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742 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 29, NO. 2, MARCH
2014
Optimal PMU Placement ConsideringControlled Islanding of Power
System
Lei Huang, Student Member, IEEE, Yuanzhang Sun, Senior Member,
IEEE, Jian Xu, Member, IEEE,Wenzhong Gao, Senior Member, IEEE, Jun
Zhang, Member, IEEE, and Ziping Wu, Student Member, IEEE
AbstractThis paper proposes an optimal phasor measurementunit
(PMU) placement model considering power system con-trolled
islanding so that the power network remains observableunder
controlled islanding condition as well as normal
operationcondition. The optimization objectives of proposed model
areto minimize the number of installed PMUs and to maximize
themeasurement redundancy. These two objectives are
combinedtogether with a weighting variable so that the optimal
solutionwith minimum PMU number and maximum measurement redun-dancy
would be obtained from the model. To reduce the number ofrequired
PMUs, the effect of zero-injection bus is considered
andincorporated into the model. Furthermore, additional
constraintsfor maintaining observability following single PMU
failure or lineloss are also derived. At last, several IEEE
standard systems andthe Polish 2383-bus system are employed to test
the presentedmodel. Results are presented to demonstrate the
effectiveness ofthe proposed method.
Index TermsControlled islanding, integer linear program-ming,
measurement redundancy, optimal phasor measurementunit (PMU)
placement, state estimation.
I. INTRODUCTION
S TATE estimator plays an important role in the security ofpower
system operation. Its main purpose is to preciselyestimate the
voltage phasors of all system buses based on aset of acquired
measurements [1]. When a measurement set al-lows a unique solution
of the state estimation (SE) problem, thepower system is said to be
observable [2]. Recently phasor mea-surement units (PMUs) have been
used as measurement devicefor SE, which will advance the
traditional supervisory controland data acquisition (SCADA) system.
PMU provides voltagephasor of the bus where it is installed and
current phasors of allbranches incident to that bus [3]. The PMU
measurements fromdifferent buses, which are synchronized by the
common clocksignal from global positioning system (GPS), can help
simplify
Manuscript received February 16, 2013; revised June 04, 2013 and
September07, 2013; accepted October 06, 2013. Date of publication
November 04, 2013;date of current version February 14, 2014. This
work was supported in part bythe National Natural Science
Foundation of China under Grant 51007067 andthe Ministry of Science
and Technology of China under Grant 2012AA050218.Paper no.
TPWRS-00198-2013.L. Huang, Y. Sun, and J. Xu (corresponding author)
are with the
School of Electrical Engineering, Wuhan University, Wuhan,
Hubei430072, China (e-mail: [email protected];
[email protected];[email protected]).W. Gao, J. Zhang, and Z. Wu are
with the Department of Electrical and
Computer Engineering, University of Denver, Denver, CO 80210 USA
(e-mail:[email protected]; [email protected];
[email protected]).Color versions of one or more of the figures in
this paper are available online
at http://ieeexplore.ieee.org.Digital Object Identifier
10.1109/TPWRS.2013.2285578
the process of state estimation and improve the accuracy of
es-timation results. If we place PMUs in all busses of a
network,all the voltage phasors can be directly measured without
run-ning any state estimator [4]. However, PMU and its
associatedcommunication facilities are costly. Furthermore, the
voltagephasor of the bus incident to the bus with PMU installed
canbe computed with branch parameter and branch current
phasormeasurement [5]. So it is neither economical nor necessary
toinstall PMUs at all system buses. Thus, one of the important
is-sues is to find the optimal number and placement of PMUs
forpower system state estimation.Optimal PMU placement (OPP) is
firstly attempted in [6],
formulating as a combinatorial optimization problem of
min-imizing the PMU number for system observability. In
recentliteratures, the OPP model has been generalized to
includeadditional constraints or contingencies. In [7], an integer
pro-gramming formulation of OPP problem is proposed with
thepresence of conventional measurements. A generalized
integerlinear programming (ILP) formulation for OPP is presentedin
[8], considering effects of zero-injection buses (ZIBs)
andconventional measurements. The model proposed in [9] takesinto
account PMU channel limitations. Contingency conditionsof line
outage or PMU loss are considered separately or si-multaneously in
the OPP model proposed in [10]. Generally,the existing OPP models
concerns about the determination ofminimum number and optimal
location set of PMUs, ensuringthat the entire power system remains
a single observable island[1]. In another word, these models can
only handle the casesin which the power system is operated as a
single and inte-grated network. However, some severe faults may
lead partsof the network to angle, frequency or voltage
instability. Inthat case, trying to maintain system integrity and
operate thesystem entirely interconnected is very difficult and may
causepropagation of local weaknesses to other parts of the
system[11]. As a solution, controlled islanding (CI) is employed
bysystem operators, in which the interconnected power systemis
separated into several planned islands prior to catastrophicevents
[12], [13]. After system splitting, wide area blackoutcan be
avoided because the local instability is isolated andprevented from
further spreading [14]. In order to operateeach island with power
balancing and stability after controlledislanding, it is essential
to provide an OPP scheme which cankeep the network observable for
the post-islanding condition aswell as normal condition.After
having determined the model of OPP, many investiga-
tors have presented different methods to solve this
optimizationproblem. These methods can be generally divided into
math-ematical and heuristic algorithms [1]. The
representativemathematical algorithm is ILP [7][10], which is
capable ofsolving large-scale problems in a short time and
achieving theglobally optimal solution. Exhaustive search [15] is
another
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ROCARSALResaltado
UsuarioNota adhesivaun modelo de colocacin optima de PMU, que
considera la observabilidad del sistema en isla y normal
UsuarioNota adhesivaLas unidades de medicin de fasores
Recientemente se han utilizado como dispositivo de medicin para la
estimacin de estado, que har avanzar el control de supervisin y
adquisicin tradicional de datos (SCADA)
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HUANG et al.: OPTIMAL PMU PLACEMENT CONSIDERING CONTROLLED
ISLANDING OF POWER SYSTEM 743
mathematical method used for OPP. However, it is not suitablefor
large-scale systems with huge search space. Among theheuristic
optimization algorithms, genetic algorithm [16], Tabusearch [17],
simulated annealing [6], differential evolution[18], particle swarm
optimization [19], immunity algorithm[20], iterated local search
[21], spanning tree search [22], andgreedy algorithm [23] have been
developed. In spite of someadvantages, the major disadvantage
associated with intelligentsearch-based methods is that they do not
guarantee that aglobally optimal solution is found [15].In this
paper, an ILP model of OPP considering controlled is-
landing (OPP-CI) is proposed. This model is able to determinethe
minimal number and optimal location set of PMUs in orderto provide
the full network observability in normal operation aswell as in
controlled islanding scenario. To distinguish multipleoptimal
solutions, measurement redundancy is incorporated intothe
optimization objective. Meanwhile the effect of ZIBs is con-sidered
for further reducing the number of required PMUs. Inaddition, to
enhance the robustness of OPP scheme, the contin-gencies of single
PMU or line outage are also incorporated intothe proposed OPP
model.The rest of this paper is structured as follows. In Section
II,
the concepts and rules of network observability are
introducedand the basic OPP formulation is described. The
derivation ofOPP model considering controlled islanding is
introduced withdetails in Section III. In Section IV, contingencies
of singlePMU or line outage are incorporated into OPP-CI model
indi-vidually or simultaneously. In Section V the performance of
theproposed new model is assessed using several IEEE
standardsystems and a practical 2383-bus system. Finally,
conclusionsare given in Section VI.
II. BASIC FORMULATION OF OPTIMAL PMU PLACEMENT
A. Observability Analysis Based on PMUs
The observability of a system is the precondition for its
stateestimation. Power system observability analysis is usually
car-ried out in two different ways, namely, numerical
observabilityanalysis and topological observability analysis [24].
Numericalobservability algorithm makes use of the gain matrix of
stateestimation in a system. When the gain matrix is of full
rank,the system is said to be numerically observable. However,
dueto high computation burden of verifying the rank of the
gainmatrix, this approach would not be preferred for practical
ap-plications [19]. On the other hand, graph concept is utilized
intopological observability methods. The network is consideredto be
topologically observable if a spanning tree of full rankcan be
found in the graph [25]. This tree connects all nodes andbranches
observed by direct measurements or calculations.Observability
analysis with phasor measurements has been
studied in a number of literatures. A method to combine
phasormeasurements and conventional measurements in
observabilityanalysis is firstly proposed in [26] and then improved
in [27]. Itis able to handle not only systems including only phasor
mea-surements, but also the ones with both phasor measurementsand
conventional measurements. Additionally, current phasormeasurements
which lead to multiple-solutions can be detectedusing the
proposedmethod. In [28], a direct numerical algorithmis presented
to determine observable islands and restore observ-ability for
power systems. The adoption of reduced networkmodel makes the
proposed method computationally attractive.Based on PMUmeasurements
as well as conventional measure-
ments, a hybrid topological/numerical method for power
systemobservability analysis is provided in [29], which shows
goodperformance in simulations.In this paper the concept of
topological observability is
adopted and the following simple rules have been applied [19].
If voltage phasor and current phasor at one end of a branchare
known, voltage phasor at the other end of that branchcan be
obtained using Ohms law.
If voltage phasors at both ends of a branch are known,
thecurrent phasor through this branch can be calculated.
The measurements such as bus voltage phasors and branchcurrent
phasors, directly obtained from PMUs, are referred toas direct
measurements; measurements derived by employingthe above two rules
are referred to as indirect measurements,or pseudo measurements as
in [30]. When the voltage phasorat a bus can be obtained either
from direct measurements orindirect measurements, this bus is
identified as observable. Inan observable network, each and every
bus must be observed atleast once by using direct or indirect
measurement.
B. Basic Optimal PMU PlacementIn a power system network, the PMU
placement at a bus can
be seen as a binary decision variable defined as
ifotherwise.
(1)
For a system with buses, therefore, the optimal PMU place-ment
problem can be formulated as an integer linear program-ming problem
as follows:
(2)
subject to constraints
(3)
where is the cost of installing a PMU at bus . Without loss
ofgenerality, cost of PMU installation at each bus is assumedto be
equal to 1 per unit in the present study.
refers to the number of times that the th bus is observedthrough
PMU measurements.
is the th entry of network connectivity matrixdefined as
ifotherwise.
(4)
For example, with (2), minimizing the number of PMUsfor the IEEE
14-bus system (Fig. 1) can be formulated asfollows:
(5)
where are PMU placement variables that are generatedby binary
integer programming. Therefore, (5) represents theminimum number of
PMUs.
UsuarioResaltado
UsuarioNota adhesivaLa observabilidad de un sistema es la
condicin previa para la estimacin de estado. Anlisis de observacin
del sistema de alimentacin se lleva a cabo por lo general en dos
formas diferentes, a saber, el anlisis de observabilidad numrica y
el anlisis de observabilidad topolgica
UsuarioNota adhesivaSe adopta el concepto de observabilidad
topolgica y aplica las siguientes reglas simples. Si se conocen
fasor de voltaje y fasor de corriente en un extremo de una rama, el
fasor de voltaje en el otro extremo de la rama se puede obtener
usando la ley de Ohm. Si se conocen fasores de tensin en ambos
extremos de una rama, el fasor de corriente a travs de esta rama se
puede calcular.
UsuarioNota adhesivaColocacin optima bsica de PMU en un sistema
de potencia. que permite observar el sistema en condiciones
normales de operacin.
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744 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 29, NO. 2, MARCH
2014
Fig. 1. IEEE 14-bus system.
The observability constraints (3) are listed explicitly
asfollows:
(6)
The inequality constraints in (6) represent the requirementsof
observed times for each of the 14 buses. For example, if atleast
one PMU is installed on bus 1, 2 or 5, constraint issatisfied, and
bus 1 will be observable. In Fig. 1, the 14-bussystem is made
completely observable by placing 4 PMUs onbuses 2, 6, 7, and 9
[31], although this is not the only optimalsolution.
III. OPTIMAL PMU PLACEMENT CONSIDERINGCONTROLLED ISLANDING
A. Controlled Islanding
Cascading failures are the most significant threats for
powersystem security. Cascading failures together with additional
linetripping can lead the system to uncontrolled splitting [11].
For-mation of uncontrolled islands with significant power
imbalanceis the main reason for system blackouts. In order to avoid
cat-astrophic wide area blackouts due to cascading failures,
con-trolled islanding has been considered as an effective
defensestrategy. The main advantages of controlled islanding of
powersystems can be listed as follows [11]: It can separate weak
and vulnerable areas from other stableparts of the system.
Compared to the whole system, small subsystems areeasier to be
handled and controlled under dynamic andemergency conditions.
To date, a lot of investigations have been conducted on
thistopic and various methods for controlled islanding
[11][14],[32][35] have been proposed; for example, a method of
con-trolled islanding with constraint of observability is presented
in[36]. Since our research is focused on the OPP problem, thispaper
does not study methods of controlled islanding in detailbut only
uses the controlled islanding results of several IEEEstandard
systems presented in [13], [34], and [35], and assumesa suitable
controlled islanding scheme for the Polish 2383-bussystem. However,
the proposed OPP method can be applied toany other controlled
islanding schemes.After establishment of planned islands, there
exist some
factors which may threat the stability and integrity of
eachisland, such as power imbalance, line overloading,
voltage,angle and frequency instabilities, etc. [11]. Therefore, to
main-tain static and dynamic stability, necessary load shedding
andother control actions may be needed in each island, whichalways
require real-time information throughout the island. Inaddition,
real-time measurements in different islands shouldbe collected and
analyzed together to determine whether andhow the power system can
be restored to normal operation. Toensure the effectiveness of all
the above actions, it is essentialto keep each island totally
observable through properly placedPMUs. In other words, the optimal
placement of PMUs shouldbe carried out in such a manner that the
network remains ob-servable under controlled islanding condition as
well as normaloperation condition.Compared to (3), the
observability constraints of OPP-CI
model are modified as follows:
(7)
where is the binary entry in the connectivity matrix
forpost-islanding network, which is defined as
ifotherwise. (8)
For instance, assuming that the controlled islanding is in
ef-fect for the IEEE 14-bus system following a cascading fault,as
shown in [34] and Fig. 1, the whole system is separatedinto two
subsystems and several lines are opened during the is-landing
process. According to (7), the observability constraintsfor OPP-CI
can be written explicitly as follows:
(9)
UsuarioNota adhesivaColocacin optima de PMU considerando el
control del sistema con separacin en islas del anterior sistema en
el cual se permita la observabilidad y el control del sistema en
estas condiciones.
UsuarioNota adhesivaventajas del islado de sistemas de
potencia
UsuarioResaltadoSe puede separar reas dbiles y vulnerables de
otras partes estables del sistema.
UsuarioNota adhesivaUnmarked definida por Usuario
UsuarioResaltadoEn comparacin con el conjunto del sistema,
subsistemas pequeos son ms fciles de ser manipulados y controlados
bajo condiciones dinmicas y de emergencia.
UsuarioNota adhesivaUnmarked definida por Usuario
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HUANG et al.: OPTIMAL PMU PLACEMENT CONSIDERING CONTROLLED
ISLANDING OF POWER SYSTEM 745
Comparing (6) and (9), it is concluded that the
observabilityconstraint shown in (7) is stricter than that in (3).
Therefore, anOPP scheme subject to (7) can keep the power network
com-pletely observable both for normal operation scenario and
con-trolled islanding condition.
B. Dealing With Multiple Optimal Solutions
PMUs placement through the objective function (2) and
in-equality constraints (7) may lead to multiple optimal
solutionswith the same minimum number of PMUs. For the 14-bussystem
in Fig. 1, installations of 5 PMUs in {1, 2, 6, 8, 9}, {1,4, 6, 7,
9}, {2, 5, 6, 8, 9}, {4, 5, 6, 7, 9}, and {4, 5, 6, 8, 9} canall
satisfy the constraints (7) and lead the system to
completeobservability in both normal operation condition and
controlledislanding scenario.In this study, thus, maximizing the
measurement redundancy
is considered as an additional objective to pick out the most
suit-able OPP scheme for power systems. Conventionally,
measure-ment redundancy is defined as the ratio of the number of
mea-surements (including direct measurements and indirect
mea-surements) to the number of states [37]. Considering that
themost important state variables in state estimation are bus
voltagephasors, the measurement redundancy can be redefined as
theratio of the number of voltage measurements to the number
ofsystem buses. Moreover, the measurement redundancy under
is-landing operation scenario as well as normal operation shouldbe
considered.To keep consistency with (2) which is a minimization
problem, the objective function of maximizing
measurementredundancy is formulated as a minimization problem as
well:
(10)
where is the total number of system buses; constant isthe
maximum number of times that the th bus can be observedin normal
operation, which equals to the number of its incidentlines plus
one; variable represents the number of times thatthe th bus is
observed by the solved OPP scheme in normaloperation; and refer to
the corresponding constant andvariable in islanding operation
condition, respectively; and
are weighting factors assigned to the two componentsof the
objective function. Since there is greater probability fora power
system to be operated in normal condition than in is-landing
condition, in this study and are set at 0.7and 0.3,
respectively.Therefore, for the 14-bus system, set {4, 5, 6, 7, 9}
is the most
suitable solution because it has smaller value of than
otherones, as shown in Table I.
C. OPP Model Considering Controlled Islanding
In this part, the problem of optimal PMU placement con-sidering
controlled islanding is modeled. The objective ofOPP-CI is to
minimize the number of installed PMUs and tomaximize the
measurement redundancy with the full networkobservability for
normal operation and controlled islandingscenarios as the
constraints.1) OPP-CI Ignoring the Effect of Zero-Injection Bus: :
The
ZIBs refer to the network nodes without generation or load
con-nected. A ZIB together with all its incident buses can be
defined
TABLE ICOMPARISON ON MEASUREMENT REDUNDANCY OF DIFFERENT
OPP SOLUTIONS FOR IEEE 14-BUS SYSTEM
as a zero injection cluster (ZIC). For a power network, if the
in-fluence of ZIBs is ignored or the network does not contain
anyZIBs at all, its OPP-CI model can be formed as
(11)
subject to observability constraints (7), where is theweighting
factor.Here the two components of the objective function, and,
stand for the considerations of PMU number (2) and mea-
surement redundancy (10), respectively. The weighting factoris
used to determine which factor is more dominant than the
other one in the OPP procedure. In this study, reducing
PMUnumber is selected as the more important objective. Note
that
is the number of incident lines to the th bus; letand . This
way
of specifying can ensure the value of to be less than1, which
guarantees the priority of minimizing PMU numberin OPP-CI problem.
As a result, the globally optimal solutionwith minimum PMU devices
installed and maximum measure-ment redundancy can be found out.2)
OPP-CI Considering the Effect of Zero-Injection Bus: If
the effect of ZIBs is considered, the total number of PMUs inOPP
problem will be reduced due to the following rules [20]: In a ZIC,
if the zero-injection bus is observable and its ad-jacent buses are
all observable except one, then the unob-servable bus will be
identified as observable by applyingKirchhoffs current law (KCL) at
ZIB.
In a ZIC, if all the buses are observable except the
zero-in-jection one, then the zero-injection bus can be also
identi-fied as observable by using nodal equations.
Combining these two cases can lead to the conclusion that aZIC
is observable when it has at most one unobservable bus.Since this
unobservable bus could become observable finallyby means of the
properties of ZIC, in this paper, it is defined aspseudo
unobservable bus.Assuming that is the index of the th
zero-injection bus
and is the th zero injection cluster, the auxiliary
binaryvariable is defined so that implies that bus isthe pseudo
unobservable bus in . Then the observabilityconstraint of can be
mathematically formulated as
(12)
Notice that the power system could be operated in bothnormal
operation condition and controlled islanding condition,and the
elements of a ZIC may change due to the line tripping in
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746 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 29, NO. 2, MARCH
2014
the process of islanding. Thus the incorporation of ZIB
effectsinto OPP-CI problem should be simultaneously considered
forthese two operation scenarios of power system.For an -bus power
network with zero-injection buses,
the observability constraints of OPP-CI considering the effectof
ZIBs are listed as follows:
(13)
Here, and indicate the numbers of times that busis observed by
means of the installed PMUs and the proper-ties of ZIC, in normal
operation condition and controlled is-landing condition,
respectively. On the other hand, the secondand fourth equations in
(13) indicate that the number of pseudounobservable buses in a ZIC
must be less than or equal to 1,for both normal scenario and
islanding scenario. Therefore, theconstraints in (13)
simultaneously ensure that one of the busesin a ZIC will be
observable when it can be reached by PMUs,or it is the only pseudo
unobservable bus in this ZIC.
IV. OPP-CI FORMULATION AGAINST SINGLEPMU LOSS OR SINGLE LINE
OUTAGE
The previous OPP-CI formulation ensures complete observ-ability
of the network assuming a fixed network topology aftercontrolled
islanding as well as absolutely reliable measurementdevices.
However, PMUs may fail to work due to loss of GPSsignal, failure of
measurement instruments or loss of communi-cation channels.
Furthermore, transmission line outages may re-sult in loss of
complete observability. Thus, operators may planto have a reliable
monitoring system by installing extra PMUsin the network.In this
section, constraints associated with OPP-CI against
single PMU or single line loss are formulated. Meanwhile,
theobjective function still remains the same as (11). Therefore,
forOPP-CI considering each contingency, it is sufficient to
replacethe previous constraints with the following related
constraints.
A. OPP-CI Formulation Considering Single PMU Outage
Outage of a PMU at bus , denoted as , can be con-sidered into
the previous OPP-CI model by setting the corre-sponding decision
variable to zero. To facilitate the formu-lation of the
optimization problem, a parameter, , is definedas follows:
ifotherwise
(14)
then, the associated constraints to OPP-CI situation
consideringsingle PMU outage are as follows.For network without
ZIBs:
(15)
For network with ZIBs:
(16)In these expressions, and are binary auxil-
iary variables whose values are equal to 1, if bus is the
pseudounobservable bus for and (the th zero injectioncluster in
controlled islanding condition), respectively.
B. OPP-CI Formulation Considering Single Line OutageOutage of a
line may cause the loss of observability for one of
its terminal buses which would otherwise be observable
usingcurrent phasor of that line [15].For a power network with
lines, single line contingen-
cies can be defined. The connectivity matrices for the networkin
normal operation condition and controlled islanding condi-tion
change in each of such defined contingencies. Let param-eters and
be defined as th entries of theconnectivity matrices for networks
in normal and islanding sce-narios, respectively, where superscript
represents loss ofline . Thus, related constraints of OPP-CI
considering singleline outage are as follows:For network without
ZIBs:
(17)
For network with ZIBs:
(18)
Similarly, and are binary auxiliary variableswhose values are
equal to 1, if bus is pseudo unobservable bus,when line is out, for
and , respectively.
C. OPP-CI Formulation Considering Single ContingenciesIn this
case, the assumed contingency for power system could
be either one of the two types, i.e., the single PMU loss or
thesingle line outage. Hence, the set of constraints defined in
thetwo previous subsections should be considered simultaneously.It
can be expected that more PMUs are required in order tomain-tain
network observability in this case than in cases with onlysingle
contingency.
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HUANG et al.: OPTIMAL PMU PLACEMENT CONSIDERING CONTROLLED
ISLANDING OF POWER SYSTEM 747
TABLE IISPECIFICATIONS OF TEST SYSTEMS
TABLE IIICONTROLLED ISLANDING SCHEMES FOR DIFFERENT SYSTEMS
TABLE IVCOMPARISON OF OPP RESULTS WITH OR WITHOUT CONSIDERATION
OF CONTROLLED ISLANDING (IGNORING THE EFFECT OF ZIB)
V. CASE STUDIES AND RESULTS
The proposed OPP-CI model was tested on the IEEE 14-, 30-,39-,
118-bus systems, and the Polish 2383-bus system, whosedetailed
information can be found in Table II. The case studieswere
performed in two parts. First, for all the test systems,
OPPsconsidering controlled islanding, with and without inclusion
ofthe effect of zero-injection bus, are carried out and the
resultsare compared with those neglecting controlled islanding.
Next,single line and single PMU contingencies are taken into
accountand their influence on the OPP-CI solution is studied. The
simu-lation results with respect to IEEE standard systems and
Polish2383-bus system are given in the following. All simulations
areexecuted in a laptop having a 2.60-GHz dual-core CPU and 4GB of
RAM. The OPP problem is modeled in MATLAB andsolved by CPLEX
Toolbox for MATLAB.
A. Case Results for IEEE Standard Test Systems1) OPP Considering
Controlled Islanding: At the first part,
optimal PMU placement is carried out so as to achieve
totally
observability of network under both normal operation
conditionand controlled islanding condition. To perform the OPP-CI
pro-cedure, the controlled islanding plans for different IEEE
sys-tems should be known a priori. In this paper, these
controlledislanding schemes are extracted from [34], [35], and
[13]. Forclarity, they are listed again in Table III. As for IEEE
14-bussystem, two islands with 6 buses and 8 buses in each
island,respectively, are included in the islanding scheme. However,
asmentioned in Section III, the proposed OPP-CI model is not
justsuitable to the above controlled islanding cases but also can
beapplied to any other controlled islanding schemes.Table IV
provides the comparison of the number and loca-
tions of required PMUs resulting fromOPPwith or without
con-sideration of controlled islanding. The effect of
zero-injectionbus is incorporated into the comparison in Table V.
the PMUsinstallation percentage in the two tables refers to the
ratio be-tween the number of PMUs and the number of system
buses.Results of Tables IV and V reveal that generally more
PMUs
are required by power network to maintain observability forboth
controlled islanding scenario as well as normal condition.
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748 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 29, NO. 2, MARCH
2014
TABLE VCOMPARISON OF OPP RESULTS WITH OR WITHOUT CONSIDERATION
OF CONTROLLED ISLANDING (INCLUDING THE EFFECT OF ZIB)
TABLE VIOPP-CI RESULTS WITH CONSIDERATION OF SINGLE
CONTINGENCIES (IGNORING THE EFFECT OF ZIB)
TABLE VIIOPP-CI RESULTS WITH CONSIDERATION OF SINGLE
CONTINGENCIES (INCLUDING THE EFFECT OF ZIB)
Comparing Tables IV and V , it is also noticed that
consideringthe influence of zero-injection bus reduces the number
of re-quired PMUs in all cases.2) OPP-CI Considering Contingencies
of Single Line and
PMU Outages: In this part, OPP-CI is implemented for IEEE14-,
30-, 39- and 118-bus systems considering contingenciesof line or
PMU outages. Corresponding to the constraints(15)(16), (17)(18) and
the combination of (15)(18), threedifferent cases are considered,
i.e., outage of a single line, lossof a single PMU, and a single
contingency of line outage orPMU loss. The influence of
zero-injection bus is neglected inTable VI, while it is considered
in Table VII.As expected, a robust measurement system against
single
PMU or line outages needs more PMUs than the case
neglectingcontingencies. Additionally, in comparison with single
line
outage, single PMU loss has more adverse impact on the net-work
observability, which can be concluded from the requirednumber of
PMUs in the relevant cases.Table VIII shows the CPU computation
times for solving
OPP-CI problems. For each IEEE test system, only the timeneeded
for the most complex calculation, i.e., OPP-CI consid-ering the
single contingency and the effect of ZIB, is listed.In all the
previous calculations in the paper, the uncertain-
ties of PMUmeasurements and network parameters are ignored,and
the values of weighting factors and are fixed. How-ever, two
appendixes are added at the end of this paper: in Ap-pendix A, the
uncertainties associated with the voltage phasorsmeasured or
computed by the proposed OPP-CI schemes areassessed, while the
variances of OPP-CI results with differentvalues of and are shown
in Appendix B.
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HUANG et al.: OPTIMAL PMU PLACEMENT CONSIDERING CONTROLLED
ISLANDING OF POWER SYSTEM 749
TABLE VIIICPU COMPUTATION TIMES FOR CALCULATIONS OF OPP-CI
CONSIDERING SINGLE CONTINGENCY AND EFFECT OF ZIB
TABLE IXOPP RESULTS AND THEIR ASSOCIATED CPU COMPUTATION TIMES
FOR POLISH 2383-BUS SYSTEM
TABLE XSTANDARD UNCERTAINTIES IN THE INDIRECT-MEASUREMENTS
OF MAGNITUDES AND PHASOR ANGLES OF VOLTAGEPHASORS FOR IEEE
14-BUS SYSTEM
B. Case Results for Polish 2383-Bus SystemThe details about the
Polish 2383-bus system can be obtained
from [38] which indicates that all the buses in this system
aredivided into different areas. Based on this, a CI scheme for
thePolish 2383-bus system is assumed, in which the partition
ofcontrolled islands is roughly consistent with that of
sub-areasdetermined in [38]. However, a few buses are repartitioned
inthe CI scheme to avoid isolated bus or areas in each
island.Finally, in the CI scenario the whole Polish 2383-bus
systemwill be divided into 5 islands with 369 buses, 281 buses,
880buses, 560 buses, 293 buses, respectively. The OPP-CI
resultswith different considerations and their associated CPU
compu-tation times are shown in Table IX.
VI. CONCLUSIONAn effective OPP scheme should ensure complete
observ-
ability of a power network under various operation conditions.To
avoid wide-area blackout following cascading failures,power system
might be operated in controlled islanding mode.
TABLE XISTANDARD UNCERTAINTIES IN THE MEASUREMENTS OF
VOLTAGEPHASORS AT PSEUDO UNOBSERVABLE BUS FOR IEEE 14-BUS
SYSTEM
In this paper, an OPP model considering controlled islandingof
power system is proposed. The proposed model guaranteescomplete
observability of power network for normal conditionas well as
controlled islanding condition, with or withoutconsidering the
effect of zero-injection bus. By introducingthe measurement
redundancy into the optimization objective,our OPP-CI model can
find the globally optimal solution withthe minimum number of PMUs
and maximum measurementredundancy. Furthermore, single PMU or line
loss is also in-corporated into the model. At last, case studies on
several IEEEstandard test systems and a large-scale practical
system provideverification of the effectiveness of the presented
OPP models.
APPENDIX AEVALUATION OF MEASUREMENT UNCERTAINTYFOR IEEE 14-BUS
AND 118-BUS SYSTEMS
The uncertainties associated with the voltage phasors mea-sured
or computed by the proposed OPP-CI configurations areevaluated in
this appendix. For each IEEE test system, the uncer-tainty
evaluation is performed only for the OPP-CI scheme con-sidering the
effect of ZIB, which implies the minimum numberof installed PMUs
and consequently the worst performance onuncertainty.Detailed
formulas of uncertainty calculation for different
types of measurements are derived in the
following.DirectMeasurements: The uncertainties of direct
measure-
ments are calculated from the manufacturers
specifications.Assuming that the probability of measurement
uncertainty isof uniform distribution and the is the bounding
limitsof the measurement of , the standard uncertainty in the
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750 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 29, NO. 2, MARCH
2014
TABLE XIIOPP-CI RESULTS WITH REGARDS TO DIFFERENT VALUES OF FOR
IEEE 30-BUS SYSTEM
measurement can be expressed as [39]
(19)
where is usually specified by PMU manufacturers
[40].Indirect-Measurements: The uncertainties for indi-
rect-measurements are evaluated by using the classical
uncer-tainty propagation theory [39].Let there be a PMU installed
at bus , with bus connected to
bus through line . With the help of PMU measurements,i.e., the
voltage phasor at bus and the currentphasor through the line , the
voltage at buscan be expressed as
(20)
where and are line resistance and reactance.By decoupling (20)
into two equations in the real-imaginary
coordinate system and solving the new equations, the magni-tude
and the phasor angle of the voltage phasor can be
obtained, which are functions of the magnitude and phase angleof
and , and the parameters of line :
(21)
Assuming that the input quantities in (21) are
uncorrelated(similar assumptions are made for the following
derivations),the combined standard uncertainty of the voltage
magnitudeand the phase angle , according to [39], can be given
by
(22)
where is the partial derivative, , , , , ,, and is the standard
uncertainty in the measurement.
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HUANG et al.: OPTIMAL PMU PLACEMENT CONSIDERING CONTROLLED
ISLANDING OF POWER SYSTEM 751
TABLE XIIIOPP-CI RESULTS WITH REGARDS TO DIFFERENT VALUES OF FOR
IEEE 39-BUS SYSTEM
Measurements Calculated With the Properties of ZIC: Forthis type
of measurements, i.e., the voltage phasors at the
pseudounobservable buses, their uncertainties can also be obtained
bythe classical uncertainty propagation theory.If the pseudo
unobservable bus is exactly the ZIB of one
ZIC, it implies that there is no bus in the ZIC with
PMUinstalled. The voltage phasor of bus is obtained from
thefollowing KCL equation:
(23)
where is the number of buses in this ZIC, is the number oflines
connecting bus and the zero-injection bus . andare resistance and
reactance, respectively, of th line betweenbus and bus .The voltage
magnitude and phasor angle of bus can then be
expressed as
(24)
where and are sets of voltage magnitudes and phasorangles,
respectively, of buses in the ZIC except the bus .and are parameter
sets of lines incident to the bus .
Thus, the standard uncertainty of and can be given by
(25)
with , , , and is the total number of variablesin .For the
situation that the pseudo unobservable bus is not
the zero-injection bus (assuming that bus is the zero-injec-tion
bus), there may exist some PMUs having been placed atthe buses
incident to bus . Another equation, thus, should beformulated to
obtain the bus s voltage phasor:
(26)
here, is the set of PMU buses in the ZIC and refers to theset of
buses without PMU installed. is the current phasorthrough the th
line from bus to bus .
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752 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 29, NO. 2, MARCH
2014
TABLE XIVOPP-CI RESULTS WITH REGARDS TO DIFFERENT VALUES OF FOR
IEEE 30-BUS SYSTEM
Similarly to (24), there exists a new pair of equations for
themagnitude and phasor angle of voltage phasor :
(27)
where and are sets of magnitudes and phasor angles,
re-spectively, of the currents from buses in to bus . andare sets
of voltage magnitudes and phasor angles of buses inexcept the bus ,
respectively. and are parameter sets oflines from buses in to bus
.Then the standard uncertainties can be obtained with the fol-
lowing formula:
(28)
Table X shows the standard uncertainties in the voltagephasors
corresponding to the indirect-measurements, whileTable XI gives the
standard uncertainties in the voltage phasorsof pseudo unobservable
buses. The uncertainties of the PMUbuses are excluded, since they
can be directly computed from(19). The typical values of maximum
uncertainties in PMUmeasurements are specified by the manufacturer
in [40], wherethe maximum uncertainties for voltage and current
magnitude
Fig. 2. Standard uncertainties in calculated voltage phasors for
the IEEE14-bus system.
are 0.02% and 0.03% of the actual values, respectively; andthe
maximum error in the measurement of phase angle is0.01 degrees. The
actual values of phasor measurements aredetermined by performing
power flow for the power system.Additionally, a 5% uncertainty is
assumed to all transmissionline parameters.
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HUANG et al.: OPTIMAL PMU PLACEMENT CONSIDERING CONTROLLED
ISLANDING OF POWER SYSTEM 753
TABLE XVOPP-CI RESULTS WITH REGARDS TO DIFFERENT VALUES OF FOR
IEEE 39-BUS SYSTEM
It is noted that under the proposed OPP-CI configuration abus
may be observed by more than one PMU, such as bus 7in Table X. In
that case, a pair of measurement uncertainties,i.e., uncertainty in
voltage magnitude and uncertainty in voltagephasor angle, can be
obtained with regard to each connectedPMU. The minimum value of
magnitude uncertainty and min-imum value of angle uncertainty
should be chosen and treatedas the final uncertainties for that
bus. Additionally, there aresome buses located on the boundary of
islands. In other words,these buses have lines incident to other
islands, such as bus 2 inIEEE 14-bus system. As shown in Table X,
the CI process maycause the loss of the observations on these buses
from the PMUslocated in other islands. Therefore, their measurement
uncer-tainties in CI condition may be different from that in
normalcondition.The standard uncertainties in calculated voltage
phasors for
IEEE 14-bus system are shown in Fig. 2. The order in whichthe
buses are selected to depict the measurement
uncertaintiescorresponds to the bus indices. In other words, the
order forIEEE 14-bus system is 1-3, 7-8, and 10-14. In Fig. 2, the
solidcircle and x-mark refer to the measurement uncertainties
fornormal operation condition, while the dashed circle and
x-mark
are used to display the measurement uncertainties in CI
opera-tion condition.Fig. 3 shows the standard uncertainties
associated with cal-
culated voltage phasors for IEEE 118-bus system. There are
4buses having different uncertainties for normal condition andCI
condition. Among them buses 26 and 65 are the boundarybuses with
tie lines incident to other islands, and buses 26 and65 are pseudo
unobservable buses. It is noted that buses 26 and65 have small
measurement uncertainties in CI condition thanthat in normal
condition. The reason is that the bus numbersof the ZICs which they
belonged to are reduced due to the CIprocess. This leads to the
reduction of uncertainty sources in thecalculation equations of
their voltage phasors, and consequentlyresults in smaller
measurement uncertainties.
APPENDIX BOPP-CI RESULTS WITH DIFFERENT AND
FOR IEEE 30-BUS AND 39-BUS SYSTEMS
Results With Different Weighting Factor : Tables XIIand XIII
show the OPP-CI results with regards to differentvalues of for IEEE
30-bus system and 39-bus system,
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754 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 29, NO. 2, MARCH
2014
Fig. 3. Standard uncertainties in calculated voltage phasors for
the IEEE118-bus system.
respectively. For all of the calculations, the values of are
setthe same as before, i.e., .In these two tables, corresponds to
the results cal-
culated in Section V. refers to the case in which onlythe
measurement redundancy of normal operation condition isconsidered,
while indicates that only the measurementredundancy of islanding
condition is activated. The cells withthe same OPP result are
joined together.Since the priority of minimizing PMU number is
ensured in
(11) by the selected value of , all the OPP results for a
givenOPP-CI strategy are with the same minimum number of PMU,as
shown in the tables. However, the PMU locations may varywith the
value of , especially for the cases associated withthe IEEE 30-bus
system. It is because that the weighted propor-tions of the two
components in (10), i.e., the measurement re-dundancy differences
in normal condition and islanding condi-tion, are changed due to
the varying . Consequently, for thosebuses with tie lines connected
to other islands, their weightedmeasurement redundancy differences
will be changed.
Results With Different Weighting Factor : The variancesof OPP-CI
solutions with the weighting factor in (11) arelisted in Tables XIV
and XV. All the calculations are accom-plished under the value of .
Similarly, for each tablethe cells with the same result are
combined together.The values of calculated from are
0.1111 for IEEE 30-bus system and 0.1429 for 39-bus system.For
the cases of , the component in (11) isan integer and the component
has the value between 0 and1. This guarantees that minimizing PMU
number is more domi-nant than maximizing measurement redundancy in
the OPP pro-cedure. Thus with a given , a robust OPP solution can
be ob-tained for each OPP-CI strategy. On the other hand, the
value
of will increase with and may exceed 1. In that case,additional
PMUmay be needed due to the trade-off between theobjective of
minimizing PMU number and the objective of max-imizing measurement
redundancy.
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UsuarioResaltadoMtodos para la formacin de islas controladas
UsuarioResaltadoMtodos para la formacin de islas controladas
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Lei Huang (S13) was born in Hunan, China, in1987. He received
the B.E. degree in electricalengineering from Wuhan University,
Wuhan, China,in 2009. Currently he is pursuing the Ph.D. degreein
the School of Electrical Engineering, Wuhan Uni-versity. He is also
a visiting scholar in the Universityof Denver, Denver, CO, USA.His
research interests include optimal PMU place-
ment, voltage stability analysis and wind power.
Yuanzhang Sun (SM01) received the Ph.D. degreein electrical
engineering from Tsinghua University,Beijing, China, in 1988.He is
currently a Professor of the School of Electric
Engineering,WuhanUniversity,Wuhan, China. He isalso an adjunct
Professor at Tsinghua University. Hismain research interests are
power system dynamicsand control, voltage stability, and renewable
energy.
Jian Xu (M08) received the B.E. and Ph.D. degreesin electrical
engineering from Wuhan University,Wuhan, China, in 2002 and 2007,
respectively.Currently he is an Associate Professor in the
School of Electric Engineering, Wuhan University.Also, he is a
visiting scholar in Washington StateUniversity, Pullman, WA, USA.
His research inter-ests are PMU application, power system
operation,and voltage stability.
Wenzhong (David) Gao (SM03) received the M.S.and Ph.D. degrees
in electric power engineering fromGeorgia Institute of Technology,
Atlanta, GA, USA,in 1999 and 2002, respectively.He is currently an
Associate Professor in the De-
partment of Electrical and Computer Engineering,University of
Denver, Denver, CO, USA. His re-search interests are renewable
energy, smart grid,and power system analysis.
Jun Zhang (M09) received the Ph.D. degree inelectric engineering
from Arizona State University,Tempe, AZ, USA, in 2008.He is
currently an Assistant Professor in the De-
partment of Electrical and Computer Engineering,University of
Denver, Denver, CO, USA. His re-search interests are smart grid and
statistical signalprocessing.
Ziping Wu (S12) received the B.E. degree inthermal power
engineering and the M.S. degreein electrical power engineering from
North ChinaElectric Power University, Beijing, China, in 2006and
2009, respectively. Now he is pursuing the Ph.D.degree in the
Department of Electrical and ComputerEngineering, University of
Denver, Denver, CO,USA.His research interests include wind power
genera-
tion, renewable energy, and smart grid.
UsuarioResaltadoMtodos para la formacin de islas controladas
UsuarioResaltadoMtodos para la formacin de islas controladas
UsuarioResaltadoMtodos para la formacin de islas controladas
UsuarioResaltadoMtodos para la formacin de islas controladas