the power systems
and to assess voltage stable state along with corrective measures to restrict
voltage collapse (Chiang et al., 1990; Schlueter
et al., 1991; Chakrabarti et al., 2002;
Kundur et al., 2004; Attia
et al., 2006) phenomena at any operating condition so as to help
system planner and operator for better system operation with better voltage
profile at load buses.
Identification of weak/weakest bus (Zambroni de Souza and
Quintana, 1994; Rahman and Jasmon, 1995; Haque,
2003a; Wang et al., 2005) some time termed
as critical bus is a major concern of voltage stability study along with development
of different voltage stability indices (Lof et al.,
1993; Dey and Chakrabarti, 2003; Augugliaro
et al., 2007; Suganyadevia and Babulal, 2009)
using load flow technique (Rahman and Jasmon, 1995;
Nizam et al., 2006; Chakrabarti
et al., 2010), transmission loss (Verbic et al.,
2006), local measurements (Lee and Ong, 1998; Smon
et al., 2006; Su and Wang, 2009), system
loadability (Chebbo et al., 1992; Zambroni
de Souza and Quintana, 1994; Turan et al., 2006;
Hamada et al., 2010), singularity of load flow
Jacobian (Lof et al., 1992; Chen
and Wang, 1997; Verbic et al., 2006) to assess
voltage stability margin (Soliman et al., 2003;
Haque, 2003b; Haque, 2006; Bedoya
et al., 2008; Fujisawa and Castro, 2008)
and system critical load. A bibliography, based on the voltage stability of
power systems is reported by Ajjarapu and Lee (1998).
Offline application being the inherent disadvantage of load flow solution compelled
the researchers to develop on line tools for voltage stability study (Ajjarapu
and Christy, 1992; Ajjarapu and Lee, 1992; Vu
et al., 1999; Vaahedi et al., 1999;
Wang et al., 2005; Smon et
al., 2006; Wang et al., 2009; Su
and Wang, 2009).
Gradually the concept of deriving two bus equivalent network of any multibus
power network comes up to obtain a quick view of voltage stable states of the
power system. In this regard several voltage stability indicators have been
developed using Thevenin equivalent circuit (Rahman and
Jasmon, 1995; Haque, 1995; Chen
and Wang, 1997; Haque, 2003b) with on line phasor
measurement (Smon et al., 2006; Wang
et al., 2009; Su and Wang, 2009; EIAmary
and EISafty, 2010) or construction of special phasor diagram (Turan
et al., 2006). Other type of single line two bus equivalents (Jasmon
et al., 1991; Jasmon and Lee, 1991b; Chakrabarti
et al., 2005; Nagendra et al., 2009)
have also been developed including non linearity of load (Chebbo,
et al., 1992; Hamada et al., 2010),
based on phasor measurement (Gubina and Strmcnik, 1995;
Liu et al., 2008). Some of the two bus equivalent
methodologies developed are also capable to predict better scheme for strategic
load shedding based on voltage stability criterion instead on the ground of
the simple voltage magnitude criterion (Wiszbuewski, 2007).
Later on development enable the equivalent system for on line study of voltage
stability using the boundary of the voltage stability region in the PQ plane
(Haque, 2002, 2003a; Paosateanpun
et al., 2006; Hamada et al., 2010)
to get quick overview on the system voltage instability.
This study presents the efforts in a nutshell to provide complete information regarding voltage stability in equivalent mode. The proposed methodologies for development of network equivalent available so far may be divided into three sections in broad sense i.e., based on Thevenin theory, by other innovative approach and based on online measurement. REVIEW OF NETWORK EQUIVALENTS
A power system consists of several buses which are connected by means of transmission
lines (Chakrabarti and Halder, 2008) and is becoming more
and more complex to meet the continuous grow of electrical load demand. In general,
the voltage stability assessment (Chakrabarti and Halder,
2008) of such a stressed system is very difficult. A fast and easier way
to assess the voltage stability of the complex system is through its network
equivalent with only one line. By using this equivalent system, the occurrence
of voltage collapse of actual system can be studied easily in global mode (Jasmon
et al., 1991; Gubina and Strmcnik, 1997;
Dey et al., 2004; Wang et
al., 2009; Nagendra et al., 2009) and
it is not necessary to consider every line of the network separately. Also,
the power system equivalents (Housos et al., 1980;
Le et al., 1997; Fu et
al., 1997) are of importance for the study of static characteristics
(Sauer and Pai, 1990) of a large system when either
the computer facilities for direct solution or the available solution time is
restricted. The use of equivalent representation of the complex system is required
frequently to simplify the lengthy calculations and analysis system stability
easily. Due to simplicity of the single line equivalent technique (Jasmon
and Lee, 1991a, b), stability analysis based on
this equivalence is much simplified making it most suitable for use in real
time power system monitoring. In addition, local methods (Verbic
and Gubina, 2004) also give a very good insight into the nature of the voltagecollapse
process and can easily be used for protection schemes.
This section provides the review of the network equivalent methodologies on the basis of Thevenin theory, other innovative approach and on line measurement, available for assessment of voltage stability in power system.
Development of network equivalent using Thevenin theory: Chebbo
et al. (1992) used the concept of the twobus theory to estimate
the maximum loading capability of a particular load bus in a power system where
the power system is first replaced by the Thevenin theory to get the twobus
equivalent model which include the effects of nonlinearity of loads and generators
in the equivalent circuit though some repetitive computation in the original
system is required to get the solution of the ultimate results.
Rahman and Jasmon (1995) also proposed a new voltage
stability index which helps to identify the critical buses using Thevenin equivalent
circuit of the power system referred to a load bus using a new load flow technique.
Haque (1995) uses the base case system information
to find special twobus equivalents of the system for analyzing the voltage
stability problem thus determines the maximum loading capability, especially
the reactive power demand of a particular load bus in a power system through
the Thevenin equivalent circuit where generators are modeled to reflect actual
operation, even for a change in operating conditions.
Vu et al. (1999) proposed a simple method of
determining the voltage stability margin of a power system using some local
measurements. The measured data are used to obtain the Thevenin equivalent of
the system. The equivalent system is then used to determine the relative strength
or weakness of the transmission network connected to a particular load bus.
Turan et al. (2006) gave the concept of a maximum
power transfer phasor diagram using local measurements and estimated parameters
of Thevenin equivalent for Nbus power system for easy evaluation the relations
between major parameters affecting voltage stability margins. Critical values
for major parameters and a voltage stability margin are evaluated from the constructed
phasor diagram.
Load shedding often becomes the last line of defense to prevent the voltage
collapse in the course of a developing disturbance. Wiszbuewski
(2007) presented a criterion by measuring the variation of apparent load
power against the change of load admittance (dS/dY) and the ratio of (Z_{L}/
Z_{S}) to initiate the load shedding instead of the voltage criterion
but on the ground of the simple Thevenin circuit.
Other innovative approach to develop network equivalent of multibus power
system: Jasmon et al. (1991) developed a
technique for reducing a radial network into a single line equivalent whose
parameters can be obtained from the results summary of any load flow study which
could be used for the practical online monitoring of power system voltage stability.
Jasmon and Lee (1991b) presented an innovative technique
for load flow calculations and voltage instability analysis of distribution
networks by reducing a radial network into a single line equivalent which simplifies
lengthy calculations of an unreduced network and thus enables the fast computations
of load flow solutions of distribution networks. The conditions for voltage
collapse being derived from the single line equivalent; no voltage computation
is required as all the voltages are eliminated from the governing equations
which indicate the superiority of the technique compared to other known methods.
Strmcnik and Gubina (1996) made an analytical approach
to find voltage collapse proximity determination based on a twobus equivalent
of a radial network where the voltage phasors at the both ends of the radial
network are transformed in order to form the voltage phasors of its twobus
equivalent. Exact voltage stability limit relations for the equivalent derived
from Jacobian matrix are established via geometrical relationship between both
voltage phasors.
Chen and Wang (1997) described the DistFlow method
to find the load flow solutions for radial power networks from which an equivalent
2bus network can be obtained during the solving process where only one feasible
voltage solution exists for a radial power network which can be judged directly
from the sign of the Jacobian determinant of the equivalent 2bus network obtained.
Moghavvemi (1997) proposed a method for determining
the voltage stability factor based on the concept of power flow thorough a single
line equivalent. Adopting the technique of reducing the power system network
into its equivalent single line system, a voltage stability factor is derived
and used to examine the overall system stability.
Haque (2003b) proposed a simple equivalent model of
the power system to generate the voltage stability boundary involving very little
computations which does not require any repetitive load flow simulations and
thus various voltage stability margins of the critical bus or area in a large
power system can be directly determined from the generated stability boundary
for different initial operating conditions with a high potential for online
application.
With the concept of network equivalencing technique, Dey
et al. (2004) developed a global voltage security indicator for assessing
the voltage stability of actual system. Improvement of global voltage security
is highlighted with the application of the static Var Compensator (SVC) in a
typical weak load bus in the considered system.
Wang et al. (2005) developed a twobus equivalent
methodology based on tracking the weakest power transmission path by defining
the weak power draining buses incorporating the electrical distance information
and reactive generation reserves so that the on line voltage stability of the
power system can be assessed correctly instead of using Thevenin equivalent
parameters.
Chakrabarti et al. (2005) developed a unique
methodology of offline diagnosis of the weakest bus in a longitudinal multibus
power network from conventional load flow technique using the concept of equivalencing
the multibus network to twobus radial system and by studying the necessary
parameters of the equivalent system. A generalized voltage stability criterion
being developed, it is applied to multibus power system adopting the developed
equivalencing technique in order to obtain global voltage stable states.
A new equivalent model using Newton method or the least square estimation method
with multiple continuous samples of PMU (Phasor Measurement Unit) measurements
is proposed by Liu et al. (2008) to analyze and
predict the voltage stability of a transmission corridor in terms of Available
Transfer Capacity (ATC). This equivalent model retains all transmission lines
in a transmission corridor, which is more detailed and accurate than traditional
Thevenin equivalent model with acceptable computation burden.
An Equivalent System Model (ESM) including the effects of both local network
and system outside the local network is developed by Wang
et al. (2009) to derive the equivalent node voltage collapse index
(ENVCI) using only local voltage phasors with accuracy in modeling and calculations
and ease in real time or online applications.
Optimal power flow technique has been used by Nagendra
et al. (2009) to develop a two bus series equivalent to diagnose
the weakest part of the multibus power system and assess the voltage stability
in a multibus power network in a global mode.
Hamada et al. (2010) described the boundary
of the voltage stability region in the PQ plane to get a quick overview on
the system voltage at a certain loading condition using equivalent twobus system
for each loading condition as fixed twobus system for all loading conditions
would not give accurate results.
Online measurement based network equivalencing techniques: Kashem
et al. (1998) used only local measurements to find a proximity index
based on the voltage to power sensitivities for predicting voltage collapse
a load bus which is a measure of closeness of the current operating point to
the stability limit point. The index can also identify the marginally stable
operating point which may extend beyond the maximum power transfer point by
estimating the network equivalent as viewed from a load bus.
Smon et al. (2006) simplified the determination
of the Thevenin’s parameters and enables derivation of the new local voltage
stability index using Tellegen’s theorem and adjoint networks which requires
the voltage and current phasors measurements only, to evaluate the system’s
voltage stability at a bus and therefore it is very appealing for PMUbased
online monitoring and protection schemes as it involves onestep calculation
procedure.
Su and Wang (2009) used Wide Area Measurement System
(WAMS) to convert the whole system into an equivalent two bus system by multiple
measured synchronized phasors of the target bus and the buses connected to it
with simple computation. An on line voltage stability index is developed based
on the relationship of the load impedance magnitude and the Thevenin’s
equivalent impedance magnitude. Early prediction of voltage collapse has great
advantages in reducing lot of reliability and economical problems.
EIAmary and EISafty (2010) simulated an early voltage
instability detector based on PMUs readings which are connected in the distribution
system. The simulated detector depends on two parallel concepts for voltage
collapse prediction, to give faster and precious response. In the first part,
the Lookup table offline calculated values for I_{Lm} and V_{Lm}
phasors using PSO technique are compared with the online PMUs readings. The
objective function of the PSO depends on the system Thevenin equivalent as seen
from the load terminal, which is connected to the PMU. The second main part
of the algorithm is the online calculation of the ratio of the Thevenin impedance
to the terminal load impedance (Zth/ZL) of the network, utilizing the readings
of the connected PMU. The variation in Thevenin voltage is also used in voltage
instability prediction. The Thevenin equivalent of the system contributes in
voltage instability detection of the system.
CONCLUSION Present literature review in the context of voltage stability assessment using network equivalent reveals that Thevenin’s theory is mostly used to develop the equivalent of any multibus power network. Other innovative approaches have also been developed and found to be capable to provide a quick overview on voltage stability status of any power system. On line methodologies along with simple on line measurements helps system operator to get a authentic insight regarding system voltage stability. Some literature suggests proper corrective measure to arrest voltage collapse using voltage stability criteria using equivalent networks. The area of voltage stability study using network equivalent is quite new as compared to conventional techniques and may be more useful if FACTS devices be incorporated in future research.

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