5-Optimal PMU Placement Considering Controlled Islanding of Power System

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)

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.


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


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


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.


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.


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.


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


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.


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 .


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.


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,


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





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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.





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