Voltage Stability Indices: Taxonomy, Formulation and Calculation algorithm

1
International Journal of
Science, Engineering and Innovative Research Vol. 8, February 2016
Voltage Stability Indices
Taxonomy, Formulation and Calculation algorithm
Ebrahim Saadati1
, Ahmad Mirzaei2
1
Yazd University, Faculty of ECE
2
Yazd University, Faculty of ECE
(1
ebrahim.saadati@gmail.com)
Abstract- Modern power systems are operating closer to their
limits, due to load growth and maximum efficiency of usage of
transmission line capacity. Operation of power system under such
condition may endangers the voltage stability of network and
leads to voltage collapse. Voltage stability indices are used as an
efficient, accurate and uncomplicated method in static voltage
stability analysis. In this paper a review of most common line
voltage stability indices which previously studied in literature, is
provided. A model of IEEE-14 Bus System has been presented as
case study. The simulation tool used was DIgSILENT
PowerFactory and the indices were computed using the DPL.
Simulation results approved the theoretical background of
indices.
Keywords- Voltage stability, Voltage collapse, Line stability
index, Calculation algorithm, DIgSILENT Programming
language
I. INTRODUCTION
Electricity consumption growth, complexity and cost of
construction new transmission lines, causing power systems to
be more vulnerable than ever. The more efficient use of
transmission network has already led to a situation in which
many power systems are operated more often longer and closer
to voltage stability limits resulting in a higher probability of
voltage instability or collapse [1]. IEEE defines voltage
collapse as: the process by which voltage instability leads to
loss of voltage in significant part of the power system [2]. In
order to prevent a voltage collapse, there is a desperate need to
identify the methods which is used for predict occurrence of
this phenomenon.
Even though voltage instability is known for long time,
active work involving voltage stability started in 80’s [3] and
attracted the attention of many researchers up to now. Although
stability studies, in general require a dynamic model of the
power system, the static approaches is used widely for
assessment of voltage stability [4]. Static analysis is accurate
and less complex as it requires low computation time to carry
out system stability analysis [5].
Several analytical methods are available for measuring
voltage stability margin and prediction of voltage collapse
points. Static and dynamic bifurcation analysis [6,7], P-V and
Q-V curves (also called nose curves) [2,8], continuation power
flow (CPF) [9] and modal analysis [10,11], are popularly used
in voltage stability research. Another set of papers have studied
the voltage stability by define an index. Voltage stability
indices (also called voltage collapse proximity/prediction
indices), can used as a tool for identifying weakest area, critical
lines and the power stability margin.
VSIs can be used for online and offline purposes. Operators
would be able to observe the monitored value of indices which
change with system parameters and take the appropriate action.
So the online value of VSIs is used to prevent voltage
instability. Furthermore, before operation of power system,
planner and designer would be able to used offline value of
indices for network stability studies.
As expressed in [12,13], VSIs classified in two main
categories: jacobian matrix based and system variable based
VSIs. Table I illustrates a comparison of these two groups.
TABLE I. COMPARISON OF VOLTAGE STABILITY INDEX
Jacobian based VSI System variable based
Offline application Online application
Planner/Designer use Operator use
More computing time Less computing time
Proximity to voltage collapse Determining weak area
Jacobian matrix based VSIs can calculate the voltage
collapse point or maximum loadability limit and determine the
voltage stability margin, so the computation time is high and
they are not suitable for online assessment. On the other hand,
system variables based VSIs, which use the elements of the
admittance matrix and some system variables such as bus
voltages or power flow through lines, require less computation
and therefore these VSIs are adequate for online monitoring.

International Journal of Science, Engineering and Innovative Research, Volume 8, February 2016 2
WWW.IJSEIR.ORG Paper ID:ISSN: 4212-513X
This paper provides a general overview of the indices and
their formulation that have been recently reported in literature.
Some of them were tested in IEEE-14 Bus test system. The
simulation tool used is DIgSILENT PowerFactory and the
stability indices are computed by DIgSILENT Programming
Language (DPL).
The paper is organized as follows: in Section II, taxonomy
of the reviewed VSIs is done. Also mathematical description
and formulation of them is presented. In section III the
programming procedure for calculating indices is given. In
order to validate the theoretical background of indices and see
how they work, some of them is tested on IEEE-14 Bus test
system in section IV. Finally, conclusions are presented in
Section V.
II. VOLTAGE STABILITY INDICES
After a literature research on voltage stability indices, a
lack of an organized, detailed and complete classification of
these indices was noticed. The main object of this section is to
give a classification and wide perspective of VSIs which is
proposed in most recent literature or the base paper.
Table II presents a taxonomy of VSIs. As shown in this
table, system variable based VSIs is classified in two group:
bus index (which use the bus parameters) and Line index
(which use line parameters). Definition of used parameters in
each index can be found in given reference.
Furthermore, the stable and unstable condition of each
index is presented. As you can see, value of line stability
indices change from “0” (in no load condition) to “1” (in
critical/ collapse condition).
In general, line VSIs are formulated base on a single line
in an interconnected network that is illustrated in Fig. 1.
Figure 1. single line in an interconnected network
Formulation process starts with the power equations and
forming a quadratic/quartic equation in terms of the receiving
end voltage. Then the condition for achieving real roots, in
order to satisfy the stability criteria, was investigated and the
index reproduced. In this section detailed formulation of some
VSIs is formulated and discussed.
A. Voltage Collapse Index (VCI)
This index derives from the load apparent power. As we
know, the following equation can be expressed for apparent
power:
𝑆𝑖 = 𝑉𝑖 𝐼𝑖 (1)
By use of Taylor’s theorem and neglecting higher order
terms, incremental change can be written as:
∆𝑆𝑖 =
𝜕𝑆𝑖
𝜕𝑉𝑖
∆𝑉𝑖 +
𝜕𝑆𝑖
𝜕𝐼𝑖
∆𝐼𝑖 (2)
When the load of a bus approaches the critical value ∆𝑆𝑖
approaches zero. So to assure stability:
𝐼𝑖∆𝑉𝑖 + 𝑉𝑖∆𝐼𝑖 ≥ 0 (3)
Or
1 +
𝐼𝑖∆𝑉𝑖
𝑉𝑖∆𝐼𝑖
≥ 0 (4)
Therefore the VCI at bus i, defined by:
𝑉𝐶𝐼𝑖 = (1 +
𝐼𝑖∆𝑉𝑖
𝑉𝑖∆𝐼𝑖
)
𝛼
(5)
It is raised to a power of (𝛼 >1) in order to give a more or
less linear characteristic to the index. The value of a may
depend on the system. Also this index has the value of “1” at
no load and reach to “0” at the voltage collapse point or
instability.
B. Stability Index (SI)
Following quadratic equation can be expressed for the
sending end voltage:
𝑉𝑗
4
+ 2𝑉𝑗
2
(𝑃𝑗 𝑅 + 𝑄𝑗 𝑋) − (𝑉𝑖 𝑉𝑗)
2
+ |𝑍|2
(𝑃𝑗
2
+ 𝑄𝑗
2
) = 0
(6)
From the above equation, line receiving end active and reactive
power can be written as:
𝑃𝑗 =
−𝑉𝑗
2
𝑐𝑜𝑠𝜃 ± √ 𝑉𝑗
4
𝑐𝑜𝑠2 𝜃 − 𝑉𝑗
4
− |𝑍|2 𝑄𝑗
2
− 2𝑉𝑗
2
𝑄𝑗 𝑋 + (𝑉𝑖 𝑉𝑗)
2
|𝑍|
(7)
𝑄𝑗 =
−𝑉𝑗
2
𝑠𝑖𝑛𝜃 ± √ 𝑉𝑗
4
𝑠𝑖𝑛2 𝜃 − 𝑉𝑗
4
− |𝑍|2 𝑃𝑗
2
− 2𝑉𝑗
2
𝑃𝑗 𝑅 + (𝑉𝑖 𝑉𝑗)
2
|𝑍|
(8)
The condition for the solution existence is therefore:
𝑉𝑗
4
𝑐𝑜𝑠2
𝜃 − 𝑉𝑗
4
− |𝑍|2
𝑄𝑗
2
− 2𝑉𝑗
2
𝑄𝑗 𝑋 + (𝑉𝑖 𝑉𝑗)
2
≥ 0 (9)
𝑉𝑗
4
𝑠𝑖𝑛2 𝜃 − 𝑉𝑗
4
− |𝑍|2 𝑃𝑗
2
− 2𝑉𝑗
2
𝑃𝑗 𝑅 + (𝑉𝑖 𝑉𝑗)
2
≥ 0
(10)
By adding to last equation, the SI defined by:
𝑆𝐼𝑗 = 2(𝑉𝑖 𝑉𝑗)2
− 𝑉𝑗
4
− 2𝑉𝑗
2
(𝑃𝑗 𝑅 + 𝑄𝑗 𝑋) − |𝑍|2
(𝑃𝑗
2
+ 𝑄𝑗
2
) (11)

C. Line Collapse Proximity Index (LCPI)
The relationship among the network parameters can be
expressed as:
Where A, B, C and D known as transmission parameters.
The current at the receiving end of the line is expressed as:
𝐼𝑗 =
𝑃𝑗 − 𝑗𝑄𝑗
𝑉𝑗
∗ = (𝑃𝑗 − 𝑗𝑄𝑗)/𝑉𝑗∠ − 𝛿𝑗 (13)
Also the sending end voltage can be written as:
𝑉𝑖∠𝛿𝑖 = (𝐴∠𝛼)(𝑉𝑗∠𝛿𝑗) + (𝐵∠𝛽)(𝐼𝑗∠0) (14)
By substituting the value of Eq. 13 into Eq. 14, we have:
𝑉𝑖∠𝛿𝑖 = (𝐴∠𝛼)(𝑉𝑗∠𝛿𝑗) + (𝐵∠𝛽)((𝑃𝑗 − 𝑗𝑄𝑗)/𝑉𝑗∠ − 𝛿𝑗) (15)
TABLE II. TAXONOMY OF VOLTAGE STABILITY INDICES
VSI Type Index Formula Stable Condition Unstable Condition Ref.
JacobianMatrixbased
𝑡 𝑐𝑐 = |𝑒 𝑐
𝑇
𝐽 𝐽𝑐𝑐
−1
𝑒 𝑐|
Quadratic Shape Linear Shape [14]
𝑖 =
1
𝑖0
𝜎 𝑚𝑎𝑥
𝑑𝜎 𝑚𝑎𝑥
𝑑𝜆 𝑡𝑜𝑡𝑎𝑙
⁄ 0 ≤ 𝑖 ≤ 1 𝑖 = 0 [15]
𝑇𝑉𝐼𝑖 = |
𝑑𝑉𝑖
𝑑𝜆
|
−1
𝑇𝑉𝐼𝑖 ≠ 0 𝑇𝑉𝐼𝑖 → 0 [16]
𝑉
𝑉0
⁄ 𝑉
𝑉0
⁄ = 1 𝑉
𝑉0
⁄ = 0 [17]
SystemVariablebased
Bus
Index
𝐿𝑗 = |
𝑆𝑗+
𝑌𝑗𝑗+
∗
𝑉𝑗
2| 𝐿𝑗 < 1 𝐿𝑗 = 1 [18]
𝑉𝐶𝐼𝑖 = (1 +
𝐼𝑖∆𝑉𝑖
𝑉𝑖∆𝐼𝑖
)
𝛼
𝑉𝐶𝐼𝑖 ≤ 1 𝑉𝐶𝐼𝑖 = 0 [19]
𝑆𝐼𝑗 = 2(𝑉𝑖 𝑉𝑗)2
− 𝑉𝑗
4
− 2𝑉𝑗
2
(𝑃𝑗 𝑅 + 𝑄𝑗 𝑋) − |𝑍|2
(𝑃𝑗
2
+ 𝑄𝑗
2
) 𝑆𝐼𝑗 ≥ 0 𝑆𝐼𝑗 < 0 [20]
𝑉𝐶𝑃𝐼 𝑘𝑡ℎ 𝑏𝑢𝑠 = |1 −
∑ 𝑉𝑚
𝑁
𝑚=1,𝑚≠𝑘
𝑉𝑘
| 𝑉𝐶𝑃𝐼𝑘𝑡ℎ < 1 𝑉𝐶𝑃𝐼𝑘𝑡ℎ = 1 [21]
Line
index
𝐿 𝑚𝑛 =
4𝑄𝑗 𝑋
(𝑉𝑖 sin(𝜃 − 𝛿))2
0 ≤ 𝐿 𝑚𝑛 < 1 𝐿 𝑚𝑛 = 1 [22]
𝐿𝑉𝑆𝐼 =
4𝑃𝑗 𝑅
(𝑉𝑖 cos(𝜃 − 𝛿))2
0 ≤ 𝐿𝑉𝑆𝐼 < 1 𝐿𝑉𝑆𝐼 = 1 [23]
𝐿𝑄𝑃 = 4 (
𝑋
𝑉𝑖
2) (
𝑋𝑃𝑖
2
𝑉𝑖
2 + 𝑄𝑗) 0 ≤ 𝐿𝑄𝑃 < 1 𝐿𝑄𝑃 = 1 [24]
𝐹𝑉𝑆𝐼 =
4𝑍2
𝑄𝑗
𝑉𝑖
2
𝑋
0 ≤ 𝐹𝑉𝑆𝐼 < 1 𝐹𝑉𝑆𝐼 = 1 [25]
𝑉𝐶𝑃𝐼(𝑃𝑜𝑤𝑒𝑟) =
𝑃𝑅
𝑃𝑅(𝑚𝑎𝑥)
0 ≤ 𝑉𝐶𝑃𝐼 < 1 𝑉𝐶𝑃𝐼 = 1 [26]
𝐿𝑆𝑍 = 2
|𝑍|. |𝑆𝑗|
|𝑉𝑖|2 − 2|𝑍|(𝑃𝑗 cos 𝜃 + 𝑄𝑗 sin 𝜃)
0 ≤ 𝐿𝑆𝑍 < 1 𝐿𝑆𝑍 = 1 [27]
𝑉𝑄𝐼𝑙𝑖𝑛𝑒 = 4
𝑄𝑗
𝐵𝑖𝑗. |𝑉𝑖|2 0 ≤ 𝑉𝑄𝐼 < 1 𝑉𝑄𝐼 = 1 [5]
𝑁𝑉𝑆𝐼 =
2𝑋√ 𝑃𝑗
2
+ 𝑄𝑗
2
2𝑄𝑗 𝑋 − 𝑉𝑖
2
0 ≤ 𝑁𝑉𝑆𝐼 < 1 𝑁𝑉𝑆𝐼 = 1 [28]
[
𝑉𝑖
𝐼𝑖
] = [
𝐴 𝐵
𝐶 𝐷
] [
𝑉𝑗
𝐼𝑗
] (12)

𝐿𝐶𝑃𝐼 =
4𝐴 cos 𝛼 (𝑃𝑗 𝐵 cos 𝛽 + 𝑄𝑗 𝐵 sin 𝛽)
(𝑉𝑖 cos 𝛿)2 0 ≤ 𝐿𝐶𝑃𝐼 < 1 𝐿𝐶𝑃𝐼 = 1 [29]
Separating Eq. 15 into real and imaginary parts and
considering 𝛿𝑖 − 𝛿𝑗 = 𝛿 we obtain following quadratic
equation from real part:
𝑉𝑗
2
(𝐴. cos 𝛼) − 𝑉𝑗(𝑉𝑖 cos 𝛿) + (𝑃𝑗 𝐵 cos 𝛽 + 𝑄𝑗 𝐵 sin 𝛽) = 0 (16)
The roots of quadratic equation will be:
𝑉𝑗
=
−𝑉𝑖 cos 𝛿 ± √(𝑉𝑖 cos 𝛿)2 − 4𝐴 cos 𝛼 (𝑃𝑗 𝐵 cos 𝛽 + 𝑄𝑗 𝐵 sin 𝛽)
2𝐴 cos 𝛼
(17)
In order to satisfy the stability criteria, the discriminant of
Eq. 17 must be greater than zero and the LCPI defined by:
𝐿𝐶𝑃𝐼 =
4𝐴 cos 𝛼 (𝑃𝑗 𝐵 cos 𝛽 + 𝑄𝑗 𝐵 sin 𝛽)
(𝑉𝑖 cos 𝛿)2
(18)
The value of LCPI change from “0” at no load to “1” at
unstable condition.
D. Novel Voltage Stability Index (NVSI)
In lossless condition, the current of line can be expressed
as:
𝐼𝑖𝑗 =
𝑉𝑖∠𝛿𝑖 − 𝑉𝑗∠𝛿𝑗
𝑗𝑋
(19)
Also the apparent power can be written as:
𝑆𝑗 = 𝑉𝑗
̅ 𝐼𝑖𝑗
∗̅̅̅̅ (20)
By substituting Eq. 19 in Eq. 20 we have a quartic equation
in term of receiving end voltage that the discriminant must be
greater than zero. So:
(2𝑄𝑗 𝑋 − 𝑉𝑖
2
)
2
− 4𝑋2
(𝑃𝑗
2
+ 𝑄𝑗
2
) ≥ 0 (21)
And the NVSI defined as:
𝑁𝑉𝑆𝐼 =
2𝑋√𝑃𝑗
2
+ 𝑄𝑗
2
2𝑄𝑗 𝑋 − 𝑉𝑖
2 ≤ 1
(22)
III. CALCULATION PROCEDURE
In order to find out effectiveness of line VSIs in a power
system, we should increase the load of network. In each step
we need to run a load flow and then by having the voltage,
current and power, calculate the index by use of their formula.
The flowchart of this algorithm is shown in Fig. 2.
DIgSILENT programming language (DPL) is used for
programming calculation procedure. The program is capable
to run on any optional power test system and it will give a
suitable conception from voltage stability of network based on
VSIs.
It’s necessary to mention that increase of load can be done in
all loads of power system or in specific load bus. If use the
second state, we can identify the weak bus in addition to the
critical lines.
Figure 2. VSIs calculation procedure algorithm
IV. TEST SYSTEM AND SIMULATION RESULT
A. Power system model
IEEE-14 Bus test system, is considered as the case study in
this paper and the network is simulated in DIgSILENT. Fig.3
illustrated single line diagram of this network.
B. Simulation result
VSIs calculation algorithm executed for considered case
study. Here the increase of reactive power in specific load bus
considered as the load pattern. In following some of plots and
results derived from executing programed algorithm are
presented.

Fig. 4 shows the variation of FVSI for critical line (10-11)
by increase of reactive power at bus 10. Furthermore the
voltage of bus 10 illustrated and as you seen when the FVSI
reached unity the voltage of bus decrease suddenly. This point
known as collapse point and more increase of load power isn’t
possible because the load flow can’t converge.
Figure 3. Single line diagram of IEEE 14 Bus test system
Fig. 5 illustrated the variation of LQP for critical line (6-
12) and voltage at bus 12 by increase of reactive power. In this
load bus when the LQP index reached unity the system
became unstable as you seen.
A comparison of some indices is conducted in Fig.6 and
Fig.7. As you observe from these figure, all stability indices
confirmed each other. Also by collation this two figure, we
realize that bus 14 is weaker than bus 12 because the indices
reached unity earlier in bus 12. In the other word the
maximum permissible load at bus 14 is less than bus 14.
V. CONCLUSION
In this paper the most common voltage stability indices
were reviewed then the formulation of some stability index is
expressed. Also the calculation procedure and its algorithm are
presented. An integrated program was provided in DPL which
is capable to execute on any optional electrical network and
compute some of indices. A simulation of IEEE-14 bus test
system was conducted in DIgSILENT. Simulation results
indicate the performance of programmed script code.
It was also observed that all indices are coherent with their
theoretical background and by use of them can identify critical
line of each bus and weakest bus in power system.
Figure 4. FVSI10-11 and voltage change by increase Q10 Figure 5. Comparison of indices for line 13-14 by increase Q14
0 5 10 15 20 25 30 35
0
0.2
0.4
0.6
0.8
1
1.1
Reactive Power (p.u)
FVSI/Voltage(p.u)
FVSI
Voltage@Bus10
0 5 10 15 20 25
0
0.2
0.4
0.6
0.8
1
1.1
Indices
LQP
LSZ
FVSI
0 20 40 60 80 100 120
0
0.2
0.4
0.6
0.8
1
1.1
LQP/Voltage(p.u)
LQP
Voltage@Bus12
0 20 40 60 80 100 120
0
0.2
0.4
0.6
0.8
1
1.1
Indices
FVSI
LQP
LSZ

Figure 6. QP6-12 and voltage change by increase Q12 Figure 7. Comparison of indices for line 6-12 by increase Q12
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BIOGRAPHY
Ebrahim Saadati was born in Isfahan, Iran. He graduated
from Yazd University, Iran, with master
degree in electrical power engineering in
2015. He is familiar with specialized
software, including Matlab/ Simulink
and DIgSILENT.
Ahmad Mirzaei was born in Yazd, Iran. He received the
Ph.D. degree in electrical engineering
from the Isfahan University of
Technology, Iran. He is with the faculty
of ECE, Yazd University, where he was
an Assistant Professor.

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