Engineering, Technology & Applied Science Research Vol. 7, No. 6, 2017, 2240-2250 2240

www.etasr.com Ravindra and Srinivasa Rao: Sensitive Constrained Optimal PMU Allocation with Complete …

Sensitive Constrained Optimal PMU Allocation with
Complete Observability for State Estimation Solution

M. Ravindra
Electrical and Electronics Engineering Department

Jawaharlal Nehru Technological University, Kakinada
Kakinada, India

ravieeejntu@gmail.com

R. Srinivasa Rao
Electrical and Electronics Engineering Department

Jawaharlal Nehru Technological University, Kakinada
Kakinada, India

srinivas.jntueee@gmail.com

Abstract—In this paper, a sensitive constrained integer linear
programming approach is formulated for the optimal allocation
of Phasor Measurement Units (PMUs) in a power system network
to obtain state estimation. In this approach, sensitive buses along
with zero injection buses (ZIB) are considered for optimal
allocation of PMUs in the network to generate state estimation
solutions. Sensitive buses are evolved from the mean of bus
voltages subjected to increase of load consistently up to 50%.
Sensitive buses are ranked in order to place PMUs. Sensitive
constrained optimal PMU allocation in case of single line and no
line contingency are considered in observability analysis to
ensure protection and control of power system from abnormal
conditions. Modeling of ZIB constraints is included to minimize
the number of PMU network allocations. This paper presents
optimal allocation of PMU at sensitive buses with zero injection
modeling, considering cost criteria and redundancy to increase
the accuracy of state estimation solution without losing
observability of the whole system. Simulations are carried out on
IEEE 14, 30 and 57 bus systems and results obtained are
compared with traditional and other state estimation methods
available in the literature, to demonstrate the effectiveness of the
proposed method.

Keywords—sensitive; bus; observability; phasor; measurement;
pmu; state; estimation; zero; injection; zib

I. INTRODUCTION
State estimation plays a vital role in real-time control of

power system providing security and reliability. It acts as a
filter between the received information and application
functions that need reliable data. In power systems, supervisory
control and data acquisition (SCADA) systems are used to
collect the raw data (bus voltage magnitudes, currents and
complex power flows) of the transmission network and this
data is processed for state estimation solution. However, this
information is not sufficient to estimate the accurate states of
voltage and phase angles at every bus in the system. One of the
recent developments in real time analysis of power systems is
an implementation of synchrophasor in the state estimation
process. Synchrophasors or PMUs are used to directly measure
phase angles associated with voltages and currents with respect
to an absolute time reference. This absolute reference is
provided by common timing signal by high accuracy clocks to
Coordinated Universal Time (UTC) such as Global Positioning

Systems (GPS) [1]. A phasor measuring unit is a device used to
store synchronized measurement data and to communicate the
same to a control center through GPS. Allocation of PMUs at
every bus to measure the complete data of the system is not a
feasible solution from economic perspective. Hence, in order to
get complete observability and identify gross errors in
measurement set, PMUs must be placed optimally at sensitive
nodes by considering ZIB in the network. In a power system
network, some of the buses are sensitive to sudden load
changes that may lead to a blackout. Blackouts occur in power
grid predominantly due to inadequate generation and state
predictability of the system [2]. Prior knowledge of the states
of a system at sensitive buses can be very useful in avoiding
blackouts. Different types of sensitive indices are proposed in
the literature for various applications in the performance
analysis of power system [3-7]. In this paper voltage sensitive
indices are used to identify sensitive buses for the placement of
PMUs.

Several approaches are proposed in literature for optimal
PMU placement. Initial work on phasor measurements using
Synchrophasors to measure phasor voltage and currents is
proposed in [8]. Authors in [9] proposed a Linear Programming
based measurement system for maintaining observability when
a single line outage occurs in the network. Authors in [10] used
GA with the immune operator to select optimal sites for PMU
placement to achieve observability with improved converging
speed and execution time. Authors in [11] implemented Non-
dominated Sorting Genetic Algorithm (NSGA) in PMU
placement problem and simultaneously optimized two
conflicting objectives such as minimization of number of
PMUs and maximization of the measurement redundancy and
obtained Pareto-optimal solutions. Authors in [12] developed
an integrated model to show the effect of ZIB and conventional
measurements in PMU allocation to enhance system
observability by considering single branch and single PMU
outage contingencies separately and simultaneously. Authors in
[13] proposed a method for multi-staging of PMUs by
modeling zero injection constraint as a linear model in integer
linear programming approach with two indices such as Bus
Observability Index (BOI) and System Observability
Redundancy Index (SORI). Author in [14] proposed a
generalized linear integer programming approach for redundant

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PMU placement, full and incomplete observability, with and
without zero injections and showed that accuracy, redundancy,
and robustness of state estimation solution will be enhanced
with PMU placement. Authors in [15] considered the PMU
placement problem as a quadratic minimization problem with
continuous decision variables and solved the problem using
non-linear weighted least squares approach. Authors in [16]
proposed a binary search algorithm to determine the minimum
number of PMUs for observability under normal conditions
and single branch outage conditions. Authors in [17]
introduced the concept of time-synchronized measurements for
PMU placement and solved the problem using integer
quadratic programming to minimize the total number of PMUs
and to increase the redundancy at power system buses. In [18],
authors formulated a binary integer linear programming
approach with binary decision variables to locate PMUs at each
bus by integrating possibility of single or multiple PMU loss in
the decision strategy while preserving observability and the
lowest metering economy. Authors in [19] introduced a hybrid
constrained state estimator in which conventional and
synchrophasor measurements are incorporated simultaneously
in the problem without using any measurement transformation
and shown that solution converges faster with small
uncertainty. Authors in [20] introduced a multistage state
estimation procedure to include synchrophasor measurements
without disturbing existing SCADA system. This procedure
requires more PMUs which opposes the economic criteria.
Authors in [21] investigated three different methods to include
PMU measurements into state estimation problem.

In this paper, a sensitive constrained integer linear
programming approach is used for optimal PMU allocation
considering sensitive buses along with ZIBs. The PMUs are
placed at sensitive buses which are identified based on voltage
stability index with complete observability. The method is
tested with different test cases and results are presented. The
rest of the paper is organized as follows: Section II deals with
traditional state estimation method; Section III covers optimal
PMU allocation considering sensitive buses; Section IV
presents state estimation with conventional and PMU
measurements. Section V provides results and analysis and
Section VI gives conclusions.

II. TRADITIONAL STATE ESTIMATION
The traditional state estimation method collects data from

SCADA systems to find states of power systems. For a given
set of measurements, weighted least squares state estimation
[22] can be represented as:

2
m

1k
kkW krMin



(1)

subjected to: kkk rxhZ  )( , k=1….m (2)

where rk = Zk-hk(x) is residual value of measurement k
calculated from the difference of the measured Zk and a
nonlinear function hk(x) related to measurement state vector
(x).

The residual vector 2kr is weighted by kkW =
2

k which is
calculated from the standard deviation of respective
measurements and m is the number of measurements. The gain
matrix is formed for better accurate state estimation,
formulated with Jacobian H matrix and covariance

}......,{ 222
2
1 mdiagR  . The Jacobian matrix for traditional

state estimation is defined as
























































v
v

QQ
v

PP
v

QQ
v

mag

flfl

inin



(3)

The gain matrix is defined as

ZRHxG T  1)( (4)

where HRHG T 1 and )(xhZZ k
From (4), the solution is obtained by updating the state

vector

xxx tt 1 (5)

The convergence of the state estimation algorithm is
obtained when Δx becomes smaller than the tolerance value 10-
5. In state estimation process observability of the system can be
obtained numerically by finding the rank of the Jacobian matrix
or through topological bus connectivity matrix. If Jacobian
matrix H has full rank the system is numerically observable
otherwise is not observable. In this paper topological
observability is considered, each bus is checked with
Redundancy Index (RI) and total system with Complete
System Observability Redundancy Index (CSORI) to show
complete observability of the system.

III. OPTIMAL PMU ALLOCATION CONSIDERING SENSITIVE
BUSES

A. Formulation for the Optimal PMU Allocation Problem
The proposed approach is formulated as an optimization

problem of allocating PMUs in the network with highest
preference at sensitive buses for complete observability
considering cost criteria which produce accurate state
estimation solution. The optimization problem for optimal
allocation of PMUs is formulated as:

k
k 1

F kMin x

 (6)

Subject to BAX  and eqeq BXA  (7)

Engineering, Technology & Applied Science Research Vol. 7, No. 6, 2017, 2240-2250 2242

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where Fk is cost coefficient of PMU installed at bus k, Aeq

matrix of order n×n consisting of sensitive buses, whose entries
are all equal to one and for other buses it is zero, n is number of

buses,  TnxxxxX ...321 is a decision variable
vector, B and Beq is observability constraints matrix which can

be written as  Tn 11....1 1 1 1  , kx is binary decision variable and
A is incidence matrix of order i×j which are defined as






otherwise 0

busat installed is PMU if 1 k
xk



 


otherwise 0

other each toconnectedor if 1
,

ji
A ji

Proper identification of sensitive buses in the system and

optimal placement of PMUs at these buses accurately estimate
the states of state estimation problem.

B. Identification of Sensitive Buses
From experiences of major blackouts occurred in India,

failure of the supply occurred due to either sudden removal of
generation or overloading beyond safety limits leading to dip in
system voltage. In order to determine the most sensitive buses
in a system which are prone to load changes, voltage stability
indexes (VSI) at each bus are formulated by increasing the load
from 5 to 50%.





N

i
avg iV

N
V

)(
1

(8)

0VVVSI avg 
(9)

where, V0 is the vector of true bus voltages obtained from the
base load, Vavg is calculated from average of voltages obtained
with increased load and N is the number of load samples. VSI
is calculated from the difference of average voltage and true
bus voltage. The bus with the highest VSI value among buses
is considered as the highest sensitive bus in the system. The
calculated values of VSI for 14-bus test case system are shown
in Table I. From Table I data, sensitive buses are arranged in
descending order i.e., B14 > B9 > B10 > B4 > B7 > B5 > B11
> B13 > B12, where B1 is slack bus B2, B3, B6, and B8 are
generator buses, B7 is ZIB and remaining are load buses. B14
and B9 buses are the most sensitive buses (in that order)
affected by load change. In this paper, sensitive buses are
considered as constraints for optimal PMU allocation. The
procedure to compute voltage stability index (VSI) and
sensitive buses at each bus is presented in Figure 1.

TABLE I. VSI DATA OBTAINED FROM LOAD FLOWS

Bus no B1 B2 B3 B4 B5 B6 B7
VSI 0 0 0 0.0062 0.0059 0 0.0062

Bus no B8 B9 B10 B11 B12 B13 B14
VSI 0 0.0095 0.0094 0.0055 0.0036 0.0051 0.0115

Fig. 1. Flowchart to estimate sensitive buses

C. Sensitive Constrained Optimal PMU Allocation
Placement of PMU at each and every bus increases the

number of PMUs in the system which is a drawback from
economic perspective. Hence, in order to optimize the number
of PMUs and to allocate PMUs at sensitive buses for accurate
state estimation solution, the problem is formulated as follows
using an eight-bus network as shown in Figure 2.

Fig. 2. Eight- bus system network diagram

The optimization problem can be expressed as

1 2 3 8..........Min x x x x   (10)

Subject to observability constraints

Bus 1: 121  xx (11)

Bus 2: 15321  xxxx (12)

Bus 3: 1352  xxx (13)

Bus 4: 154  xx (14)

Bus 5: 15432  xxxx (15)

Bus 6: 176  xx (16)

Bus 7: 18765  xxxx (17)

Bus 8: 187  xx (18)

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where }1,0{kx in sensitive constrained integer linear
programming formulation. For instance, consider bus 5 and 7
are the most sensitive buses in the eight-bus system. In order to
allocate PMU at sensitive buses, locations suggested
are 1,1 75  xx . As a result of substituting these values in
observability constraints (11-18), inequalities for buses 2, 3, 4,
5, 6, 7, 8 are satisfied. After eliminating satisfied constraints,
residual observable constraints subjected to optimization are
shown below. The sensitive constrained optimization problem
is written as




1k
kxMinimize (19)

Subjected to observability constraints 121  xx (20)

With application of integer linear programming the
solution  01010010 which means PMUs need
to be allocated at 2, 5, 7 to make the system completely
observable.

D. Optimization of Sensitive Constrained PMUs Allocation
with Zero Injection (ZI) Modeling

In a power system when there is no generation or load at
any particular bus, power injection to rest of the network is
zero. Such buses are considered as ZIBs. When ZIBs are
considered, the size of connectivity matrix will be reduced,
thus it is possible to minimize the number of PMUs in the
network. In this problem of zero injection modeling, every bus
connected to ZIB is considered. Permutation matrix P is
formulated from an array of the vector that includes buses not
associated with ZIB and array of buses associated with ZIB.
ZIB constraint matrix is formulated with the number of buses
not associated with ZIB and buses associated with ZIB. The
buses associated with only ZIB can be written as Zas.

ZIB constraint matrix is written as

0
L L

M
as

I
Z

Z
   

 
(21)

where IL×Lis identity matrix of L number of buses not
associated with ZIB. The bus connectivity matrix developed for
optimal allocation of PMU can be presented as

* *pmu MZ A P Z (22)

where, A is bus incidence matrix defined from the bus system,
P is permutation matrix of the bus system and Zm is ZIB
constraint matrix formulated in (21). Bus constraint matrix bcon
is defined as the vector of buses (not associated with ZIB
which are all equal to one) and the number of buses associated
with each ZIB. Bus constraint matrix is vector formed to check
observability of the system. The optimization of sensitive
constrained PMUs with zero injection modeling is formulated
as

k
k 1

F kMin x

 (23)

subjected to observability constraints

pmu conZ X b (24)

eq eqA X B (25)

where: X=[x1 x2 …… xN]
T, xkє(0,1), k=1,2,3,….,n , Zpmu is bus

connectivity matrix defined in (22), bcon and Beq is bus
constraint matrix formed to check observability of the system.
Aeq is a matrix of n×n order which consists of sensitive bus
elements whose entries are equal to one and for other bus
elements it is zero. Equation (25) ensures that xk variable must
be one for sensitive buses in the system. It gives priority to
sensitive buses for allocation of PMUs in the bus network. For
instance, in Figure 2, bus-2 is a ZIB which is optimized with
modeling of bus connectivity matrix. Permutation matrix of the
system is



































00010000

00000100

00000010

00000001

10000000

01000000

00100000

00001000

From (21) ZIB constraint matrix is written as

























11110000

00001000

00000100

00000010

00000001

Bus constraint matrix is developed such that every bus of the
system is measured by at least two PMUs.

T
conb ]31111[

From (22), bus connectivity matrix for optimal allocation of
PMUs can be calculated as

























01031342

11110000

01100000

00011000

pmuZ

Bcon, Aeq, F, Zpmu and number of non- zero elements in Aeq
are inputs for sensitive constrained integer linear programming
method, which results in a decision variable vector [0 0 0 0 1 0
1 0]. This gives optimal allocation of PMUs at 5 and 7 buses.

E. Optimization of Sensitive Constrained PMUs Allocation
under Single Line Contingency

The optimization of sensitive constrained PMUs with zero

injection modeling in case single line contingency is

formulated as

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k
k 1

F kMin x

 (26)

subjected to observability constraints

2pmu conZ X b (27)

eq eqA X B (28)

In case of single line contingency, the problem is formulated in

such a way that each network bus is observed by at least two

PMUs

F. Complete Observability of the System
The performance of optimization method depends on the

observability of the power system. The performance of every
bus of the system is measured by Bus Observability Index
(BOI). The maximum BOI is limited to maximum connectivity
(χ) of the bus plus one [13].

1k k   (29)

For bus-k, (  ) BOI gives the number of times the bus is
observed by the PMU. To know the performance of the total
system, Complete System Observability Redundancy Index
(CSORI) is determined and it can be presented as the sum of
BOI of every bus in the system.

k
k

CSORI


  (30)

Maximum redundancy of the bus can be formulated as

n
T

k pmu k
k

Max b Z x

 (31)

subjected to following constraints

0
1

k
k

x 


 (32)

pmuZ X b (33)

where μ0 is the minimum number of PMUs obtained for
complete system observability, Zpmu is bus connectivity
matrix obtained from the proposed sensitive constrained
approach. b is the vector of observability constraints which is
written as transpose of vector [1 1 1 1 . . .1].

IV. STATE ESTIMATION WITH CONVENTIONAL AND OPTIMAL
SENSITIVE CONSTRAINED PMU MEASUREMENTS

Newton-Raphson load flow is considered for calculation of
conventional measurements and true state values (Vtru, δtru) of
the system. For integration of PMU measurements into state
estimation, the method followed in this paper is widely
investigated in [19], [21], [25-27] and this process exhibits
good results. For the joint optimality of PMU and SCADA
measurements, the optimization method used for solving

nonlinear equations is WLS method in which the errors are
weighted with standard deviation. The proposed method
includes optimal PMU measurements at sensitive buses along
with ZIBs obtained from sensitive constrained ILP approach
formulated in this paper. The procedure to find states of the
power system with sensitive constrained state estimation
method is presented in Figure 3. The measurement function for
state estimation is written as

kkk rxhZ  )( (34)

where Zk
is a vector of conventional and optimal PMU

measurements, x is state vector, hk is the measurement vector
of non-linear function related to state vector x and rk is
measurement error vector. The residual vector obtained from
the measurement function can be written as

rk = Zk-hk(x) , k=1,2….m (35)

The objective function for sensitive constrained state
estimation is formulated as

( ( ))
( )

m
k k

k kk

Z h x
J x Min

R


  (36)

Rkk is diagonal matrix formed with inverse of variances of
conventional and PMU measurements which is written as

Rkk=diagonal of
2 2 2
1 21 / ,1 / ,.......1 / m     (37)

where 2m is covariance of the measurements. The state
estimation method integrates PMU measurements with
conventional measurements by designing Jacobian matrix i.e.
first order derivatives of PMU and conventional measurements
are formulated in the state estimation process. The Jacobian
matrix with PMU measurements at sensitive buses and
conventional measurements can be formulated as

in in

fl fl

pmu
pmu

pmu

rpmu rpmu

ipmu ipmu

P P

v
Q Q

v
P P

v
Q Q

v
H

I I

v
I I








  
   
  

  
 
  

  
   
 
  
 
   
  
   
   

(38)

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Fig. 3. Flow chart for state estimation with optimal allocation of PMUs
considering sensitive buses

Gain matrix is defined for the best solution of the system
state accuracy which is written as

1( ) Tpmu pmuG x H R r
   (39)

where 1Tpmu pmu pmuG H R H
 , ( )r Z h x   . State

estimation solution is obtained by updating state vector which
is defined as

1 1 1( ) ( ( ))T T tpmu pmu pmux H R H H R Z h x
     (40)

1 1 1 1( ) ( ( ))t t T T tpmu pmu pmux x H R H H R Z h x
      (41)

where xt is the vector of state variables at tth iteration. The
convergence tolerance (ε) for state estimation is selected as 10-5
and state error is obtained from the difference of estimated
states (Vest, δest) and true values of the system. Voltage
magnitude error (Verr) and phase angle error (δerr) are written as

err est truV V V  , err est tru    (42)

V. RESULTS AND ANALYSIS
In order to allocate PMUs and estimate the states of a

power system, the proposed method is applied to 14, 24, 30 and

57-bus test systems. The single line diagrams of 14 and 57-bus
system are shown in Figures 4 and 5 respectively. Simulations
are carried out on the test systems with MATLAB and results
are presented and compared. MATLAB programs are run on
Intel(R) core(TM), i3 processor at 2.20 GHz with 4 GB of
RAM. Table II shows the zero injection buses, and sensitive
buses generated from formulation (8), (9).

Fig. 4. Sensitive constrained PMU locations in 14 bus network

TABLE II. SENSITIVE AND ZERO INJECTION BUSES

IEEE test
systems

Sensitive
buses

ZIB buses

14 bus 14,9 7

24 bus 22,21 11,12,17,24

30 bus 30,26,24,19 6,9,22,25,27,28

57 bus 31,33,29,32,25
4,7,11,21,22,24,26,34,
36,37,39.40,45,46,48

Fig. 5. Sensitive constrained PMU allocation for 57 bus system.

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A. Optimal PMU Allocation with Sensitive Constrained
Integer Linear Programming

General optimal PMU locations without sensitive
constraints using BILP method [30] is shown in Table III.

TABLE III. PMU ALLOCATIONS WITHOUT SENSITIVE CONSTRAINTS

IEEE
Test

Systems

No. of
PMUs

PMU Locations

14bus 4 2,6,7,9
24 bus 7 2, 3, 8, 10, 16 , 21,23
30 bus 10 1,7,9,10,12,18,24,25,27,28
57 bus 17 1,4,6,13,19,22,25,27,29,32,36,39,41,45,47,51,54

Binary integer programming is used to model sensitive

constrained ILP method for optimization. Using bus constraint
vector matrix and bus connectivity matrix, zero injection
modeling is framed in sensitive constrained ILP to optimize
PMU allocation in the network. The sensitive buses are
arranged according to the priority of higher sensitive buses.
Table IV shows sensitive constrained PMU allocation with and
without out zero injection modeling. Table V shows sensitive
constrained PMU allocation with and without zero injection
modeling under single line contingency. With ZIB modeling,
allocation of PMUs in the network is reduced to meet the cost
criteria. The method’s performance is analyzed through the
system redundancy. Redundancy Index (RI) measures the
observability of each bus of the system covered under PMUs.
System redundancy increases with the optimal location of
PMUs in the system.

Tables VI and VII show the redundancy index of sensitive
constrained PMU locations under no line and single line
contingency. It can be observed from Table VI that under
single line contingency PMUs are placed in the network in a
way that each branch is observable by at least 2 PMUs. Table
VIII shows the comparison of CSORI that is observable with
the optimal number of PMUs under no line and single line
contingency in the system.

TABLE IV. SENSITIVE CONSTRAINED PMU ALLOCATIONS WITH AND
WITHOUT ZI MODELING

IEEE
Test

systems

Without ZI Modeling With ZI Modeling

No. of
PMUs

PMU locations
No. of
PMUs

PMU locations

14bus 5 2,6,7,9,14 4 2,6,9,14
24 bus 8 3,4,8,10,16,21,22,23 7 1,4,6,8,19,21,22

30 bus 10
3,6,7,9,10,12,19,24,

26, 30
9

3,6,7,10,12,19,24,
26,30

57 bus 20
1,4,9,12,20,24,25,28,
29,31,32,33,36,38,39,

41,45,46,50,53
17

3,4,9,12,15,20,24,
25,29,31,32,33,36,

38,50,54,56

Table IX shows the number of PMUs allocated with
proposed method and other methods [10, 12-17]. The proposed
approach optimizes and allocates PMUs at sensitive buses that
are prone to load changes. PMU allocation at sensitive buses is
given the highest preference to provide accurate states of the
system. All other methods cited in [10, 12-17] optimized the

PMUs without considering the constraints of the buses and
allocation of PMUs at sensitive buses which results in
inaccurate system measurements.

TABLE V. SENSITIVE CONSTRAINED PMU ALLOCATIONS WITH AND
WITHOUT ZI MODELING UNDER SINGLE LINE CONTINGENCY

IEEE
Test

systems

Without ZI Modeling With ZI Modeling

No. of
PMUs

PMU locations
No. of
PMUs

PMU locations

14bus 10 2,4,5,6,7,8,9,11,13,14 8 2,4,5,6,9,10,13,14

24 bus 15
1,2,3,7,8,9,10,11,15,
16, 17,20,21,22,23

12
1,2,7,8,9,,10,16,
18,19,21,22,23

30 bus 21
1,2,3,5,6,8,9,10,11,12,

13,15,16,18,19,22,24,25,
26,27,30

17
1,3,5,6,7,10,12,13,
15,17,19,20,22,24,

26,27,30

57 bus 33

1,3,4,6,9,12,15,19,20,
22,24,25,27,28,29,31,32,
33,35,36,37,38,39,41,43.
45,46,47,50,51,53,54,56

1,2,4,6,9,12,15,18,
20,22,24,25,28,29,
30,31,32,33,35,36,
38,41,45,48,49,50,

51,53,54,56

TABLE VI. RI OF BUS WITH SENSITIVE CONSTRAINED PMU ALLOCATIONS

Bus No B1 B2 B3 B4 B5 B6 B7
RI 1 1 1 3 2 1 2

Bus No B8 B9 B10 B11 B12 B13 B14
RI 1 3 1 1 1 2 2

TABLE VII. RI OF BUS WITH SENSITIVE CONSTRAINED PMU ALLOCATIONS
UNDER SINGLE LINE CONTINGENCY

Bus No B1 B2 B3 B4 B5 B6 B7
RI 2 3 2 5 4 4 4

Bus No B8 B9 B10 B11 B12 B13 B14
RI 2 4 2 2 2 3 3

TABLE VIII. CSORI FOR SENSITIVE CONSTRAINED PMU LOCATIONS

IEEE
Test

systems

No Line Contingency Single Line Contingency

CSORI
with ZI

Modeling

CSORI
without

ZI
Modeling

CSORI
with ZI

Modeling

CSORI
without ZI
Modeling

14 bus 18 22 36 42
24 bus 24 33 48 62
30 bus 39 43 67 82
57 bus 72 79 120 128

TABLE IX. COMPARISON OF OPTIMIZATION METHODS FOR COMPLETE
OBSERVABILITY

Methods
14 bus
system

24 bus
system

30bus
system

57bus
system

Generalized ILP
Programming[14]

10 17

WLS[15] #4
#10 #17

Integer quadratic [17] 4 10 17
BILP [18][30] 4 10 17

CRO[28] 4 - 10 17
BPSO[29] 4 - 10 17
BGO[31] 4 - 10 -

Proposed sensitive
constrained ILP

*4 7
*9

*17

* Optimal number of PMUs allocated at sensitive buses.

# Optimal number of PMUs obtained solely with PMU measurements.

Engineering, Technology & Applied Science Research Vol. 7, No. 6, 2017, 2240-2250 2247

www.etasr.com Ravindra and Srinivasa Rao: Sensitive Constrained Optimal PMU Allocation with Complete …

B. State Estimation with Sensitive Constrained PMU
Allocations

In state estimation process, load flow solution is considered
as a true case for generating measurements with errors.
Measurement noise (Gaussian random variable) has been added
to measurement function to limit the noisy measurements. To
obtain measurement accuracy for each measurement, the error
variance of the measurements is added to each type of the
measurement. The Gaussian measurement noise is set to 10-4.
In this proposed state estimation method, voltage phasors are
measured in polar coordinates and current flow measurements
are measured in rectangular coordinates. As current flow
measurements in polar coordinates lead to very large and
uncertain for a certain range of terminal voltages and phase
angles that result in an ill condition of the state estimation gain
matrix [21]. Hence current measurements are considered in
rectangular coordinates in the following test cases.

1) Test case 1: 14-bus system
To define the accuracy of estimation with PMU allocation

at sensitive buses and to show the difference between with and
without sensitive bus locations, state estimation method
integrating PMU measurements without sensitive buses is
shown. Three scenarios i.e., Traditional state estimation
without PMU allocations, state estimation integrating PMU
measurements and state estimation with proposed sensitive
constrained PMU allocations are compared. In the state
estimation method without considering sensitive buses, four
PMUs are allocated at 2, 6, 7 and 9 to obtain 38 measurements.
Total 85 (38+47) measurements are used in state estimation
method without considering sensitive buses. In this 14-bus
system shown in Figure 4, four PMUs are allocated at bus 2, 6,
9 and 14 to obtain 36 measurements and 47 conventional
measurements to make the system completely observable. The
traditional state estimation method is performed with 47
conventional measurements and state estimation method with
proposed sensitive constrained PMU locations utilizes a total of
83 (36+47) measurements to get the accurate performance of
the state estimation.

(i) Conventional measurements:

Power injections: {1, 2, 3, 4, 7, 8, 10, 11, 12, 14}

Power flows: {1-2, 3-2, 3-4, 4-2, 4-7, 4-9, 5-2, 5-4, 5-6,6-
13, 7-9, 11-6, 12,-13}

(iii) PMU measurements without considering sensitive
buses:

Voltage phasors: {2, 6, 7, 9}

Current flows: {2-1, 2-3, 2-4, 2-5, 6-5, 6-11, 6-12, 6-13, 7-
4, 7-8, 7-9, 9-4, 9-7, 9-10,9-14}

(iii) PMU measurements considering sensitive buses:

Voltage phasors: {2, 6, 9, 14}

Current flows: {2-1, 2-3, 2-4, 2-5, 6-5, 6-11, 6-12, 6-13, 9-
4, 9-7, 9-10, 9-14, 14-9, 14-13}

State estimation accuracy of the system depends on
variance of the estimated states. The smaller variance value

implies better accuracy. For the proposed state estimation,
variances of power injections and power flows are set to
0.0001, and 0.00064. For PMU measurements, variance for
voltage phasors is set to 0.00001 and for real and imaginary
current flows it is set to 0.001. The state estimation solution
with optimal PMU allocations converged after 6 iterations.
Voltage magnitude error and phase angle error are obtained
from the difference of estimated and true values (load flow
calculated values are considered as true values). The voltage
magnitude error and phase error obtained using traditional state
estimation method without PMUs (scenario 1) and proposed
state estimation with PMU (scenario2) and state estimation
with proposed sensitive constrained PMU locations (scenario 3)
is presented in Figure 6 and 7 respectively. Allocation of PMUs
at sensitive buses has impact on measurement accuracy with
complete observability of the system. That can be observed
with voltage and phase angle errors obtained and presented in
scenario 3. State estimation with PMU allocation at sensitive
constrained buses i.e. at 2, 6, 9, and 14 shows very small angle
error deviation when compared to traditional state estimation
method and state estimation without sensitive constrained
locations as shown in Figure 7. Comparison of scenarios 1, 2
and 3 presented in Figures 6 and 7 gives better accuracy results
for complete observability of the system with the proposed
sensitive constrained state estimation.

Fig. 6. Comparison voltage magnitude error of 14-Bus system

Fig. 7. Comparison of Phase angle error of 14-bus system

2) Test case 2: 57-bus system
The topology of the 57-bus system with PMUs is shown in

Figure 5. The optimal PMUs obtained are 17 and are located at
{3, 4, 9, 12, 15, 20, 24, 25, 29, 31, 32, 33, 36, 38, 50, 54, 56}
with sensitive constrained ILP method to make system
completely observable. There are a total of 145 conventional
measurements (voltage, power flow, and power injection

Engineering, Technology & Applied Science Research Vol. 7, No. 6, 2017, 2240-2250 2248

www.etasr.com Ravindra and Srinivasa Rao: Sensitive Constrained Optimal PMU Allocation with Complete …

measurements) and with 17 PMUs allocated in the network,
140 measurements (Voltage phasor measurements and real and
imaginary current flow measurements) are obtained to get the
accurate performance of the state estimation. PMU locations
obtained through ILP method with and without sensitive
constrained locations are shown as follows.

(i) Conventional measurements:

Power injections: {1, 2, 3, 4, 5, 6, 7, 8, 9, 14, 15, 16, 20, 25,
32, 33, 37, 40, 50, 53, 56}

Power flow measurements: {2-1, 2-3, 3-4, 3-15, 4-5, 4-18,
5-6, 6-7,6-9, 7-8, 7-29, 9-10,10-12,10-51,11-13,11-41,11-43,
12-17, 13-49, 14-46, 15-45, 18-19, 19-20, 21-22, 22-23, 23-24,
24-25, 24-26, 27-28, 28-29, 29-52, 30-31, 32-33, 34-35,35-36,
36-40, 37-38, 37-39, 38-48, 39-57, 41-42, 41-43, 44-45, 46-47,
47-48, 48-49,50-51, 52-53, 54-55, 57-56}

(ii) PMU measurements obtained through general ILP
Method

Voltage phasors: {1, 4, 6, 13, 19, 22, 25, 27, 29, 32, 36, 39,
41, 45, 47, 51, 54}

Current flows: {1-2, 1-15, 1-16, 1-17, 4-3, 4-5, 4-6, 4-18, 6-
4, 6-5, 6-7, 6-8, 13-9,13-12,13-14,13-15, 13-49,19-18,19-
20,22-21,22-23,22-38, 25-24, 25-30, 27-26,27-28, 29-28, 29-
52, 32-31, 32-33, 32-34, 36-35, 36-37, 36-40,39-37,39-57,41-
10,41-11,41-42,41-43,41-56,45-15,45-44,47-48,47-46,47-
49,51-10,51-50,54-53,54-55}

(iii) PMU measurements obtained through sensitive
constrained ILP method

Voltage phasors: {3, 4, 9, 12, 15, 20, 24, 25, 29, 31, 32, 33,
36, 38, 50, 54, 56}

Current flow measurements: {3-2, 3-15, 4-3, 4-5, 4-6, 4-18,
9-10, 9-11, 9-12, 9-13, 9-55, 12-9, 12-13, 12-16, 12-17, 15-3,
15-1,15-13, 15-14, 15-45,20-19, 20-21, 24-23, 24-26, 25-24,
25-30, 29-7, 29-28, 29-52, 31-32, 31-30, 32-33, 32-34, 33-
32,33-34,36-35, 36-37,36-40, 38-22, 38-37, 38-44, 38-48, 38-
49,50-49, 50-51, 54-55, 54-53, 56-57, 56-41,56-42, 56-40}

The PMU measurements obtained through general ILP
method are 140 and with including conventional measurements,
total measurements are 285 (145+140). With the integration of
PMU measurements obtained through sensitive constrained
ILP and conventional measurements, we obtain total 278
(145+133) measurements to perform state estimation. For the
proposed method, variances of power injections, power flows
and voltage phasors of PMU measurements are set to 0.00001
and for real and imaginary current flow measurements it is set
to 0.001. The state estimation solution with optimal PMU
allocations converged after 7 iterations. The voltage magnitude
error and phase error obtained for 57 bus system using
traditional state estimation method without PMUs (scenario 1),
state estimation with PMU (scenario 2) and state estimation
with proposed sensitive constrained PMU allocation (scenario
3) is presented in Figures 8 and 9 respectively. With PMUs
allocation at sensitive buses, state error obtained in the
measurement is very low which means improved accuracy of
the state estimation.

Fig. 8. Comparison voltage magnitude error of 57-Bus system

Phase angle measurement is very important in power system
state estimation, as a small deviation of angle leads to
instability of the system. Comparison of scenario 1
(Traditional state estimation without PMU), scenario 2 (State
estimation with PMU) and scenario 3 (State estimation with
proposed sensitive bus constrained PMU allocation) in Figure
9 shows the state estimation method with sensitive constrained
PMU allocation (Scenario 3) results in very small phase angle
error deviation showing the impact of PMU measurement
accuracy with complete observability of the system.

Fig. 9. Comparison of Phase angle error of 57-bus system

C. Performance of State Estimation with Proposed Sensitive
Constrained Allocation with Root Mean Square Error
(RMSE) Indicator

In order to quantify accuracy and performance of state
estimation, an RMSE is formulated and error is computed. The
states of 14- and 57-bus systems with optimal sensitive
constrained PMUs allocation are examined with RMSE and
compared with traditional state estimation and state estimation
method without considering sensitive buses. RMSE is
formulated as

RMSE 



n

k
truest kxkx

n
1

2))()((
1

(43)

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www.etasr.com Ravindra and Srinivasa Rao: Sensitive Constrained Optimal PMU Allocation with Complete …

where xest
is estimated state, xtru is the true state of the system,

and n is the number of buses in the system. RMSE for voltage
and phase angles are computed separately because the scale of
measurements is different in each case. RMSE measures the
accuracy of state estimation voltage and phase angle errors of
the system. RMSE with PMU for the proposed sensitive
approach has better accuracy compared to traditional state
estimation and all other methods. RMSE values of 14- and 57-
bus systems are shown in Table X. When compared to state
estimation solution given in [19, 21, 26, 27], the solution with
proposed method is better.

TABLE X. COMPARISON OF RMSE INDEX OF PROPOSED METHOD WITH
OTHER METHODS

Methods
RMSE of Voltage (p.u)

IEEE 14-Bus
system

IEEE57-
Bus System

Traditional State Estimation
Method[22]

0.0006 0.007

Ref[19] 0.000076 0.002
Ref[21] 0.00011 -
Ref[27] 0.00099 -

Proposed sensitive constrained
state estimation method

0.000027 0.0009

VI. CONCLUSION
This paper presented a sensitive constrained integer linear

programming approach for optimal allocation of PMUs
considering sensitive buses giving the highest priority in the
system for state estimation solution. This approach for optimal
PMU allocation with zero injection modeling is able to
minimize the number of PMUs in case of single and no line
contingency without losing complete observability of the
system. With the optimal allocation of PMUs at sensitive
buses, redundancy of the system is improved and resulted in
economic benefit. Allocation of PMUs at sensitive buses
enhanced the accuracy and performance of the state estimation
solution. Results obtained from the simulations of 14, 30 and
57 bus systems show the effectiveness of the proposed method
when compared with other methods.

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