Energy and Power Engineering, 2013, 5, 1497-1502
doi:10.4236/epe.2013.54B283 Published Online July 2013 (http://www.scirp.org/journal/epe)
The Mechanism of Voltage Instability Analysis
Considering Load Characteristic
Jundong Duan, Jiaxing Huang
School of Electrical Engineering & Automation, Henan Polytechnic University, Jiaozuo, China
Received 2013
ABSTRACT
Load characteristics are the main factor to affect voltage stability. In addition, load modeling reflecting the actual load
characteristics has been a well—known difficult problem and unsolved so far, It is mainly due to the fact that the load
composition, amount and characteristics are always changing. This paper introduces the slip into the equivalent imped-
ance load model to analyze load characteristic, the varying slip is e mployed to indicate time-varying load characteristic
precisely, and considering the dissimilar load behaviors, discusses node voltage, power during the changes of load cha-
racteristic, obtains voltage inflexion and power inflexion, and then analyzes the mechanism of power system voltage
instability based on static voltage stability region. An example indicates the feasibility of the method.
Keywords: Power System; Voltage Instability; Dynamic Load Ch aracter; Three Element Analysis Method; The Equiv-
alent Impedance Load Model
1. Introduction
Load characteristic is one of the most active, direct fac-
tors affecting voltage stability, which causes a dynamic
change of voltage and even determines the process of
voltage collapse in the extreme environment. Therefore,
more and more scholars try to investigate load character-
istic.
Reference [1] shows the relationship between load
characteristic and voltage stability by stud ying the power
exponent relationship between the voltage stability re-
gion and the power function model. Reference [2] ana-
lyzes voltage stability with load modeling according to
the load characteristic space, which has a very strong
generalization capability. However, the generalization
capability is not enough for unknown sample space. Ref-
erence [3] investigates the influence of stalling of induc-
tion motor to PV curve and voltage static stability.
However, it doesn’t consider the change of PV curve
along with the change of the operating condition. Refer-
ence [4] transforms the induction motor parameters to
line parameters and describes load characteristic of in-
duction motor with th e constant power load model, b ut it
does not take the instable character of induction motor
into account. Reference [5] shows that the adjusting
proportion of the dynamic load model will causes the
changes of the instable mode by simulation, and there are
different effects on the voltage stability under the differ-
ent instable mod e. Reference [6] studies the node critical
state according to the static voltage stability theory. It
analyses the shortages of the p-v description of voltage
critical state by using an equivalent circuit of the simple
system. In addition, it discusses the influence on the node
voltage critical position when the system operating con-
dition changes, and describes the node critical state with
the load impedance angle, the maximum power and the
critical voltage. Therefore, it establishes a more intuitive
voltage stability region to make a stability judgment.
On the basis of references[5，6] ,This paper introduces
the slip into the equivalent impedance load model to
analyze load characteristic, which can reflect the change
of load characteristic clearly, meanwhile, discusses the
changes of the node voltage and power when the load
dynamic characteristic changes. By taking advantage of
three element analysis method which deduces the voltage
stability region of the transmission network, this paper
analysis the mechanism of vo ltage instability con sidering
load characteristic. At the same time, the research on
load dynamic characteristic provides the basis for load
modeling which could reflect load characteristic more
actually.
2. System Model
Based on the mechanism of static voltage stability, it
only consider the changes of node load in the research of
static voltage stability, regardless of the dynamic charac-
teristics of the synchronous generator, on-load tap chan-

J. D. DUAN, J. X. HUANG
1498
ger(OLTC) and the reactive power compensation devices.
Therefore, the system can be equivalent to a two-node
system from the testing load node to the system by The-
venin equivalent theory. It shows in Figure 1.
3. The Transmission Power Characteristic of
Power System
Because of the shortages that the P-V curve changes
along with the present operating condition and it's diffi-
cult to get a clear steady boundary, In addition, the varia-
tion quantity of voltage amplitude and an gle isn't equ al to
the variation quan tity of active power and reactive power.
This paper takes three element analysis method put for-
ward by references [6] to make a stability judgment.
Three-element analysis method presents a clear voltage
boundary and power boundary in view of the changes of
the operating condition. It describes critical state with
load impedance angle, power limit, and critical voltage.
The following conclusion will be held from Figure 2 two
nodes system: generator-transmission-load.
1) The upper area of Vcr- curve is voltage stability
region. The area under Vcr- curve is voltage instability
region that the value of voltage is less than the critical
value of nod e v ol tages.
2) The upper area of Pmax- curve is power instability
region that the value of node power is more than trans-
mission power limit. The area under Pmax- curve is
power stability region.
4. Load Model Analysis
At present, the r egional power grid adopts the parallel of
the induction motor and constant impedance load to si-
mulate comprehensive load. Because parallel and serial
is equivalent, it does not affect the electric circuit by
Figure 1. The two nodes system.
(a) (b)
Figure 2. (a) Vcr- curve and (b) Pmax- curve.
taking the induction motor and constant impedance load
in series. So, comprehensive load _type model equiva-
lent circuit can be shown in F igure 3.
Where: r1 + x1 is series constant impedance load,
''
12 12
cr jc x
sis dynamic load. The following analysis of
the proportion of constant impedance load means the
proportion of series constant impedance load.
It’s necessary to analyze the change of comprehensive
load impedance, because it's th e substantial cause for the
change of load characteristics. According to Figure 3,
the comprehensive load impedance, the power factor and
the critical slip can be written as follows:
''
12
11
()(
cr 12
)
Z
rjxc
s
 x (1)
'
12
1
()
cos
cr
r
s
Z


 (2)
'
12
2
1112
()
cr cr
srxcx '
2
(3)
where: 1
11
m
x
c
x
 .
Figure 4 can be given according to (1), it shows that
modulus of impedance decreases quickly with slow in-
crease of the slip in 0<s<scr. however, module of load
impedance changes a little with the quick increase of the
slip in scr> s> 1.
Figure 5 can be given according to (2), it shows that
power factor decreases quickly with slow increase of the
slip in 0<s<s, while modulus of impedance changes a
little with quick increase of the slip in s>s>1. Here, the
s defines as the abrupt change point of power factor.
Power factor changes acutely in s<s, and in s>s , the
changes of power factor are unobvious.
The increase of proportion of constant impedance load
makes the Z-s curve, cos-s curve move up, improved
the overall quality of modulus of impedance and power
factor.
Figure 3. -type model equivalent circuit.

J. D. DUAN, J. X. HUANG 1499
Figure 4. Z-s curve.
Figure 5. cos-s curve.
Figure 6. -s curve
Thus, the different slip and proportion of constant
impedance dynamic load will alter the impedance of the
comprehensive load, moreover affect load characteristic.
Figure 6 can be given according to (2). With micro
increase of the slip, impedance angle increases rapidly,
therefore, it’s convenient for further analysis to take -s
coordinate change, that it’s clear to analyze the sharp
change of load characteristic causes by the module of
impedance and power factor in large interval of , but
not in small interval of the slip.
5. The Node Voltage and Active
Poweranalysis
In Figure 1, the node voltage and active power can be
written as:

2
'2
2'
12
111
2
e
Tcr s
Vrxc
Nr s

2
x





 






(4)

' 212
1
2
'2
'
12
1112
cr V
Pr
s
cr
rxcx
s



 





(5)
V- curve is given by (2) and (4) when torque is con-
stant, which shows in Figure 6. Along with increase of
the impedance angle (increase of the slip), voltage drops,
and it reach minimum at s .Later, the slip keeps on
increasing, on the contrary, voltage rises, P- curve is
given by (2) and (5),which shows in Figure 7. Along
with increase of the impedance angle (increase of the
slip), the node active power increases, and it reach max-
imum at scr .Later, the slip keeps on increasing, on the
contrary, the active power decreases.
In Figure 7, the increase of the proportion of constant
impedance load leads V- curve to move up, and the
more the proportion of constant impedance load is, the
more powerful the up warp will be, the less s will be,
the more beneficial to the voltage recovery will be. At
this time, the high current caused by the starting or
blocking of motor load can't make the node voltage
worse. In Fig. 8, P- curve move down with increase of
proportion of constant impedance load, moreover, the
more the proportion of constant impedance load is, the
less scr will be; It means that the active power could be
easier to comes to its maximum, but the smaller the
maximum will be.
Therefore, the node voltage inflexion point influenced
by load characteristic is s point (load voltage critical
turning point); the amount of s determines what time it
begins to recover voltage and the amount of the mini-
mum node voltage. The node active power inflexion
point influenced by load characteristic is scr point (load
power critical turning point); the amount of scr deter-
mines what time the active power begins to reach maxi-
mum and the amount of the maximum node power.
6. Example Analysis
This paper takes Figure 1: two nodes system as an ex-
ample to analyze, and choose Category 6 industrial and

J. D. DUAN, J. X. HUANG
1500
civil motor parameters recommended by the IEEE.
Figure 7. V- curve.
Figure 8. P- curve.
Figure 9. V- stability.
Figure 10. P- stability.
According to the Figure 2(a), 7 and Figure 2(b), 8,
the node V- stable curve and P- stable curve can be
obtained respectively, as shown in Figure 9 and Figure
10.
6.1. The Analysis of the Influence of
Comprehensive Load Characteristic on
Static Voltage Stability
Load characteristic varies along with the slip. In Fig. 9,
the node voltage drops along with increase of the slip,
eventually V- curve and Vcr- curve intersect, and then
if the slip continues increasing, the system will be insta-
bility, but it's un certain to cause voltage collaps e. In s>s ,
voltage recovers gradually alo ng with increase of the slip.
If it could recover to the critical voltage, the system will
recover the voltage stability. Conversely, it will be volt-
age collapse. In Fig. 10, the node power increases along
with increase of the slip, eventually P-  and Pmax- curve
intersect, and then if the slip continues increasing, the
power will be unbalance, and the unbalance will last
forever.
6.2. The Analysis of the Influence of Proportion
of Constant Impedance Load and Dynamic
Load on Static Voltage Stability
Load characteristic varies along with the change of pro-
portion of constant impedance load and dynamic load. In
Figure 9, the intersection of V- curve and Vcr- curve
may be zero, one or two. If the proportion of constant
impedance and dynamic load is 2:1, V- curve and Vcr-
curve don’t intersect, the system has always been in vol-
tage stability. However, with increase of proportion of
dynamic load, V- curve move down, the s increases,
and the value of maximum power decreases, V- curve
and Vcr- curve will get an intersection, thus, there will

J. D. DUAN, J. X. HUANG 1501
be parts of V- curve falling in the unsteady region; In
Figure 10, the intersection of P- and Pmax- curve may
be zero or one. If the proportion of constant impedance
and dynamic load is 2:1, P- and Pmax- curve don’t in-
tersect, the system has always been in power balance.
Nevertheless, with increase of proportion of dynamic
load, P- curve moves up, the scr increases, and the value
of maximum power increases, P- and Pmax- curve will
get an intersection, thus, there will be parts of P- curve
falling in the unsteady region.
7. Instability Mechanism Analysis
7.1. Instability Causes by the Slip
The node voltage drops along with increase of the slip.
And the system stay in the edge of voltage stability until
V- and Vcr- curve intersect. If the slip continues in-
creasing, the system will be instable, the voltage start
recovery until the slip reach s .At this moment if s is
lower, there will be enough space for voltage to recover,
it won’t cause voltage collapse. Conversely, if s is larger,
voltage can’t return to the critical voltage, because the
slip can’t increase again. And finally, it will be voltage
collapse.
The node power increases along with increase of the
slip. And the system stay in the edge of power stability
until P- and Pmax- curve intersect. If the slip continues
increasing, the power will be unbalance. Th e node power
is the same to the node voltag e which has recovery char-
acteristic. In scr< s< 1, with increase of the slip, the node
power decrease, however if the power system transfer
limits can’t increases, the system will always be in power
unbalance state.
7.2. Instability Causes by the Proportion of
Dynamic Load and Constant Impedance
Load
Along with increase of the proportion of dynamic load,
V- curve moves down, s increases, and then the more
curve will fall in unstable region. However the increase
of s means the voltage recovery space is reduced, the
decrease of the voltage minimum corresponding to s
means there is bigger distance between the value of vol-
tage instability and the value of critical voltage. It’s un-
favorable to voltage recovery, thereby, the system volt-
age stability weakens. On the contrary, the increase of
proportion of constant impedance load will enhance the
system voltage stability. Further more, if the proportion
of constant impedance load increases to an extent, V-
and Vcr- curve won’t intersect, load characteristic which
changes along with the slip has no effect on voltage sta-
bility.
With increase of proportion of dynamic load, P-
curve moves upward, scr increases, the node power
maximum corresponding to scr increases, and the more
curve will fall in unstable region. Conversely, with in-
crease of proportion of constant impedance load, P-j
move downward, scr decreases, and the more curve will
fall in unstable region, but the node power maximum
corresponding to scr decreases. Here the amount of scr
means the positional closeness of Pmax- curve and the
node power maximum. Due to the change of load pro-
portion, P- may be intersect with Pmax- before scr, or
may be intersect with Pmax- after scr. However, if they
get intersection before scr, the more transmission power
will be transferred under the power balance conditions.
Further more, with regard to heavy load line, we hope
that it can reduce transmission power under the power
balance conditions, so the proportion of dynamic load
should be increased. With regard to heavy load line, we
hope that it can transfer more power under power balance
conditions, so the proportion of constant impedance load
should be increased. But the overlarge proportion of con-
stant impedance load is unfavorable to transfer more
power, and the overlarge proportion of dynamic load is
unfavorable to power balance. Hence, a reasonable load
proportion should be taken, and then a reasonable power
will be transferred under power balance conditions,
which benefits to avoiding power unbalance and enhance
transmission efficiency.
8. Conclusions
This paper takes the equivalent impedance load model to
analyze load characteristic and study the mechanism of
voltage instability considering load characteristics based
on static voltage stability region. The following conclu-
sion will be obtained through the above analysis:
1) The change of th e slip result in the change of power
factor and module of impedance, consequently, causes
the change of load characteristic. And the system stabil-
ity worsens along with increase of th e slip.
2) The different proportions of constant impedance
load and dynamic load result in different comprehensive
load characteristics which generate different impacts on
static voltage stability. The amount of s determines the
strength of system voltage stability. The amount of scr
determines the maximum of power transmission.
3) The increase of proportion of Constant impedance
load is helpful to voltag e stability, however if the pro por-
tion is overlarge, it’s not conducive to power transfer.
Therefore, a reasonable load proportion could avoid
power unbalance and enhance transmission efficiency.
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