2012 NIRMA UNIVERSITY INTERNATIONAL CONFERENCE ON ENGINEERING, NUiCONE-2012, 06-08DECEMBER, 2012
1
Optimal Placement of Phasor Measurement Unit
under Multiple Constraints Condition : An
Approach
J. S. Bhonsle1 and A. S. Junghare2
Abstract-- Synchronized measurement technology facilitates the
promising tool for future monitoring, protection and control
(WAMPAC)
of a power systems. Synchronized Phase
Measurement Unit (PMU) is a monitoring device, which
provides the realization of the real-time wide area monitoring of
power system. This paper presents an approach of placing the
optimum no. of PMU at strategic locations for multiple
constraints. The PMU placement problem is formulated as a
binary integer linear programming, in which the binary decision
variables (0, 1) determine optimum no. of PMU installation with
their locations, preserving the system set observability, such as
full or partial, measurement redundancy and zero injections.
The proposed approach integrates the impacts of both existing
conventional power flow measurements (if any) and the possible
failure of single or multiple PMU / communication line into the
decision strategy of the optimal PMU allocation. Proposed
approach is tested on sample 7-bus system, IEEE 14-bus system,
21-bus VKM system and IEEE 30-bus systems.
Index Terms-- Binary integer linear programming, optimal
PMU placement, maximum redundancy, failure of PMU /
Communication line.
I.
for PMU placement. The author in [9,16] developed optimal
PMU placement algorithm using integer programming. In Ref
[8,11] multistage scheduling frame work for PMU placement
was proposed. In Ref [4] a heuristic approach was presented
to obtain a PMU placement solution. Particle swarm based
optimization technique for PMU placement was proposed in
Ref[12,13]. In Ref [15] author considered one line/ PMU
outage case with GA. In Ref [6] author addressed PMU
placement solution against measurement loss and branch
outage. In Ref [5,7,10,11] techniques are proposed for
complete and or incomplete observability.
The placement of a minimal set of phasor measurement units
at a strategic location to make the power system observable at
set observability for multiple constraints is the main
objective of this paper. This paper has proposed a modified
topological observability based PMU placement formulation
with and without conventional power flow and power
injection measurements, with one depth of unobservability,
maximum redundancy and failure of PMU or communication
line.
INTRODUCTION
S
YNCHRONIZED Phase Measurement Unit (PMU) is a
monitoring device, which was first introduced in mid-1980s.
Phasor measurement units (PMU) are devices, which use
synchronization signals from the global positioning system
(GPS) satellites and provide the phasors of voltage and
currents measured at a given substation. PMU is considered
as the most promising measurement technology for
monitoring the power systems. This unit is composed of a
number of phasors that capture measurements of analog
voltage, current waveform and the line frequency. As the
PMUs become more and more affordable, their utilization
will increase not only for substation applications but also at
the control centers for the EMS applications. The Phasor Data
Concentrator (PDC), which is located in the control center,
gathers all the phasor data received from geographically
distributed PMUs via the communication network.
The
communication network is one of the components that can be
investigated and upgraded, to provide more accurate real-time
information between the PMU and the control center.
Several PMU placement algorithms using different techniques
have been proposed by different researchers. In the Baldwin’s
pioneering work [1] PMU placement problem was solved by
using simulated annealing method. In Ref [3] a special
tailored non-dominated sorting genetic algorithms developed
978-1-4673-1719-1/12/$31.00©2013IEEE
II.
PMU PLACEMENT PROBLEM
PMUs provide two types of measurements : bus voltage
phasors and branch current phasors. In this paper, it is
assumed that each PMU has enough channels to record the
bus voltage phasor at its associated bus and current phasors
along all branches that are incident to this bus. Also the cost
of all installed PMUs are assumed to be equal. Costs of
PMU differ and increases as the number of channels.
Unequal cost of the PMUs can be formulated in the problem.
Three laws that can be used for PMU placement are [15] –
Ohm’s Law, (V=IR) – Any bus that is incident on an
observed line connected to an observed bus is observed.
Ohm’s Law , (I=V/R) – Any line joining two observed buses
is observed.
Kirchhoff’s Law – If all the lines incident to an observed bus
are observed, except one, then all the lines incident to that bus
are observed.
The objective of the PMU placement problem is to render an
observable system by using a minimum number of PMUs.
1) Only PMU Measurements for Complete Observability
The optimal placement of PMU is formulated as
follows
2012 NIRMA UNIVERSITY INTERNATIONAL CONFERENCE ON ENGINEERING, NUiCONE-2012
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Subject to the constraints
CX ≥ b
Where Ci,j = 1 if i = j
= 1 if i and j are connected
= 0 otherwise
X = [x1 x2 ….xn]
If the case of failure of PMU/ Communication line is to
be considered, then entry for vector b is updated and set
for maximum measurement redundancy.
where,
C Connectivity matrix of the system
xi PMU placement variable
b Vector of length n
According to the approaches introduced in above mentioned
case (1) and (2), the following two inequalities are as follows
yp + yk ≥ 1 and yl + yp + yk + yq ≥ 3
In order to satisfy yp + yk ≥ 1 , the first inequality needs to be
subscribed from the second inequality corresponding to the
injection measurement and consider that its right-hand side
needs to be reduced by one due to the injection I kas shown
below :
(yl + yp + yk + yq - yp - yk ) ≥ 3 – 1
The second inequality becomes yl + yq ≥ 2
2) Conventional and PMU based measurement for
complete Observability
The optimal placement of PMU is formulated as follows
If bus s is not associated to any conventional measurement,
then the corresponding constraint of the minimization
problem is still kept ys ≥ 1
3) Only PMU based measurement for one-depth of
UnObservability
Subject to the constraints
T P C X ≥ b2
T =
I M×M
0
The optimal placement of PMU is formulated as
follows
0
Tmeas
where,
M is the no. of buses not associated to conventional
measurements
P Permutation matrix
T Measurement matrix
b2 Vector defined according to cases of conventional
measurements
Subject to the constraints
A C X ≥ b1
where,
A Branch to node incident matrix
b1 Vector of length m
4) Conventional and PMU based measurement for onedepth of UnObservability
For Conventional measurements, three cases needs to be
analyzed as below suggested in Ref [9,16]
The optimal placement of PMU is formulated as
follows
a) If a power flow measurement is on line ij
Subject to the constraints
B A C X ≥ B b1
where,
B Branch associated matrix that keeps the branches
that are associated with zero injection measurement
and removes the branches that are not associated with
zero injection measurement
then the following needs to be held : yi + yj ≥ 1
which means one bus voltage can be solved from this
measurement and the other needs to be covered by
PMU
b) Suppose that an injection measurement is at bus k as
follows
then the following inequality needs to be held :
yl + yp + yk + yq ≥ 3
c) The power flow measurements and the injection
measurements are associated
III. BINARY INTEGER LINEAR PROGRAMMING
The algorithm searches for an optimal solution to the binary
integer programming problem by solving a series of LPrelaxation problems, in which the binary integer requirement
on the variables is replaced by the weaker constraint
0 ≤ x ≤ 1. The algorithm



Searches for a binary integer feasible solution
Updates the best binary integer feasible point found
so far as the search tree grows
Verifies that no better integer feasible solution is
possible by solving a series of linear programming
problems
2012 NIRMA UNIVERSITY INTERNATIONAL CONFERENCE ON ENGINEERING, NUiCONE-2012, 06-08DECEMBER, 2012
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Steps to be follow for optimum PMU placement for
multiple constraints :
1.
2.
3.
4.
5.
6.
7.
8.
Define the minimization objective function
Form the Connectivity matrix C
Form the branch to node incident matrix A
Form the Measurement matrix T
Consider the zero injection bus entry if any
Accordingly form the Permutation matrix P
Form branch associated matrix B
Set the entry of vector according to set
constraints
9. Find optimum solution for objective function
using binary linear optimization
10. Display Result
IV. TEST SYSTEM DETAILS
Fig. 2 IEEE 14 bus System
Table I gives the system detail and size of matrix for the
considered cases as follows. Table II provides the
measurement details. Fig. 1 and Fig. 2 shows single line
diagram for sample 7- bus[16] and IEEE 14 bus system
respectively.
21-bus VKM system is a 400 KV Maharashtra Grid having 5
Generators and 48 transmission lines for generating,
transmitting and distributing the electrical power. Generation
from private company is not considered in analysis.
V. RESULT AND DISCUSSION
Fig. 1 Sample 7-bus system
TABLE I
SYSTEM DETAIL AND SIZE OF MATRIX
System
7-bus
sample
system
IEEE 14bus
system
21 bus
VKM
system
IEEE 30bus
system
No.
of
variables
7
14
X
C
A
b
b1
7
7×7
8×7
7×1
8×1
14
14×14
20×14
14×1
20×1
Program is developed in MATLAB software using binary
integer programming for determining optimum no. of PMU’s
and their location for the multiple constraints to ensure tight
monitoring of power system. The proposed approach attempts
to provide a local PMU redundancy to allow for the loss of
PMUs while preserving the global network observability. The
cost of all PMUs are assumed to be equal at 1 p.u. Unequal
cost of PMU can be easily formulated. PMU is assumed to
have sufficient no. of channels. The vectors can be updated
and set according to the heaviness of connectivity of each
bus in the system. The approach is made adaptive.
Considering all the constraints, the optimum no. of PMU
required for meeting objective function is tabulated in table
III. Graphical representation for the optimum no. of PMUs
verses no. of buses and branches for the test system under
consideration is shown in Fig. 3.
TABLE III
OPTIMUM NO. OF PMU
21
21
21×21
48×21
21×1
48×1
System
30
30
30×30
41×30
30×1
41×1
TABLE II
MEASUREMENT DETAILS
System /
Conventional
Measurement
7-bus
Sample
system
IEEE
14-bus
system
21 bus
VKM
system
IEEE
30-bus
system
Zero Injection bus
2
7
18
10
Power Flow Meas.
2-3
-
3-15
23-24
978-1-4673-1719-1/12/$31.00©2013IEEE
No. of buses
Optimum no. of PMU
for set constraints
7-bus Sample system
7
4
IEEE 14-bus system
14
9
21 bus VKM system
21
13
IEEE 30-bus system
30
21
The optimum no. of PMUs is approximately found to be 2/3 rd
of the no. of buses depending upon network topology. No. of
optimum PMU increases with the dense network, complexity
and constraints. The PMU placement at zero-injection buses
will help find the optimal solution. PMUs at zero-injection
buses measure current phasors of corresponding lines; thus,
2012 NIRMA UNIVERSITY INTERNATIONAL CONFERENCE ON ENGINEERING, NUiCONE-2012
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the KCL at zero-injection bus provides no additional
information. The removal of PMUs from zero-injection buses
will reduce the search space which could enhance the solution
speed. Thus with zero injection bus in system reduces no. of
PMU. Only PMU based, optimum no. of PMUs with their
locations, with and without considering failure of
PMU/Communication line is shown in table IV and VI
whereas that of with Conventional and PMU based is shown
in table V and VII for the considered test system.
measurement both , the optimum no. of PMU required for the
same set constraint reduces.
TABLE V
CONVENTIONAL AND PMU BASED MEASUREMENT FOR
COMPLETE OBSERVABILITY
System
Constraints
7-bus
Sample
system
w/o failure of PMU or
Communication line
With the failure of
PMU
or
Communication line
w/o failure of PMU or
Communication line
With the failure of
PMU
or
Communication line
w/o failure of PMU or
Communication line
With the failure of
PMU
or
Communication line
w/o failure of PMU or
Communication line
With the failure of
PMU
or
Communication line
IEEE
14-bus
system
21 bus
VKM
system
Fig. 3 Graphical representation for the optimum no. of PMUs verses no. of
buses and branches
TABLE IV
ONLY PMU BASED MEASUREMENT FOR COMPLETE
OBSERVABILITY
System
Constraints
7-bus
Sample
system
w/o failure of PMU
or Communication
line
With the failure of
PMU
or
Communication
line
w/o failure of PMU
or Communication
line
With the failure of
PMU
or
Communication
line
w/o failure of PMU
or Communication
line
With the failure of
PMU
or
Communication
line
w/o failure of PMU
or Communication
line
With the failure of
PMU
or
Communication
line
IEEE
14-bus
system
21 bus
VKM
system
IEEE
30-bus
system
IEEE
30-bus
system
Location of PMU
2,4
System
Constraints
4
1,2,4,5
7-bus
Sample
system
w/o failure of PMU or
Communication line
With the failure of
PMU
or
Communication line
w/o failure of PMU or
Communication line
With the failure of
PMU
or
Communication line
w/o failure of PMU or
Communication line
With the failure of
PMU
or
Communication line
w/o failure of PMU or
Communication line
With the failure of
PMU
or
Communication line
2,6,7,9
9
2,4,5,6,7,8,9,11,13
6
1,6,8,15,17,20
13
1,3,5,6,7,8,10,12,15,16,
17, 19,20
10
1,7,9,10,12,18,24,25,27,
28
21
1,3,5,7,8,9,10,11,12,13,
15,17,19, 20,22, 24,25,
26,28, 29,30
From the table IV and V, it is seen that for the same
constraint case, even though the optimum no. of PMU
remains same, their locations changes. Also it can be
concluded that with conventional and PMU based
Location of PMU
3
2,4,5
3
2,6,9
7
2,4,5,6,9,10,13
5
2,6,8,15,17
12
2,3,5,6,7,8,10,11,14,
16,17,20
8
3,5,6,9,12,18,25,30
18
1,3,5,7,8,9,11,12,13,
15,17,19, 20,25, 26,28,
29,30
2,5
TABLE VI
ONLY PMU BASED MEASUREMENT FOR ONE-DEPTH OF
UNOBSERVABILITY
Optimum
No. of PMU
2
4
Optimum
No. of
PMU
2
IEEE
14-bus
system
21 bus
VKM
system
IEEE
30-bus
system
Optimum
No. of PMU
1
Location of PMU
2
2,4
2
4,6
4
4,5,6,9
3
8,15,20
5
1,6,8,15,18
4
2,10,15,27
8
3,5,6,10,12,18,24,
27
3
VI. CONCLUSION
The proposed approach uses binary integer linear
programming which gives the PMU placement and its
location. The proposed approach integrates the impacts of
both existing conventional power flow measurements (if any)
and the possible failure of single or multiple PMU /
communication line into the decision strategy of the optimal
PMU allocation. This paper proposed PMU placement
formulation with and without conventional power flow and
power injection measurements, with one depth of
2012 NIRMA UNIVERSITY INTERNATIONAL CONFERENCE ON ENGINEERING, NUiCONE-2012, 06-08DECEMBER, 2012
[2]
TABLE VI I
CONVENTIONAL AND PMU BASED MEASUREMENT FOR ONEDEPTH OF UNOBSERVABILITY
[3]
System
Constraints
7-bus
Sample
system
w/o failure of PMU
Communication line
With the
failure
PMU
Communication line
w/o failure of PMU
Communication line
With the
failure
PMU
Communication line
w/o failure of PMU
Communication line
With the
failure
PMU
Communication line
w/o failure of PMU
Communication line
With the
failure
PMU
Communication line
IEEE
14-bus
system
21 bus
VKM
system
IEEE
30-bus
system
Optimum No.
of PMU
Location of
PMU
or
1
4
of
or
2
3,4
or
1
6
of
or
2
10,13
or
1
18
of
or
1
18
or
1
27
of
or
2
25,27
[5]
[6]
[7]
[8]
[9]
[10]
[11]
unobservability, maximum redundancy and failure of PMU
or communication line. PMU placement solution for multiple
constraints objective for the set observability is achieved.
Approach is simple and adaptive . Formulation can be
extended to restricted no. of channels, any depth of
unobservability, unequal cost of PMUs. Further PMU can be
placed in 2-stages as shown in table VIII and suggested in
papers [15,16]. Approximately one-third no. of PMU needs to
be placed in I-stage and rest are placed in II-stage. The no. of
PMU placement depends on network topology. Proposed
approach can be efficiently used in practice.
TABLE VIII
2-STAGE PLACEMENT
System
Stages
No. of PMUs
IEEE 14bus system
I- Stage
4
2,6,7,9
II-Stage
5
4,5,,8,11,13
21
bus
VKM
system
I- Stage
6
1,6,8,15,17,20
II-Stage
7
3,5,7,10,12,16,19
IEEE 30bus system
I- Stage
10
1,7,9,10,12,18,24,25,27,28
II-Stage
13
Locations
3,5,8,11,13,15,17,19,20,22,
26,29,30
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J.S.Bhonsle and A.S.Junghare, “A Novel Approach for the Optimal
PMU
placement
using
binary
integer
programming
technique”,(IJEEE) ISSN (Print) 2231 – 5284, Vol-1, Iss-3, 2012, pp.
no. 67-72