This paper presents the assessment methodology for voltage stability using Phasor Measurement Unit (PMU) with complete system observability. For full power system observability, the PMU placement is considered with and without conventional power flow as well as injection measurement such that minimum number of PMU’s is used. Data obtained by PMU’s are used for voltage stability assessment with the help of L-Index. As the PMU gives real time voltage and current phasors and L-index is dependent on voltage and admittance values, thus the L-index so obtained can be used as real time voltage stability indicator. The case study has been carried out on IEEE-14 bus system.

1. Abstract-- This paper presents the assessment methodology for
voltage stability using Phasor Measurement Unit (PMU) with
complete system observability. For full power system
observability, the PMU placement is considered with and without
conventional power flow as well as injection measurement such
that minimum number of PMU’s is used. Data obtained by
PMU’s are used for voltage stability assessment with the help of
L-Index. As the PMU gives real time voltage and current phasors
and L-index is dependent on voltage and admittance values, thus
the L-index so obtained can be used as real time voltage stability
indicator. The case study has been carried out on IEEE-14 bus
system.
Index Terms-- Voltage Stability, L-index, phasor measurement
unit
I. INTRODUCTION
he stability and security of electrical power system are of
much important issues for planning and operation
engineers as it supports the world economy to a great
extent. Power system is critical interdependent infrastructure
and, also, it is important to transmit electrical power from
generation to load end reliably, so its performance should be
effective [1]. To maintain these features, voltage stability is
one of the most important areas for engineers to maintain the
operation of power system within contractual, steady voltage
limits before and after any disturbances. These disturbances
are, may be, sudden loss of generation or lines, or changing
loads, which affects the operating point of system and
frequency. So, it is necessary to rapidly monitor and adjust to
system changes to attain a new operating point or an
equilibrium point keeping generation to load balance. This
ability of system is the goal of voltage stability assessment and
control [2].
In recent years, series of blackouts have been encountered in
power system. These blackouts have been occurred because
either of voltage or angle instability or both together was not
detected within time and progressive voltage or angle
instability further degraded the system condition, because of
increase in loading [3, 4]. Synchronized phasor measurements
are very important for wide area measurement systems used in
advanced power system monitoring, protection, and control
Authors are with the Department of Electrical Engineering, Indian
Institute of Technology, BHU, Varanasi. (e-mail: engi.saurabh@gmail.com,
satya.989@gmail.com, sps5957@indiatimes.com )
applications. Phasor measurement unit (PMU) becomes more
and more attractive to power engineers because it can provide
time synchronized measurements of voltage and currents
phasors [5]. Synchronization is achieved by same-time
sampling of voltage and current waveforms using timing
signals from the Global Positioning System Satellite (GPS).
The present and possible future applications of phasor
measurement units have been well documented [6]. Time
synchronization is not a new concept or a new application in
power systems. As technology advances, the time frame of
synchronized information has been steadily reduced from
minutes, to seconds, milliseconds, and now microseconds.
Several algorithms and approaches have been published in
the literature for the optimal placement of PMUs in power
system. Initiating work in PMU development and utilization is
done by Phadke et al. [5, 7]. An algorithm which finds the
minimal set of PMU placement needed for power system has
been developed in [8, 9] where the graph theory and simulated
annealing method have been used to achieve the goal. In [10]
a strategic PMU placement algorithm is developed to improve
the bad data processing capability of state estimation by taking
advantage of PMU technology. Providing selected buses with
PMUs can make the entire system observable. This will only
be possible by proper placement of PMUs among the system
buses. The authors in [11, 12, 13] developed an optimal
placement algorithm for PMUs by using integer programming.
However, the proposed integer programming becomes a
nonlinear integer programming under the existence of
conventional power flow or power injection measurements. In
[13, 15] had presented a generalized integer linear
programming formulation and solution approach for
placement.
Voltage and current phasors, obtained from PMU buses, can
be used to get full system data in control center computer
using simple KVL and KCL equations. These voltage and
current phasors are used to estimate the voltage stability via L-
index. L-index is, also, helps to determine the margin between
present operating point and the voltage instable operating
point.
In this paper a similar formulation of optimal PMU
placement problem is done by integer linear programming
with and without conventional power flow and power
injection measurement. L-index has been used to identify the
voltage stability. As the PMU gives real time voltage and
current phasors, L-index so obtained can be used for real time
voltage stability indication. Therefore, the voltage stability
Voltage Stability Assessment using Phasor
Measurement Units in Power Network with Full
System Observability
S. Kesherwani, Satyendra P. Singh, Student Member, IEEE, and S. P. Singh, Senior Member, IEEE
T

2. assessment problem using Phasor measurement unit is more
efficient and can be used in practice.
II. PMU TECHNOLOGY IN POWER SYSTEM
Phasor measurement technology (for application in the
power industry) was developed near the end of 1980s and the
first product appeared on the market in the early 1990s. Phasor
Measurement Units (PMUs) are used for Wide-Area
Measurement (WAMS) applications by power engineers and
system operators as a time-synchronized tool. The PMUs
measure time-synchronized voltage and current phasors that
are time-stamped with high precision. PMUs are equipped
with Global Positioning Systems (GPS) receivers that allow
for the synchronization of the several readings taken at distant
points. Recursive algorithm is used for calculating
symmetrical components of voltages and currents [16]. PMU-
based measurements support real-time measurements of
voltage and incident current phasors at observable system
buses. The voltage phasors contain enough information to
detect voltage-stability margin directly from their
measurements. A PMU at a substation measures voltage and
current with microsecond accuracy when the measurement
was taken. It also computes power from the measurements
(MW/MVAR) and frequency. Measurements are reported at a
rate of 20-60 times a second.
The architecture of PMU is shown in fig. 1, here, the
voltage or current in analog form is given to A/D converter
followed by anti-aliasing filter, these digital values are then
given to phasor microprocessor for computation of voltage or
current phasors. GPS receiver sends signal to phasor
microprocessor so that synchronized phasors are the output of
Modem. Phase-Locked Oscillator is used here to check the
frequency of voltage or current phasor and lock the frequency
for DFT calculation. The PMUs at different locations sends
the data, through communication channels, to Data
concentrators at RTUs and then to central utilities for advance
applications for power system control and protection, and/or
for other applications, as shown in fig. 2.
Fig.1. PMU Architecture
Fig.2. PMU utilization in a power system
The various features of PMUs are given below as follows:
• PMUs Measures 50/60 Hz AC waveforms (voltage and
current) typically at a rate of 48 samples per cycle.
• PMUs are then computed using DFT-like algorithms, and
time stamped with a GPS.
• The resultant time tagged PMUs can be transmitted to a local
or remote receiver at rates up to 60 samples per cycle.
The synchronized phasor measurement technology is
relatively new, and consequently several research groups
around the world are actively developing applications of this
technology. It seems clear that many of these applications can
be conveniently grouped as follows:
• Power System Real Time Monitoring
• Advanced network protection
• Advanced control schemes
III. PROBLEM FORMULATION
A. Topology Based Formulation Method
In this study, the OPP (optimal PMU placement) problem is
described to find a scheme with minimal PMUs and locations
to install such that the entire system becomes observable. The
rules which are followed to make the network observable-
• For PMU installed buses, voltage phasor and current
phasor of all its incident branches are known. These are
called as direct measurements.
• If voltage and current phasors at one end of a branch are
known then voltage phasor at the other end of the branch
can be obtained. These are called pseudo measurements.
• If voltage phasors of both ends of a branch are known
then the current phasor of this branch can be obtained
directly. These measurements are also called pseudo
measurements.
• For a zero-injection bus i in a N-bus system :
0 1
Where Yij is the ij-th element of admittance matrix of the
system and Vj is the voltage phasor of j-th bus.
Therefore, if there is a zero-injection bus without PMU, whose
incident branches current phasors are all known except one,
then the current phasor of the unknown one could be obtained
using KCL equations.

3. B. Integer Linear Method for PMU Placement
The OPP formulation based topological observability
method finds a minimal set of PMUs such that a bus must be
reached at least once by the PMUs. The optimal placement of
PMUs for an N bus system is formulated as follows [13]:
2
. .
… … … . 3
0,1
Where, N is total no. of system buses wk is weight factor
accounting to the cost of installed PMU at bus k, X is a binary
variable vector whose entries are defined as Eq. 4 and AX is a
vector function that its entries are non-zero if the
corresponding bus voltage is observable using the given
measurement set and according to observability rules
mentioned above, it ensure full network observability while
minimizing the total installation cost of the PMUs, otherwise
its entries are zero.
1
0
4
The entries in A are defined as follows:
,
1
1
0
5
And b is a vector whose entries are all ones as shown in
Eq. (6).
1
1..
1
6
The procedure for building the constraint equations will be
described for three possible cases where there are (1) no
conventional measurement, (2) flow measurements or (3) flow
measurements as well as injection measurements (they may be
zero injections or measured injections).
C. Voltage Stability Indicator
The voltage stability problem is mostly reactive power
related problem. So, it is necessary to identify the buses in
multi-bus power system that can provide reactive power to
support voltage magnitude of the bus. In multi-bus power
system, basically, all buses are divided into two categories
as Generator bus (PV bus and Slack bus) and Load bus (PQ
bus). Generator buses provide reactive power to maintain
voltage magnitude.
The power system can be represented as:
7
Subscript L means Load bus, and G means Generator bus.
The above equation can also be represented as,
8
When we consider the voltage at load node j, we know that,
9
or,
10
Multiplying at the both sides of the equation,
11
1
12
Here, ∑ , and
∑
The term V0j includes the contribution of all generators and
S shows the contribution of other loads at the node j.
13
The L index can be given as [14],
1 14
Thus, this value of index L indicates the proximity of
voltage collaps, including the contribution of all generators
and also the contributionof other loads.
The process for praposed methode, that is to search the
optimal location of PMUs for full observability of power
network using Linear integer programming technique and to
assess the voltage stability of system using L index, is given in
flow chart shown above in fig. 8.

4. Fig. 8. Flow chart for proposed case
IV. SOLUTION METHOD
A. PMU Placement
Case1: A system with no conventional measurements
In this case, the flow measurement and the zero injection
are ignored. In order to form the constraint set, the binary
connectivity matrix A, whose entries are defined below, will
be formed first:
,
1,
1,
0,
Matrix A can be directly obtained from the bus admittance
matrix by transforming its entries into binary form.
Fig. 3. 5-bus system
Consider the 5-bus system and its measurement
configuration shown above. Building the A matrix for the 5-
bus system of Fig (3) yields:
1 1 1 0 0
1
1
0
0
1
0
1
0
0
1
1
0
1
1
1
1
0
0
1
1
15
The constraints for this case can be formed as:
1
1
1
1
1
16
The use of 1 in the right hand side of the inequality ensures
that at least one of the variables appearing in this will be non-
zero. The constraint f4 ≥ 1 implies that at least one PMU must
be placed at either one of buses 4 or 5 (or both) in order to
make bus 4 observable. Similarly, the second constraint f2 ≥ 1
indicates that at least one PMU should be installed at any one
of the buses 1, 2, or 4 in order to make bus 2 observable.
Case 2: A system with some flow measurements
This case considers the situation where some flow
measurements may be present. The modifications needed in
the formulation for this case will again be illustrated using the
5-bus example, where a flow measurement (P and Q) is added
for branch 1-2. In this case, the constraints for bus 1 and 2 will
have to be modified accordingly. The constraint equations
associated with the terminal buses of the measured branch can
be merged into a single constraint. So, for the example shown
above, the constraints for buses 1 and 2 are merged into a joint
constraint as follows,
1
1
_ 1
Which implies that if either one of the voltage phasors at
bus 1 or 2 is observable, the other one will be observable.
Applying this modification to the constraints for the shown
example of 5-bus system, the following set of final constraints
will be obtained:
_ 1
1
1
1
Here, the constraints corresponding to buses 1 and 2 are
merged into a single constraint.
Case 3: A system with both injection measurements and flow
measurements.
This case considers the most general situation where both
injection and flow measurements may be present, but not
enough to make the entire system observable. Injection
measurements whether they are zero injections or not, are
treated the same way.
Consider again the 5-bus system shown in Fig 3, where bus
4 is assumed to be a zero injection bus. In this case, it is easy
to see that if the phasor voltages at any three out of the four
buses 2, 3, 4 and 5 are known, then the fourth one can be
calculated using the Kirchhoff’s Current Law applied at bus 4
where the net injected current is known. Hence, the constraints
associated with these buses will have to be modified
accordingly as shown below:

5. _ · · 1
_ · · 1
_ · · 1
If two sets are A and B, where set A is a subset of set B,
Then A+B=B and A·B=A, where ‘.’ serves as the logical
“AND” and ‘+’ as logical “OR”. So, substituting the
expression for f3 in the expression for f1_new , we can write f1_new
as:
· ·
· ·
· · · · · ·
Proceeding with the simplifications, the product x1· f4· f5 is
eliminated because it is the subset of x1, which already exists
in the expression. Using similar reasoning, x3· f4 ·f5 and x4 ·f4
·f5 are also eliminated. Finally, the expression for f1_new
simplifies to the following:
_ 1
Applying similar simplification logic to all other
expressions will give modified constraints.
Note that the constraints corresponding to all other buses
will remain the same as given in equation (7). One exception
is the constraint for bus 4 where the injection is measured (or
known). This constraint will be eliminated from the constraint
set. The reason for removing the constraints associated with
injection buses is that their effects are indirectly taken into
account by the product terms augmented to the constraints
associated with the neighboring buses.
B. L-index
For the 5-bus system shown in fig. 3 there are two load
buses (bus 4 and bus 5) and three generator buses (bus 1, slack
bus, and bus 2 and bus 3). So for load bus 4, the value of L-
index can be examined using equation (14) as shown below,
Form equation (11),
Here, ∑
In which ∑ and,
. Thus, by using above equation L-indicator can be calculated
as,
1
V. TEST RESULTS
The proposed formulation has been tested on IEEE-14 bus
system. Binary integer programming under MATLAB has
been used to solve this problem expressed by equations (2)
and (3). The proposed integer linear programming algorithm
has been tested on different cases.
IEEE 14-bus system is shown in Figure (4) The Information
of the system and zero injections are given in the Table I. The
results without and with zero injection measurement are
displayed in Tables II.
Fig. 4. IEEE 14-bus system without PMUs
Fig. 5. IEEE 14-bus system with PMUs
TABLE I
SYSTEM INFORMATION OF IEEE BUS SYSTEMS
System Total
number of
branches
Total number
of zero
injection
Zero
injection
buses
IEEE 14-bus 20 1 7
TABLE II
SIMULATION RESULTS FOR THE 14-BUS SYSTEM WITH AND
WITHOUT CONSIDERING ZERO INJECTIONS
System Ignore zero injection Consider zero injection
Number
of PMUs
Location
of PMUs
Number
of PMUs
Location of
PMUs
IEEE
14-bus
4 2, 6, 7, 9 3 2, 6, 9
Effect of considering zero injections—
In this case, Integer Programming method expressed by
equations (2) and (3) has been used to solve the optimal PMU
placement Problem with and without considering zero
injections. Results are given in Table II.
The voltage stability assement methdology is tested on
IEEE-14 bus sytem. In this paper, L-index method discussed
in section III.C is used to check the voltage stability of
different load buses. The effect of loading at any load bus is
assessed at increasing the load on that bus it self , to its

6. neighbouring bus connect to the bus under c
to the other weak load buses using L-index
shown in fig. 6 and fig. 7.
Fig. 6. Voltage/L-index V/s loading at Bus
bus system
Fig. 7. Voltage/L-index V/s loading at Bus 4
system
As shown in fig. 6, value of L-Index t
voltage of Bus 14 is approaching to colla
increase in load at bus 14. The effect of incr
Bus 14 on L-index of a neighboring Bus 9 a
also shown in same fig. 6. The L-index of b
0.0677 to 0.2597 and that of bus 5, varies
0.0356. Variation in L-index at buses 4, 5
increase in load at bus 4 is demonstrated in
observed from this figure that the L index of
unity as the voltage of bus 4 approaches co
effect of increase in loading at Bus 4
neighboring Bus 5 and a far from Bus 13,
same fig. 7. The variation in L-Index of Bus
to 0.404 and of Bus 13 is from 0.032 to 0.063
Thus from above figures it can be conclud
of loading at any bus is more on its neigh
respect to a bus which is far from the bus und
As the effect of increase in load at bus 1
neighboring bus 9 with respect to a far bus 5
and the effect of increase in load at bus
neighbor bus 5 with respect to a far bus 13, sh
VI. CONCLUSION
This paper proposes a simple algori
placement of PMUs in power system for ful
network for voltage stability assessment. The
formulated using topology based algorithm
integer linear programming. Besides the pl
consideration and
x. The results are
s 14 for IEEE 14-
4 for IEEE 14-bus
tends to unity as
apse point due to
rease in loading at
and a far Bus 5, is
bus 9 varies from
s from 0.0197 to
5, and 13 due to
n fig.7. It can be
f bus 4 approaches
ollapse point. The
on L-index of
is also shown in
s 5 is from 0.0197
38.
ded that the effect
hboring bus with
der consideration.
4 is more on its
5, shown in fig 6,
4 is more on its
hown in fig. 7.
ithm of optimal
ll observability of
e OPP problem is
and solved using
lacement of mere
PMUs, this study also considers the
conventional measurements are pr
voltage stability assessment proble
index method and solved using MA
present case also accomplished the tw
to develop practical methods for dete
for PMUs with voltage stability ass
develop methods for implementation
Simulation results on IEEE-14 bus
the proposed placement method
assessment is satisfactorily prov
measurements with minimum numb
index to determine the stability of po
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